The landscape law for tight binding Hamiltonians
Douglas N. Arnold, Marcel Filoche, Svitlana Mayboroda, Wei Wang, Shiwen Zhang
TTHE LANDSCAPE LAW FOR TIGHT BINDING HAMILTONIANS
DOUGLAS ARNOLD, MARCEL FILOCHE, SVITLANA MAYBORODA, WEI WANG, ANDSHIWEN ZHANGA bstract . We study the integrated density of states (IDS) of a discrete Schr¨odingeroperator H = − ∆ + V in Z d . We introduce a box counting function, defined throughthe discrete landscape function of H . For any deterministic bounded potential, we giveestimates for the IDS from above and below by the landscape box counting function.For the Anderson model, we get a refined lower bound for the IDS, throughout thespectrum. In particular, near the bottom of the spectrum, we show that both the IDS andthe box counting function can be estimated by the common distribution of the randompotential, which yields an improved bound on the so-called Lifschitz tails for disorderedsystems. The work extends the recent landscape law on R d in [DFM] to tight-bindingmodels on Z d . C ontents
1. Introduction and main results 22. Preliminaries 113. Landscape law in the Deterministic case 153.1. Upper bound 163.2. General lower bound in the non-scaling case 183.3. Lower bound in the scaling case 243.4. Lower bound for the periodic potential 274. Landscape law for the Anderson model 284.1. Lifshitz tails lower bound 294.2. Lifshitz tails upper bound 314.3. In comparison with the IDS 444.4. Dual landscape and Energy near the top. 46Appendix A. Discrete Laplacian and harmonic functions 48A.1. Maximum principles for discrete sub-solutions 48A.2. The discrete Poincar´e inequality 50A.3. Discrete cut-o ff functions 51A.4. Dirichlet problem on a cube 52A.5. Estimates on the Green’s function and the Poisson kernel 55A.6. Harnack type inequalities 62Appendix B. Cherno ff bound 65References 66 a r X i v : . [ m a t h - ph ] J a n D. ARNOLD, M. FILOCHE, S. MAYBORODA, W. WANG, AND S. ZHANG
1. I ntroduction and main results
In this paper, we consider the discrete Schr ¨odinger operator H = − ∆ + V on a periodiclattice in Z d . We are interested in the finite volume integrated density of states (IDS) ofthe system. In order to describe our main results, we first introduce some notations. Let Λ = ( Z / K Z ) d (cid:27) { ¯1 , · · · , ¯ K } d be an integer torus, where K ∈ N and ¯ k , 1 ≤ k ≤ K , is thecongruence class, modulo K . For simplicity, we will omit the bar from ¯ k when it is clear.Let V = { v n } n ∈ Λ ∈ (cid:96) ∞ ( Λ ) be a real-valued, non-constant, non-negative potential. Wedenote by V max = max n ∈ Λ v n the strength of the potential. The tight binding Hamiltonian H is the linear operator on H : = (cid:96) ( Λ ) (cid:27) R K d defined by( H ϕ ) n = − (cid:88) | m − n | = ( ϕ m − ϕ n ) + v n ϕ n , n ∈ Λ , (1.1)where | n | : = (cid:80) di = | n i | is the 1-norm on Λ . We may think of ϕ either as a periodicsequence ϕ n indexed by n ∈ Z d or as a periodic function ϕ ( n ) on Z d . We are interested inthe normalized integrated density of states of H , i.e., the eigenvalue counting functionper unit volume: N ( µ ) : = K − d × (cid:8) the number of eigenvalues λ of H such that λ ≤ µ (cid:9) . (1.2)The IDS function is important in condensed matter physics. It counts the number ofenergy levels below a given value µ , and both it and its limit as K tends to infinity (thethermodynamic limit) are fundamental for characterizing the physical properties of thesystem, in addition to being of great mathematical interest. The literature on relatedtopics is huge, and we do not give a thorough review here. We refer the readers to, e.g.,[Iv] for problems related to the Weyl law, [BoKl] for deterministic local bounds in thecontinuous or discrete case, [Ki] for problems related to disordered systems, and to themany references therein.In 2012, a new concept called the localization landscape was introduced in [FM]. Ina series of papers following [FM], e.g. [ADFJM1, ADFJM3], numerous numerical ex-periments suggested astonishingly precise estimates for the IDS could be derived usingthe landscape theory. In a more recent paper [DFM], the authors gave the first rigorousmathematical treatment relating the landscape function and the IDS in the continuoussetting. In the present work, we extend this connection to the discrete model (1.1).Let us be more precise. Given an operator H as above, there is a unique positive solu-tion u = { u n } n ∈ Λ ∈ H to the equation ( Hu ) n =
1, which we call the discrete localizationlandscape function. Next, we define the landscape box counting function N u . For this,we first define, for any positive integer s , a partition P ( s ) of the set { , · · · , K } d into sub-sets which are mostly boxes of side length s , as follows. Writing K = qs + r (where thequotient q and the remainder r are non-negative integers and r < s ), we define a partition P ( s ) of the set { , · · · , K } into q subsets of s consecutive elements, and, if r >
0, oneadditional subset of cardinality r . The partition P ( s ) then consists of the boxes definedby the Cartesian products of d subsets from P ( s ), see Figure 1. For given µ >
0, we
HE LANDSCAPE LAW 3 F igure
1. The partition P (2) for Λ = { , · · · , } .then set s ( µ ) = (cid:6) µ − / (cid:7) , and define N u ( µ ) as the number of boxes on which the minimumof 1 / u n does not exceed µ , normalized by the size of the set Λ : N u ( µ ) = K − d × (cid:26) the number of Q ∈ P ( s ( µ )) such that min n ∈ Q u n ≤ µ (cid:27) . (1.3)Our goal is to estimate the IDS N in terms of the landscape box counting function N u .For this we will establish several estimates, stated below as Theorems 1, 2, and 3, whichwe collectively refer to as the Landscape Law .The first result, which will be proven in Section 3.1, shows that, after a proper scaling, N u provides an upper bound for N ( µ ) over the whole range of µ . Theorem 1.
Suppose that K ≥ and V ∈ (cid:96) ∞ ( Λ ) is a non-constant, non-negative poten-tial. Then there is a dimensional constant C > , such thatN ( µ ) ≤ N u ( C µ ) for all µ > . (1.4)In saying that C is a dimensional constant in the theorem, we mean that it depends onlyon d , and, in particular, is independent of K and V max . Because of the many constantsthat enter our estimates, we may use the same notation for di ff erent constants when theyappear in di ff erent inequalities.The next theorem, proved in Section 3.2, contains the key estimate for obtaining alower bound for N ( µ ). Theorem 2.
Retain the hypotheses in Theorem 1. Then there are dimensional constantsc , c , C , α (in particular, independent of K and V max ), such thatN ( µ ) ≥ c α d N u ( c α d + µ ) − C N u ( c α d + µ ) , for all µ > and < α < α . (1.5) D. ARNOLD, M. FILOCHE, S. MAYBORODA, W. WANG, AND S. ZHANG
In order to obtain a useful lower bound from (1.5) we need additional hypotheseson the potential. One possibility is to assume that the landscape function u = { u n } n ∈ Λ satisfies a scaling condition. Namely, we say that u satisfies the scaling condition withconstant C S > (cid:88) n ∈ Q u n ≤ C S (cid:32)(cid:88) n ∈ Q u n + (cid:96) d + (cid:33) (1.6)for every cube Q ⊂ Λ of side length (cid:96) . Here, 3 Q is the tripled cube concentric with Q (see the definition in (2.12)). In this case we obtain a stronger lower-bound, which willbe proven in Section 3.3. Theorem 3.
Retain the hypotheses of Theorem 1 and assume that u satisfies the scalingcondition (1.6) with constant C S . Then there exist a constant c > depending only ond and C S and a dimensional constant c > such thatN ( µ ) ≥ c N u ( c µ ) for all µ > . (1.7)To e ff ectively apply Theorem 3, we need to control the constant C S in the scalingassumption (1.6). This assumption is analogous to doubling hypotheses which are com-monly used in the continuous case for elliptic PDEs. For some equations, they canbe established by standard Harnack and De Giorgi–Nash–Moser arguments. In the pres-ence of a potential, such an estimate is more di ffi cult to establish, but holds, for instance,in the continuum for all Kato potentials. See the discussion in [DFM]. In the discretecase, the scaling condition (1.6) holds whenever V is periodic with C S depending on theperiod and strength of the potential. Indeed, suppose that { v n } is periodic in each of the d coordinate directions with period vector (cid:126) p = ( p , · · · , p d ) ∈ N d . Assume that K isdivisible by each p i so that the domain Λ consists of several copies of the fundamentalcell of V . Then, as we show in Section 3.4, the scaling condition C S in (1.6) depends on d , V max , and (cid:126) p only, but not on K . Combining this results with Theorem 1 and Theorem3, we obtain the following result, which will be proven in Section 3.4. Corollary 1.
Let H = − ∆ + V be as in (1.1) , with a non-trivial periodic potentialV = { v n } n ∈ Λ as above. Then (1.8) c N u ( c µ ) ≤ N ( µ ) ≤ N u ( C µ ) for all µ > , where C , c are dimensional constants and c depends on d , V max , and (cid:126) p only. The scaling condition (1.6) is not the only way to obtain the improved estimate akinto (1.7) from (1.5). We can use (1.5) to convert upper and lower bounds on N u into alower bound on N . As a simple example, suppose that for µ belonging to some intervalon the positive half-line, we have the bounds a µ β ≤ N u ( µ ) ≤ b µ β , HE LANDSCAPE LAW 5 where the power β > d /
2. Substituting these bounds into (1.5) and choosing α suf-ficiently small, it is easy to deduce the lower bound (1.7) on N ( µ ) for µ in the sameinterval. We note that this argument does not apply when N u ( µ ) behaves like µ β for β ≤ d / N ( µ ), or, more precisely,for the expectation of N ( µ ), in the case of Anderson potentials. For such a potential thevalues { v n } n ∈ Λ are given by independent, identically distributed (i.i.d.) random variables,with common probability measure P on R . Denote by F ( δ ) = P ( v n ≤ δ ) the commoncumulative distribution function of v n and bysupp P = { µ ∈ R : P (( µ − ε, µ + ε )) > , ∀ ε > } the support of the measure P . We assume that inf supp P = , sup supp P = V max ,and that supp P contains more than one point. We denote by E ( · ) the expectation withrespect to the product measure on R | Λ | generated by P . In this case we obtain thefollowing upper and lower bounds for the expectation of N ( µ ), which will be proven inSection 4. Theorem 4.
Suppose K ≥ and V = { v n } n ∈ Λ is an Anderson-type potential as above.Let C be as in Theorem 1. Then there are constants c , c > depending on d, theexpectation of the random variable, and V max , such that for all µ > ,c E N u ( c µ ) ≤ E N ( µ ) ≤ E N u ( C µ ) . (1.9) Furthermore, there is a constant µ ∗ > depending on d and expectation of the randomvariable, such that if, in addition, µ ≤ µ ∗ , then (1.9) holds with the constants c , c independent of V max . A direct consequence of the above theorem is the exponentially decaying behaviorof N near zero, which is typical for a disordered system. This phenomenon is usuallyreferred to as the Lifshitz tail. We will discuss it thoroughly later, but for now let us statethe results. Corollary 2.
Retain the setting of Theorem 4. There are constants µ , ¯ C , ¯ c , ¯ C , ¯ c , ¯ γ , ¯ γ , and K ∗ depending only on the dimension and the expectation of the random vari-able, (in particular, independent of V max ), such that for all µ < µ , E N ( µ ) ≤ ¯ C µ d / F ( ¯ C µ ) ¯ γ µ − d / . (1.10) If, in addition, µ > K ∗ / K , then E N ( µ ) ≥ ¯ c µ d / F (¯ c µ ) ¯ γ µ − d / . (1.11) Remark . The restriction µ > K ∗ / K is necessary for the lower bound (1.11). Thefirst eigenvalue of H is bounded from below by O (1 / K ). For µ (cid:46) / K , N ( µ ) = µ >
0. The restriction is only needed for thisLifshitz tail lower bound and disappears in the limit of an infinite domain. The Lifshitz
D. ARNOLD, M. FILOCHE, S. MAYBORODA, W. WANG, AND S. ZHANG tail upper bound and all the rest estimates in the Landscape Law hold for all µ > K ≥ Remark . In all the above results for Anderson model, the dependence of the constantson the expectation of the random variable can be made more explicit. The constantsactually only depend on the tail behavior of F . We will state more precise versions ofthese results in Section 4. Remark . The bounds in (1.10) and (1.11) are for the expectation of N ( µ ) on a finitedomain. We shall see that in the limit as K → ∞ , they become deterministic, see (1.15).Having surveyed the main results of the paper, in the remainder of the introduction,we discuss some aspects in more detail, and explain the relation to prior results.In the continuous model, the asymptotic behavior of N ( µ ) as µ → + ∞ has beenextensively studied since Hermann Weyl proposed the so-called Weyl law in 1911. Thereare countless extensions and conjectures about the Weyl law for various domains andoperators. However, there are also obvious shortcomings. For instance, the volume-counting principle of the Weyl law fails for a simple system where the potential is givenby two coupled harmonic oscillators, see e.g. [Fe] page 143. Some related issues havebeen addressed, still in the continuum, by the Uncertainty Principle of Fe ff erman andPhong in [Fe], where the authors suggested to work with the number of disjoint cubes ofside length µ − / and such that ( (cid:62) Q | V | P ) / p ≤ C µ . A more universal solution was o ff eredby the first Landscape Law paper [DFM], which is analogous to the present work, butin a continuous setting. To the best of our knowledge, neither the Weyl law nor theUncertainty Principle of Fe ff erman and Phong has an appropriate discrete analogue. Onecould say that the question is not quite applicable to the discrete setting: indeed, on anylattice of a finite size the potential is finite and the spectrum is contained in the interval(0 , d + V max ). However, it is important to notice that the estimates in Theorem 1 donot carry any dependence on V max or the size of the system, and hence, various limitingsituations provide a rich ground for examples. In particular, it is clear that one can set upa discrete system approximating a given continuous setting and derive, via our theorems,a meaningful discrete analogue of the estimates “at infinity” for the integrated densityof states. This is one of the signatures of the universality and novelty of the LandscapeLaw established in the present paper, as it applies to all such settings directly.Stemming from this philosophy, one interesting source of examples are the approxi-mations of Kato potentials (again, we refer the reader to [DFM] for the definitions anddiscussion; see also [Ku, HL]). As we have already mentioned, another clearcut appli-cation of the second part of Theorem 3 is to periodic potentials. These potentials are,in principle, well-understood, based on a thorough analysis of the symmetry properties,see e.g. [Ea, RS]. Here we just want to give an explicit example of validity of the scalingcondition (1.6) in the discrete setting which is not an approximation of the continuousone. In principle, many well-behaved potential will yield (1.6), however, what exactlyis “well-behaved” is a somewhat tricky question. Big changes of the potential are not HE LANDSCAPE LAW 7 friendly to (1.6). For instance, if on a 1-dimensional lattice we have a big region of V = V =
1, that would correspond to a region of − ∆ u = − ∆ u + Vu =
1, in the first case u being quadratic and in the second po-tentially exponential, which clearly destroys the “doubling” required by (1.6). It is quiteamazing that an improved estimate (1.9) nonetheless holds, basically due to the fact thatthe aforementioned exponential nature of the Lifshitz tails “beats” the polynomial cor-rection in (1.5). And hence, as we pointed out above, the Landscape Law yields sharpresults in this scenario as well.An interesting question which is, by nature, more meaningful for specific rather thangeneric types of the potential that we are emphasizing here, is in which regimes theresulting Landscape Law is particularly powerful and precise. This is the subject offorthcoming publications in physics and in computational mathematics, but we can al-ready make a few remarks here. One simple observation is that for µ > V max , N u ( µ ) ≈ min( µ d / , , (1.12)and hence, while all the estimates between N and N u still hold, they are satisfied in arather trivial way (see, e.g., Figure 2).Hence, the Landscape Law is richer when the potential is reasonably large. This wasless relevant in the continuous setting since no scale was fixed by the step of a lattice. Onthe other hand, there is also an interesting improvement specific to the discrete set-up,the so-called concept of a dual landscape.Contrary to the continuous case, the spectrum of the discrete Schr¨odinger operator H = − ∆ + V is a compact subset in [0 , d + V max ]. The eigenvalue counting near thetop of the spectrum for (cid:101) µ close to 4 d + V max can be converted into the counting near thebottom of the spectrum for µ = d + V max − (cid:101) µ close to 0. Such a conversion can befurnished through a dual model (cid:101) H = − ∆ + V max − V , see [LMF]. One can define a duallandscape function (cid:101) u as a solution to ( (cid:101) H (cid:101) u ) n = N (cid:101) u using(1.3), leading to the so-called dual landscape theory. Then, invoking (1.9), (1.10), weobtain Corollary 3.
Retain the definitions in Theorem 4. Suppose K ≥ is even. There arepositive constants a i , b i , (cid:101) µ ∗ , (cid:101) K ∗ depending on expectation of the random variable only (inparticular, independent of V max ), such that for µ > , − a E N (cid:101) u ( a (cid:101) µ ) ≤ E N ( µ ) ≤ − a E N (cid:101) u ( a (cid:101) µ )(1.13) and for µ > (cid:101) µ ∗ , K > (cid:101) K ∗ (cid:101) µ − / , − b (cid:101) µ d / (cid:2) − F ( V max − b (cid:101) µ ) (cid:3) b (cid:101) µ − d / ≤ E N ( µ ) ≤ − b (cid:101) µ d / (cid:2) − F ( V max − b (cid:101) µ ) (cid:3) b (cid:101) µ − d / (1.14) where (cid:101) µ = d + V max − µ and (cid:101) u is the landscape function for − ∆ + V max − V. D. ARNOLD, M. FILOCHE, S. MAYBORODA, W. WANG, AND S. ZHANG F igure
2. One-dimensional discrete Schr ¨odinger operator on a periodiclattice Z / Z , with a random potential V = { v n } uniformly distributed in[0 , N , the scaled landscape box-counting function N u and the dual landscapebox-counting function N (cid:101) u . The first plot shows the whole spectrum, whilethe second zooms in on the bottom spectrum.The concept of the dual landscape was first introduced in [LMF] for a one-dimensionaldiscrete lattice with the zero boundary condition. It was later generalized in [WZ] tohigher dimensional lattices to study the Agmon type of localization for the high energymodes. We will formulate and discuss the dual landscape more precisely in Section 4.4.Returning to the potential precision of our estimates, we remark that strictly speaking,the constants in (1.9) are far from optimal. However, the formulas emphasize the correctfeatures of the spectrum and in numerical experiments we can obtain much more satis-factory results. In [DMZDAWF] and accompanying numerical work still in preparation,we show that there are very stable constants c , c such that a practical landscape lawholds: N ( µ ) ≈ c N u ( c µ ), see Figure 2.Finally, let us discuss the new bounds for the Lifshitz tails (1.10) and (1.11). To thisend, let us slightly change the notation and denote the IDS on the finite box Λ = ( Z / K Z ) d by N K ( µ ). It is well known that for random Schr¨odinger operators, the thermodynamiclimit of the finite volume integrated density of states N K ( µ ) exists almost surely and is adeterministic function: N ∞ ( µ ) = lim K →∞ E N K ( µ ) a . s . = lim K →∞ N K ( µ ) . (1.15)This is the (infinite volume) IDS for H = − ∆ + V on (cid:96) ( Z d ), which also can be definedby direct functional analytic arguments, without a limiting procedure. More generally,the infinite volume IDS can also be well defined for periodic and ergodic Schr¨odingeroperators in an appropriate sense, either for the continuous or for the discrete case. HE LANDSCAPE LAW 9
Moreover, the definition is independent of the boundary conditions. The infinite volumeIDS defined by a limiting procedure using Dirichlet, Neumann, or periodic boundaryconditions all coincide, see e.g. [CL, PF, KM1, BoKl].Notice that all the scaling constants in Corollary 2 are independent of K . Therefore,(1.10) and (1.11) also hold for N ∞ ( µ ) for all µ →
0. In other words, near the bottom ofthe spectrum, we have C µ d / e − C | log F ( C µ ) | µ − d / ≤ N ∞ ( µ ) ≤ C µ d / e − C | log F ( C µ ) | µ − d / , µ (cid:38) . (1.16)In particular, this implies that • if F has an atom at 0, e.g., F is a Bernoulli distribution, then C e − C µ − d / ≤ N ∞ ( µ ) ≤ C e − C µ − d / ;(1.17) • if c µ c ≤ F ( µ ) ≤ c µ c for µ near 0, e.g., F is a uniform distribution, then C e − C | log µ | µ − d / ≤ N ∞ ( µ ) ≤ C e − C | log µ | µ − d / . (1.18)The expressions (1.16)-(1.18) are all in the form of the so-called Lifshitz tails, sug-gested by the physicist I. Lifshitz in 1960s [Li] for disordered systems. Lifshitz tailshave been studied extensively since then for numerous continuous and discrete models.Some of the early work made specific assumptions on the potential V . For example,the first (rigorous) proofs [DV, Pa, Na] of Lifshitz tails were obtained for the Poissonrandom potential and it has been proved thatlim µ (cid:38) log N ∞ ( µ ) µ − d / = − C . (1.19)These proofs strongly relied on large deviations via the Donsker-Varadhan technique,and allowed one to obtain more precise asymptotics than (1.16)–(1.18). On the otherhand, they seemed di ffi cult to apply beyond the actual Poisson model. Later Kirsch andMartinelli [KM2] and Simon [Si1] gave a proof close to Lifshitz’s intuition for moregeneral models roughly speaking treating all F which are not “too thin” at the bottom,but obtained only the weaker, so-called “double logarithmic” asymptoticslim µ (cid:38) log | log N ∞ ( µ ) | log µ = − d . (1.20)We see that estimates (1.10) and (1.11) not only recover (1.20) for all F ( µ ) (cid:29) e − µ − ε , µ (cid:38)
0, but are also “exponentially” more precise. They provide more accurate upper andlower finite volume criterion for the IDS, identifying, for instance, the logarithmic cor-rection for power-bounded F , and treating more general distributions essentially withoutany restrictions on F other than its non-triviality.Another notable source of information about Lifshitz tails comes from the so-calledparabolic Anderson model (PAM), see, e.g., [GaMo, BiKo, Ko]. The PAM is essentiallythe heat equation for the discrete random Schr¨odinger operator. By the moment analysisof the long time behavior of the solution to the heat equation, it is possible to obtain the Lifshitz tails for the corresponding Schr¨odinger operator, in particular for some un-bounded type of potentials, see [BiKo]. A typical example is the so-called “Bernoullitrap” where v n takes the value 0 with probability p and the value + ∞ with probability1 − p . By the moment analysis method, it was showed in [BiKo] that the IDS of theBernoulli trap has tail asymptotics in the strong form (1.19), where the constant C canbe computed explicitly only in terms of p and the dimension d .To the best of our knowledge, the Poisson potential [DV, Na] and the examples com-ing through the PAM (carrying additional assumptions on the tail distribution) [BiKo]are the only models where one can obtain the strong form (1.19). It is still open forexample, whether for the Anderson Bernoulli model, (1.19) holds. The best one can sayis that both in the continuous case [KM2] and in the discrete case [Si1] − C µ − d / ≤ log N ∞ ( µ ) ≤ − C µ − d / . (1.21)Furthermore, while for the Anderson model with a uniformly distributed random po-tential, the log µ correction in (1.18) was discussed in the theoretical physics literature(see e.g. Luck, Nieuewenhuisen [LN], Politi, Schneider [PS]), the rigorous proof of thelogarithmic correction for the discrete case, that is, the estimate − C | log µ | · µ − d / ≤ log N ∞ ( µ ) ≤ − C | log µ | · µ − d / , (1.22)was obtained in [Me]. We see that Theorem 4 and Corollary 2 recover (1.22) as a partic-ular case of a simple and universal law, which can be applied to all bounded potentials.There are no analogues of [Me] in the continuous setting. The best one could previouslyget [KM2] is − C | log µ | · µ − d / ≤ log N ∞ ( µ ) ≤ − C µ − d / , (1.23)and the logarithmic correction in the upper bound was first established in [DFM] as aby-product of the Landscape Law for the continuous model. It is still open whether forthe uniform case a strong form of the asymptoticlim µ (cid:38) log N ∞ ( µ ) | log µ | · µ − d / = − C (1.24)holds either in continuous or discrete case.The original intuition of Lifshitz relied on the idea that the asymptotics of N ∞ ( µ ) aredetermined by the quantum kinetic energy, referred to as the quantum regime followingthe terminology of [PF]. Most of the examples that we have discussed so far are in thisregime for short range potentials and “fat” single site distributions. Pastur [Pa] observedthat for long range potentials or “thin” single site distributions, the asymptotics of N ∞ ( µ )is determined by the potential instead, which is referred to as the classical regime. Sup-pose F is a “thin” single site distribution in the sense that F ( µ ) ∼ e − C µ − α , µ (cid:38)
0, wherethe ∼ sign means estimates from above and below modulo multiplicative constants. Then HE LANDSCAPE LAW 11 (1.16) implies the so-called Pastur tailslim µ (cid:38) log | log N ∞ ( µ ) | log µ = − d − α. (1.25)In other words, in this terminology, the Landscape Law covers the quantum and classicalregime alike. It would be interesting to see that whether the Landscape Law holds forlong range interaction (or correlated) potentials.There are also studies of Lifshitz tails for internal band edges [Me, Si2, Kl1], studiesof Lifshitz tails for random Landau Hamiltonians and magnetic fields [Er, Wa, KR], anda recent work of Lifshitz tails for an operator originated from representation-theory [FS].Let us mention that the Lifshitz tails are not only interesting on their own right, but alsocan be used as an important input to prove Anderson localization, see [Kl2, El, AW].In the weak disorder regime of the Anderson model, there is evidence that the zone ofthe Lifshitz tail goes almost up to the average of the potential, [Kl2]. It would be veryinteresting to quantify the Lifshitz tail zone for the landscape box counting function N u and to understand what happens for the energy higher up.The rest of the paper is organized as follows. We state preliminaries for tight-bindingHamiltonians and the discrete landscape theory in Section 2. In Section 3, we studydeterministic potentials and prove Theorem 1, 2, 3, and Corollary 1. In Section 4, weconcentrate on the Anderson model. We prove the Lifshitz tail estimates for N u in Sec-tions 4.1, 4.2 and finally conclude Theorem 4 in Section 4.3. Section 4.4 is a discussionfor the dual landscape theory. In the Appendix, we include some technical details for dis-crete harmonic functions and a well known probability result called Cherno ff -Hoe ff dingbound. The argument in [DFM] strongly relies on the foundations of the theory of el-liptic PDEs. Many of these estimates are challenging and require di ff erent techniquesspecific to harmonic analysis on Z d . For instance, because of the lack of rotation sym-metry and dilation invariance, many estimates for the Poisson kernel and the Green’sfunction are not known on a discrete lattice, and are technically di ffi cult to prove. Asubstantial portion of the paper is devoted to the discrete analogues of these ellipticestimates, and we hope they will be of independent interest.Acknowledgments. Arnold is supported by the NSF grant DMS-1719694 and SimonsFoundation grant 601937, DNA. Filoche is supported by Simons Foundation grant 601944,MF. Mayboroda is supported by NSF DMS 1839077 and the Simons Collaborations inMPS 563916, SM. Wang is supported by Simons Foundation grant 601937, DNA. Zhangis supported in part by the NSF grants DMS1344235, DMS-1839077, and Simons Foun-dation grant 563916, SM. 2. P reliminaries In the tight-binding model, the Hilbert space is taken as the sequence space (cid:96) ( Z d ) = { { φ i } i ∈ Z d | (cid:80) i ∈ Z d | φ i | < ∞ } where we may think of φ = { φ i } i ∈ Z d either as a function φ ( i ) on Z d or as a sequence φ n indexed by i ∈ Z d . The Z d lattice is equipped with the 1-norm: | n | : = d (cid:88) i = | n i | (2.1)which reflects the graph structure of Z d . The infinity (maximum) norm is also neededfrequently: | n | ∞ : = max ≤ i ≤ d | n i | . (2.2)Two vertices m = ( m , · · · , m d ) , n = ( n , · · · , n d ) ∈ Z d are called nearest neighbor, if | m − n | =
1. We also say nearest neighbors m , n are connected by an edge of the discretegraph Z d . We denote by e i = (0 , · · · , , , , · · · , , i = , · · · , d the canonical base of Z d . For φ = { φ n } n ∈ Z d , its i -th directional (right) derivative ∇ i φ : Z d → R is defined as ∇ i φ n = φ n + e i − φ n , i = , · · · , d , (2.3)and its gradient ∇ φ : Z d → R d is ∇ φ n = ( ∇ φ n , ∇ φ n , · · · , ∇ d φ n ) . The discrete (graph) Laplacian ∆ on Z d is defined as usual, acting on φ = { φ n } n ∈ Z d ,( ∆ φ ) n = (cid:88) | m − n | = ( φ m − φ n ) = (cid:88) ≤ i ≤ d (cid:0) φ n + e i + φ n − e i − φ n (cid:1) . (2.4)For a real sequence { v n } n ∈ Z d on Z d , the potential V is a multiplication operator actingon φ ∈ (cid:96) ( Z d ) as ( V φ ) n = v n φ n . − ∆ + V is called the discrete Schr¨odinger operator on Z d . If one takes v n = v n ( ω ) as independent, identically distributed random variables(in some probability space), the random operator − ∆ + V ( ω ) is usually referred to asthe Anderson model. We refer readers to [Ki, AW] for more details and a completeintroduction to tight-binding Hamiltonians and the Anderson model.Throughout the rest of the paper, we consider the discrete Schr¨odinger operator − ∆+ V restricted on a finite domain in Z d . Let Λ = ( Z / K Z ) d (cid:27) { ¯1 , ¯2 , · · · , ¯ K } d , where K ∈ N and ¯ k , 1 ≤ k ≤ K , is the congruence class, modulo K . We consider Λ as a subset in Z d for simplicity. We abuse the notation and denote by | · | the induced 1-norm of Z d on the congruence class Λ , where for example we consider two points (1 , n , · · · , n d )and ( K , n , · · · , n d ) are nearest neighbor and have distance one to each other in Λ . Fromnow on, we will concentrate on the finite dimensional subspace (cid:96) ( Λ ) of (cid:96) ( Z d ). Wefrequently write H : = (cid:96) ( Λ ) (cid:27) R K d for simplicity. H is equipped with the usual innerproduct on R K d , which is denoted by (cid:104)· , ·(cid:105) = (cid:104)· , ·(cid:105) H . It is easy to check that for φ ∈ H φ n = φ n + Ke i , n ∈ Λ , i = · · · , d , (2.5)which is equivalent to say that φ satisfies the periodic boundary condition. HE LANDSCAPE LAW 13
The restriction of V on Λ is still denoted by V for simplicity. We assume that v n ≥ V max = max Λ v n >
0, and assume min Λ v n =
0. We denote by H = H Λ the restriction of − ∆ + V on (cid:96) ( Λ ):( H Λ φ ) n = − ( ∆ φ ) n + ( V φ ) n = − (cid:88) | m − n | = ( φ m − φ n ) + v n φ n , n ∈ Λ . (2.6)Similar to the di ff erential case, the discrete Hamiltonian can be written in its Dirichletform on the periodic lattice Λ : (cid:104) φ, H φ (cid:105) H = (cid:88) n ∈ Λ (cid:107)∇ f n (cid:107) + (cid:88) n ∈ Λ v n φ n , (2.7)where (cid:107)∇ f n (cid:107) : = (cid:80) di = |∇ i f n | .It is easy to check that all eigenvalues of H in (2.6) are contained in (0 , d + V max )for any finite K . For the Anderson model H ∞ = − ∆ + V ( ω ) acting on the entire space (cid:96) ( Z d ), it is well known that its spectrum σ ( H ∞ ) is (almost surely) the non-random set[0 , d ] + supp V ⊂ [0 , d + V max ].Discrete Schr¨odinger operators acting on a finite dimensional space are essentiallymatrices. Take H Λ = − ∆ + V in (2.6) on Z for example, it can be identified as the sumof the following K × K matrices, − ∆ = − · · · − − . . . ... . . . . . . . . . ... . . . . . . − − · · · − K × K , V = v · · · v . . . ... . . . . . . . . . ... . . . . . . v K − · · · v K K × K , (2.8)where we abuse the notations for the operators and their matrix representations.It is easy to verify that H is invertible as long as V is not identically zero. More-over, by the maximum principle (see Lemma A.2), all the matrix elements of its inverseare positive, H − ( i , j ) > i , j ∈ Λ . Therefore, there is a unique positive vector u ∈ (cid:96) ( Λ ) solving the equation Hu = (cid:126)
1, where (cid:126) landscape equation and the solu-tion u = { u n } n ∈ Λ , will be called the landscape function . u is the discrete analogue of thelandscape function in [FM] in the continuum setting. The discrete landscape functionwas first introduced in [LMF], for a one dimensional lattice with zero boundary condi-tions. It was studied on a higher dimensional lattice with periodic boundary conditionsin [WZ]. The following result can be found in [WZ]. Theorem 5 (Theorem 2.10, Lemma 2.12 in [WZ]) . Assume that v n ≥ and is notidentically zero. The inhomogeneous boundary value problem ( H φ ) n = − ( ∆ φ ) n + v n φ n = , n ∈ Λ (2.9) F igure
3. The centric cube (in red) Q = { , , } has side length 3. Q issurrounded by its translations (in blue) Q + ae + be , a , b = , ±
3. Theunion of all the small cubes is 3 Q = { , · · · , } of side length 9. has a unique solution u = { u n } ∈ (cid:96) ( Λ ) . If, in addition, v n ≤ V max , then min n ∈ Λ u n ≥ / V max > . (2.10)As shown in [ADFJM2] for the continuous case and in [WZ] for the discrete case,1 / u : = { / u n } n ∈ Λ serves as an e ff ective potential satisfying the following landscape un-certainty principle: Theorem 6 (Lemma 2.14 in [WZ]) . For any f ∈ (cid:96) ( Λ ) , (cid:104) f , H f (cid:105) H = (cid:88) n ∈ Λ (cid:88) ≤ i ≤ d u n + e i u n · (cid:18) ∇ i f n u n (cid:19) + (cid:88) n ∈ Λ u n f n ≥ (cid:88) n ∈ Λ u n f n , (2.11) where ∇ i f n u n = f n + e i / u n + e i − f n / u n . Let us introduce a few more notations for simplicity. For a , b ∈ Z , a ≤ b , we denoteby (cid:126) a , b (cid:127) = { a , a + , · · · , b } consecutive integers from a to b . We will frequently workwith cubes in Z d , and also the corresponding congruent class of cubes in Λ = ( Z / K Z ) d .For r ∈ N , Q = (cid:126) , r (cid:127) d , we say that Q is a cube in Z d of “side length” (cid:96) ( Q ) = r , whichis the cardinality of Q projected in each direction. We denote by | Q | = Card ( Q ) thetotal cardinality of Q and call it “the volume” of Q whenever it is clear. For any a ∈ Z d , a + Q is the translation of Q in Z d of the same side length and volume. For a cube Q , wedenote by 3 Q the cube concentric with Q of side length 3 (cid:96) ( Q ), see Figure 3,3 Q : = (cid:91) ≤ i ≤ dk i = , ± r ( Q + k e + k e + · · · + k d e d ) . (2.12)Let ∂ Q be the (inner) boundary of Q : ∂ Q = { n ∈ Q : n + e i (cid:60) Q or n − e i (cid:60) Q for some 1 ≤ i ≤ d } . (2.13) HE LANDSCAPE LAW 15 F igure
4. The e ff ective potential { / u n } n = is plotted in blue. The hori-zontal reference line (in black) is µ = /
9. The partition P (3) = { Q j } j = contains four disjoint cubes. On Q , Q , Q , min(1 / u n ) falls below µ .Let dist ( n , ∂ Q ) = min {| m − n | : m ∈ ∂ Q } be the distance function to the boundary of Q measured in | · | norm. We denote by Q / Q , defined as: Q / = { n ∈ Q : dist ( n , ∂ Q ) ≥ (cid:100) (cid:96) ( Q ) / (cid:101) } (2.14)where (cid:100) x (cid:101) is the ceiling function. It is easy to verify that Q / Q in the sense that (cid:96) ( Q / = (cid:96) ( Q ) − (cid:100) (cid:96) ( Q ) / (cid:101) ≤ (cid:96) ( Q ) / Q / ⊆ Q . Theinequality becomes an equality if 3 | (cid:96) ( Q ).3. L andscape law in the D eterministic case In this section, we study the deterministic case in Theorem 1, 2, and 3. Let us recallsome of the notations first. Let Λ (cid:27) (cid:126) , K (cid:127) d be the periodic domain of side length K .Let N ( µ ) be the (finite volume) integrated density of states (IDS) of H on Λ , as definedin (1.2).Let u = { u n } be the landscape function of H , given in Theorem 5. For µ >
0, let s ( µ ) = (cid:6) µ − / (cid:7) and let N u ( µ ) be the landscape box counting function, defined w.r.t. thepartition P = P ( s ( µ ); Λ ) as in (1.3), see Figure 4,5 for examples on Z , Z . Remark . The landscape box counting function N u = N P u is defined w.r.t. to thepartition P = P ( r ; Λ ) , r = (cid:6) µ − / (cid:7) . Consider any translation of P by a ∈ (cid:126) , r − (cid:127) d , i.e., P a = { a + Q , Q ∈ P} . P a is also a partition of Λ of size r since Λ is a periodic torus.Each a + Q ∈ P a can be covered by finitely many cubes Q (cid:48) ∈ P , and vice versa. Thenumber of the cubes needed to cover each other is at most 3 d −
1, which only depends onthe dimension. Therefore, if we use P a to define a landscape box counting function N P a u ,the counting of the boxes will only di ff er by a factor of 3 ± d , i.e., for any a ∈ (cid:126) , r − (cid:127) d − d N P a u ≤ N P u ≤ d N P a u . For a similar reason, if r < r (cid:48) , then P ( r ; Λ ) is a finer partition than P ( r (cid:48) ; Λ ). Each Q (cid:48) ∈ P ( r (cid:48) ) can be covered by at most ( r (cid:48) / r + d many Q ∈ P ( r ; Λ ). Therefore, thecounting of Q (cid:48) ∈ P ( r (cid:48) ) such that min n ∈ Q (cid:48) u n ≤ µ will di ff er from the number of Q ∈ P ( r ) F igure
5. The e ff ective potential (in blue) and box-counting function ona Z lattice. The reference energy (in pink) µ = /
36. The partition P (6) contains 9 disjoint boxes. The four regular boxes have side length6. There are five irregular boxes next to the boundary.such that min n ∈ Q u n ≤ µ , by a factor of ( r (cid:48) / r + d . In other words, for r < r (cid:48) , N P ( r (cid:48) ) u ≤ N P ( r ) u ≤ (cid:18) r (cid:48) r + (cid:19) d · N P ( r (cid:48) ) u . Based on the above discussion, we are allowed to estimate N u ( µ ) by either shifting theoriginal partition P ( (cid:6) µ − / (cid:7) ) or tackling the side length of the partition slightly. Thechange of the partition will result a di ff erent box-counting function, but the new countingfunction will di ff er from N u by some multiplicative dimensional constants. This will bevery useful in the proof.3.1. Upper bound.
We start with proving the upper bound (1.4)
Theorem 7.
Suppose that K ≥ and V ∈ (cid:96) ∞ ( Λ ) is a non-constant, non-negative poten-tial. For any µ > , one has N ( µ ) ≤ N u (4 d µ ) . (3.1) Proof.
Let H = − ∆ + V be as in (2.6) acting on H = (cid:96) ( Λ ) (cid:27) R K d . We denote by (cid:104)· , ·(cid:105) the inner product on (cid:96) ( Λ ) induced by the usual one on R K d . We consider the followingtwo cases either µ <
1, or µ ≥ HE LANDSCAPE LAW 17
Case I: µ <
1. To get an upper bound for N ( µ ), it is enough to bound (cid:104) f , H f (cid:105) frombelow on some subspace of H . For r = (cid:6) µ − / (cid:7) ≤ µ − / , let P ( r ) = P ( r ; Λ ) be thepartition of side length r . Let F : = (cid:26) Q ∈ P ( r ) : min n ∈ Q u n ≤ µ (cid:27) . Let S be the linear subspace of H such that the average restricted on each Q ∈ F iszero, i.e., S = (cid:40) f ∈ H : 1 | Q | (cid:88) n ∈ Q f n = , Q ∈ F (cid:41) . The subspace S has Card ( F ) many linear independent constrains since all Q ∈ P aredisjoint. Therefore, S has codimension Card ( F ).By the landscape uncertainty principle (2.11), (cid:104) f , H f (cid:105) = (cid:88) n ∈ Λ (cid:107)∇ f n (cid:107) + v n f n ≥ (cid:88) n ∈ Λ u n f n , which implies 2 (cid:104) f , H f (cid:105) ≥ (cid:88) n ∈ Λ (cid:107)∇ f n (cid:107) + u n f n . Therefore, for f ∈ S , one has2 (cid:104) f , H f (cid:105) ≥ (cid:88) Q ∈P ( r ) (cid:88) n ∈ Q (cid:18) (cid:107)∇ f n (cid:107) + u n f n (cid:19) ≥ (cid:88) Q ∈F (cid:88) n ∈ Q (cid:107)∇ f n (cid:107) + (cid:88) Q (cid:60) F (cid:88) n ∈ Q u n f n . (3.2)In the second sum, 1 / u n ≥ min 1 / u n > µ since Q (cid:60) F . Therefore, (cid:88) Q (cid:60) F (cid:88) n ∈ Q u n f n ≥ µ (cid:88) Q (cid:60) F (cid:88) n ∈ Q f n . To bound the first gradient term on the r.h.s. of (3.2), we need the discrete version of thePoincar´e inequality (see a proof in Lemma A.4 in Appendix A): for any cube Q of sidelength (cid:96) ( Q ) = r = (cid:6) µ − / (cid:7) , (cid:88) n ∈ Q (cid:107)∇ f n (cid:107) ≥ (cid:96) ( Q ) · d (cid:88) n ∈ Q ( f n − ¯ f Q ) ≥ (cid:6) µ − / (cid:7) · d (cid:88) n ∈ Q f n ≥ µ d (cid:88) n ∈ Q f n . Notice that the last Q ∈ P in each direction may not be a regular box of equal sidelength, the above estimate remains the same since the side length of the irregular boxdoes not exceed r = (cid:6) µ − / (cid:7) , see (A.6) in Lemma A.4.Putting these two parts together, one has for f ∈ S (cid:104) f , H f (cid:105) ≥ (cid:88) Q ∈P (cid:88) n ∈ Q (cid:18) (cid:107)∇ f n (cid:107) + u n f n (cid:19) ≥ (cid:88) Q ∈F µ d (cid:88) n ∈ Q f n + µ (cid:88) Q (cid:60) F (cid:88) n ∈ Q f n ≥ µ d (cid:88) n ∈ Λ f n = µ d (cid:104) f , f (cid:105) . Therefore, the number of eigenvalues of H below µ d is bounded from above by thecodimension of the subspace S , which is Card ( F ) as discussed previously. In otherwords, we have showed that N (cid:16) µ d (cid:17) ≤ Card ( F ) | Λ | = N u ( µ ) , for µ < . Equivalently, N ( µ ) ≤ N u (4 d µ ) , for µ < d . (3.3) Case II: µ ≥ . The constructions are similar to the previous case. Let F = (cid:26) n ∈ Λ : 1 u n ≤ µ (cid:27) , S = { f ∈ H : f n = n ∈ F } . Almost the same computation shows that for any f ∈ S (cid:104) f , H f (cid:105) ≥ (cid:88) n ∈F (cid:107)∇ f n (cid:107) + (cid:88) n (cid:60) F u n f n ≥ (cid:88) n ∈F (cid:88) ≤ i ≤ d f n + e i + (cid:88) n (cid:60) F µ f n ≥ µ (cid:88) n ∈F f n + (cid:88) n (cid:60) F µ f n = µ (cid:104) f , f (cid:105) . Notice that in this case, we do not need the Poincar´e inequality since for f n =
0, we have( ∇ i f n ) = ( f n + e i − f n ) = f n + e i ≥ = µ f n trivially.Therefore, for all µ ≥ N ( µ ) ≤ N u ( µ ). Since N ( µ ) is non-decreasing, it implies that N ( µ/ (4 d )) ≤ N ( µ ) ≤ N u ( µ ) for µ ≥
1. Equivalently, N ( µ ) ≤ N u (4 d µ ) , for µ ≥ d . (3.4)Combing (3.3) and (3.4), we finished the proof for (3.1). (cid:3) General lower bound in the non-scaling case.
In this part, we prove the generallower bound for N by N u , without any additional assumptions on V = { v n } (or u = { u n } ). Theorem 8.
Suppose that K ≥ and V ∈ (cid:96) ∞ ( Λ ) is a non-constant, non-negative poten-tial. Then there are dimensional constants c , c , C , α such thatN ( µ ) ≥ c α d N u ( c α d + µ ) − C N u ( c α d + µ ) , for all µ > < α < α . (3.5)Similar to the upper bound, if one can bound (cid:104) f , H f (cid:105) from above on a subspace of H (cid:27) R K d , then the eigenvalue counting function will be bounded from below by thedimension of this subspace. HE LANDSCAPE LAW 19
Proof.
Let r = (cid:108) ( c H / µ − (cid:109) , (3.6)where (cid:100)·(cid:101) is the ceiling function and 0 < c H < < α < ( c H / − / , let R = (cid:6) α − µ − / (cid:7) ≥ r . Consider the following cases: Case I : r | R | K , i.e., K = K · R , R = R · r for some K , R ∈ N .Clearly, K ≥ R ≥ r . We assume R ≥ r ≥ r = R ≥ r ≥ c H / µ − ≤ r < c H / µ − , α − µ − / ≤ R < α − µ − / . The latter twoimply that 12 α − (32 / c H ) / ≤ R = Rr ≤ α − (32 / c H ) / . (3.7)Now for K = K R , consider the partition P ( R ; Λ ) for Λ of side length R . Then foreach Q of side length R = R r , we consider the finer partition P ( r ; Q ) of side length r .Clearly, the collection of all q ∈ P ( r ; Q ) for all Q ∈ P ( R ; Λ ) also forms a partition for Λ of size r : P ( r ; Λ ) = ∪ Q P ( r ; Q ) = { q : q ∈ P ( r ; Q ) , Q ∈ P ( R ; Λ ) } . (3.8)In each Q , let ˇ q be the q ∈ P ( r ; Q ) concentric with Q , such that 3 ˇ q ⊂ Q /
3, where 3 A and A / A ⊂ Z d are defined as in (2.12),(2.14). This construction requires R ≥
3, which can be fulfilled by restricting α < (32 / c H ) / / K (cid:38) α − µ − / . We will see later that when K is small, (3.5) holds in a rather trivial way,and the largeness condition on K can be removed eventually.For any 0 < α < (32 / c H ) / /
6, let F (cid:48) = (cid:26) Q ∈ P ( R ; Λ ) : min n ∈ ˇ q u n ≤ µ and min n ∈ Q u n ≥ α µ (cid:27) . (3.9)Given Q ∈ P ( R ; Λ ), let Q / ⊂ Q be the middle third of Q as usual. Let χ Q = { χ Qn } n ∈ Λ ∈H be a discrete cut-o ff function supported on Q such that χ Qn = , if n ∈ Q / , = , if n (cid:60) Q , ∈ (0 , , otherwise , (3.10)and | χ Qn + e i − χ Qn | (cid:40) ≤ R , if n , n + e i ∈ Q , = , if n or n + e i (cid:60) Q , for i = , · · · , d . Such cut-o ff function is the discrete analogue of a smooth bump function in the continu-ous case. We include the explicit construction in Appendix A.3 for reader’s convenience. Let S (cid:48) be the linear subspace of H (cid:27) R K d which is spanned by the cut-o ff of u on each Q in F (cid:48) . More precisely, let x Qn be given as above. We define S (cid:48) = span (cid:8) u Q = { u Qn } n ∈ Λ ∈ H : u Qn = u n · χ Qn , Q ∈ F (cid:48) (cid:9) . The subspace S (cid:48) has dimension Card ( F (cid:48) ) since all Q are disjoint and u n > (cid:10) u Q , H u Q (cid:11) / (cid:10) u Q , u Q (cid:11) from above for each u Q in S (cid:48) . First, bythe landscape uncertainty principle (2.11), (cid:10) u Q , Hu Q (cid:11) = (cid:88) n ∈ Λ (cid:88) ≤ i ≤ d u n + e i u n (cid:18) u Qn + e i u n + e i − u Qn u n (cid:19) + (cid:88) n ∈ Λ u n (cid:0) u Qn (cid:1) = (cid:88) n ∈ Λ (cid:88) ≤ i ≤ d u n + e i u n (cid:0) χ Qn + e i − χ Qn (cid:1) + (cid:88) n ∈ Q u n (cid:0) u n χ Qn (cid:1) ≤ (cid:88) ≤ i ≤ d (cid:88) n , n + e i ∈ Q u n + e i u n (cid:18) R (cid:19) + (cid:88) n ∈ Q u n ≤ d · R d − · sup Q u n + R d · sup Q u n . (3.11)On the other hand, recall that 3 ˇ q ⊂ Q /
3, then (cid:10) u Q , u Q (cid:11) = (cid:88) n ∈ Q (cid:0) u n χ Qn (cid:1) ≥ (cid:88) n ∈ Q / u n ≥ (cid:88) n ∈ q u n . Since − ( ∆ u ) n = − v n u n ≤
1, applying the discrete Moser-Harnack inequality (seeLemma A.11 in Appendix A.6) to u n on the smaller cubes ˇ q ⊂ q with (cid:96) ( ˇ q ) = r , one has, (cid:88) n ∈ q u n ≥ r d (cid:18) c H sup ˇ q u n − r (cid:19) . (3.12)By the definition of F (cid:48) in (3.9), one hassup n ∈ Q u n ≤ α − µ − , sup n ∈ ˇ q u n ≥ µ − . (3.13)Notice that r = (cid:108) ( c H / µ − (cid:109) implies ( c H / µ − ≤ r < c H / µ − , i.e., r ≤ c H µ − , r − ≤ (cid:112) / c H · µ. (3.14)Therefore, putting (3.11) and (3.12) together where R = R r , (cid:10) u Q , Hu Q (cid:11) (cid:104) u Q , u Q (cid:105) ≤ dR d − · sup Q u n + R d · sup Q u n r d (cid:0) c H sup ˇ q u n − r (cid:1) ≤ dR d − r d − · α − µ − + R d r d · α − µ − r d (cid:0) c H µ − − c H µ − (cid:1) HE LANDSCAPE LAW 21 = dR d − r − · α − µ − + R d · α − µ − c H µ − ≤ d · c − H · R d − · ( (cid:112) / c H ) · µ · α − + c − H R d · α − µ ≤ C (cid:0) R d − α − + R d · α − (cid:1) µ (3.15) ≤ C α − d − µ, (3.16)where C = C ( d , c H ). From (3.15) to (3.16), we used (3.7) where R (cid:46) α − .Then, by the orthogonality of the u Q for Q ∈ F (cid:48) , we get N : = Card (cid:8) eigenvalues λ of H such that λ ≤ C α − d − µ (cid:9) (3.17) ≥ dimension of S (cid:48) = Card( F (cid:48) ) = Card (cid:26) Q ∈ P ( R ; Λ ) : min n ∈ ˇ q u n ≤ µ and min n ∈ Q u n ≥ α µ (cid:27) ≥ Card (cid:26) Q ∈ P ( R ; Λ ) : min n ∈ ˇ q u n ≤ µ (cid:27) (: = N ) − Card (cid:26) Q ∈ P ( R ; Λ ) : min n ∈ Q u n ≤ α µ (cid:27) . (: = N )(3.18)Let e i , i = , · · · , d be the canonical base of Z d . Given a integer | j | ≤ (cid:98) R / (cid:99) , weconsider a translation T i , j : Λ → Λ , by the vector jre i , i.e., T i , j ( n ) = n + jre i forany n ∈ Λ . For the partition P ( R ; Λ ), denote by P i , j ( R ; Λ ) the translated partition by T i , j , which is a partition of Λ with the same side length R . Recall that P ( r ; Λ ) is thefiner partition of side length r , see (3.8). Denote by P i , j ( r ; Λ ) the translation of P ( r ; Λ ),which again is a refinement for P i , j ( R ; Λ ). For any ˇ q ⊂ Q ∈ P ( R ; Λ ), it is easy to checkthat Q ⊂ (cid:91) i , j T i , j ( ˇ q ) . In other words, the collection of all T i , j ( ˇ q ) will cover the entire Q , provided enough j (atmost (cid:98) R / (cid:99) many).For each translated centric cube T i , j ( ˇ q ) in the corresponding T i , j ( Q ) ∈ P i , j ( R ; Λ ), werepeat the construction of F (cid:48) and S (cid:48) starting from (3.9). By exactly the same argumentfor (3.16), we can obtain the translated version of (3.18), which is N : ≥ N i , j − N i , j , (3.19)where N is the same as in (3.17) since the eigenvalue counting will be the same for allthe translations, and N i , j = Card (cid:26) T i , j ( Q ) ∈ P i , j ( R ; Λ ) : min n ∈ T i , j (ˇ q ) u n ≤ µ (cid:27) , N i , j = Card (cid:26) T i , j ( Q ) ∈ P i , j ( R ; Λ ) : min n ∈ T i , j ( Q ) u n ≤ α µ (cid:27) . Recall ˇ q has side length r and locates at the center of each Q . We repeat the aboveprocess exact σ times so that (cid:83) j , i T i , j ( ˇ q ) = Q , and therefore, (cid:83) Q (cid:83) j , i T i , j ( ˇ q ) = Λ ,which is exactly the finer partition P ( r ; Λ ) of the entire domain. One translated T i , j ( ˇ q )corresponds to exactly one small cube q in the original partition P ( r ; Λ ). The number σ of the translations we need can be bounded from above by σ ≤ (2 · (cid:98) R / (cid:99) ) d ≤ R d ≤ C · α − d . Notice that for all j , i , N i , j < C · N for some dimensional constant C becauseof Remark 3.1. Then we sum up (3.19) w.r.t. all possible translations T i , j ( · ), one has σ · N ≥ (cid:88) i , j N i , j − σ · N i , j (3.20) = (cid:88) j , i Card (cid:26) T i , j ( Q ) ∈ P i , j ( R ; Λ ) : min n ∈ T i , j (ˇ q ) u n ≤ µ (cid:27) − σ · N i , j = (cid:88) j , i Card (cid:26) T i , j ( ˇ q ) : min n ∈ T i , j (ˇ q ) u n ≤ µ (cid:27) − σ · N i , j ≥ Card (cid:26) q ∈ P ( r ; Λ ) : min n ∈ q u n ≤ µ (cid:27) − σ · C · N . Therefore, C α − d · N ≥ Card (cid:26) q ∈ P ( r ; Λ ) : min n ∈ q u n ≤ µ (cid:27) − (cid:101) C α − d · N . (3.21)Notice that the partition in the counting of N is of side length R = (cid:6) α − µ − / (cid:7) = (cid:6) ( α µ ) − / (cid:7) , which is exactly the side length needed in the definition of N u ( α µ ). On theother hand, the partition in the counting of N is of side length r = (cid:108) ( c H / µ − (cid:109) . Theside length needed in the definition of N u ( µ ) should be r (cid:48) = (cid:6) µ − / (cid:7) , which is larger than r used in N . But the two counting functions defined by P ( r ) or P ( r (cid:48) ) will only di ff er bya dimensional factor since 1 ≤ r (cid:48) / r ≤ / c H ) / , see Remark 3.1. Therefore,Card (cid:26) q ∈ P ( r ; Λ ) : min n ∈ q u n ≤ µ (cid:27) ≥ C − · Card (cid:26) q (cid:48) ∈ P ( r (cid:48) ; Λ ) : min n ∈ q (cid:48) u n ≤ µ (cid:27) . Then by (3.21) α − d N ≥ c · Card (cid:26) q (cid:48) i ∈ P ( r (cid:48) ; Λ ) : min n ∈ q (cid:48) i u n ≤ µ (cid:27) − C · α − d · Card (cid:26) Q ∈ P ( R ; Λ ) : min n ∈ Q u n ≤ α µ (cid:27) , HE LANDSCAPE LAW 23 which implies that N ( C α − d − µ ) ≥ c α d N u ( µ ) − CN u ( α µ ) , (3.22)provided that 0 < α < (32 / c H ) / / K ≥ (cid:6) α − µ − / (cid:7) and R ≥ r ≥ K , R , r . First for K , notice that if3 r < K < (cid:6) α − µ − / (cid:7) , then we can repeat the above construction by setting R = K directly, the proof for (3.22) is exactly the same. If K ≤ r (cid:46) µ − / , there are at most3 d boxes in the partition. Therefore, N u ( µ ) ≤ d / K d . On the other hand, the argumentfor (3.11) can be used to show that the ground state eigenvalue E of H is bounded fromabove by E ≤ C µ , with a dimensional constant C . Therefore, N ( C µ ) ≥ / K d (cid:38) N u ( µ ).Then (3.22) holds trivially by picking α small, and the smallness only depends on thedimension.Then we discuss r and R , especially the restriction on Q , whose side length R has tobe at least 3 to construct the cut o ff function χ Q in (3.10). If r = α − µ − / >
3, then R = R ≥
3. It is easy to check that all the constructions and estimates for N , N , N still work. If α − µ − / ≤
3, then r = R ≤
3. Then we proceed the construction of F (cid:48) , S (cid:48) with cube Q of side length (cid:101) R =
9. Notice this will not change the counting for N and N . The change will only result a di ff erent counting for N , which we denoteby (cid:101) N . Since 1 ≤ R (cid:48) / R ≤
3, one has (cid:101) N ≤ N ≤ d (cid:101) N , due to Remark 3.1. Then weobtain similar estimates for N , N , N by tackling the dimensional constants slightly.In conclusion, (3.22) holds without any largeness restriction on K , r and R , as long as r | R | K .Equivalently, for any µ >
0, if we repeat the proof of (3.22) for (cid:101) µ = c α d + µ where c = C − , then we will obtain N ( µ ) ≥ c α d N u ( c α d + µ ) − CN u ( c α d + µ ) , provided 0 < α < (32 / c H ) / / Case II: either r (cid:45) R or R (cid:45) K , where r = (cid:6) ( c H / / · µ − / (cid:7) and R = (cid:6) α − µ − / (cid:7) arethe same in Case I. W.L.G. we assume that K = ( K − R + (cid:101) R , < (cid:101) R < R , and R = ( R − r + (cid:101) r , < (cid:101) r < r . The other two cases where either (cid:101) R = R or (cid:101) r = r are similar. Recall the construction ofthe partition for P ( R ; Λ ) and P ( r ; Q ), in the last row / column of each direction, we needto use a rectangular box instead of a cube. We denote the regular cube of side length R or r still by Q and q , and denote the remaining special rectangular boxes by (cid:101) Q and (cid:101) q ,whose side length is (cid:101) R and (cid:101) r respectively in at least one direction. And we write P ( R ; Λ ) = { Q } ∪ { (cid:101) Q } , P ( r ; Q ) = { q } ∪ { (cid:101) q } . (3.23)Notice in this case, the union (cid:101) P = ∪P ( r ; Q ) is not the original partition P ( r ; Λ ), buta finer one. Therefore, the counting function defined through (cid:101) P will be bounded frombelow by the counting function defined through P ( r ; Λ ). In this case, we define F (cid:48) only using the regular cube Q and ignoring all the (cid:101) Q . Thenby exact the same construction, we obtain (3.18), i.e., N ≥ N − N . Next, we need totranslate the partition P ( R ; Λ ) and P ( r ; Q ) by vectors of length r in each direction severalsteps, so that the centric small cubes ˇ q can cover the large cubes Q . In the previous case,we need at most j ∼ (cid:98) R / (cid:99) steps in each direction. In Case II, we want to continue thetranslation up to (cid:101) j ∼ (cid:98) R (cid:99) steps in each direction, where the total number of translationsin all directions is at most (cid:101) σ ≤ (2 · R ) d ≤ C · α − d . By doing this, the translated centriccubes T i , j ( ˇ q ) will cover 3 Q . In particular, it covers all the irregular boxes (cid:101) Q near theboundary of the domain. Notice for the regular cubes Q , translations up to (cid:101) j ∼ (cid:98) R (cid:99) steps will cause an overlap, which leads to an overestimate in the corresponding sum (cid:80) i , j N i , j as in (3.20). But since (cid:101) j ≤ R and the T i , j ( ˇ q ) are contained in 5 Q for each Q ,the over counting will be at most 5 d times more. In conclusion, we can obtain (cid:101) σ · Card (cid:8) eigenvalues λ of H such that λ ≤ C α − d − µ (cid:9) ≥ (cid:88) j , i Card (cid:26) T i , j ( ˇ q ) : min n ∈ T i , j (ˇ q ) u n ≤ µ (cid:27) − (cid:101) σ · Card (cid:26) Q ∈ P ( R ; Λ ) : min n ∈ Q u n ≤ α µ (cid:27) ≥ d Card (cid:26) q ∈ (cid:101) P : min n ∈ q u n ≤ µ (cid:27) − (cid:101) σ · Card (cid:26) Q ∈ P ( R ; Λ ) : min n ∈ Q u n ≤ α µ (cid:27) . Note that in the counting in the last line, q or Q already includes the irregular boxes (cid:101) q or (cid:101) Q respectively. Together with the fact that (cid:101) P is finer than P ( r ; Λ ), we obtain that α − d · Card (cid:8) eigenvalues λ of H such that λ ≤ C α − d − µ (cid:9) ≥ − d Card (cid:26) q ∈ P ( r ; Λ ) : min n ∈ q u n ≤ µ (cid:27) − C α − d Card (cid:26) Q ∈ P ( R ; Λ ) : min n ∈ Q u n ≤ α µ (cid:27) , which implies again N ( C α − d − µ ) ≥ c · α d N u ( µ ) − C · N u ( α µ ). The remaining argumentfor (3.5) is exactly the same as in Case I. (cid:3) Lower bound in the scaling case.
We proceed for a refined lower bound for N ( µ )under an additional scaling assumption on the landscape function. Theorem 9.
Suppose that K ≥ and V ∈ (cid:96) ∞ ( Λ ) is a non-constant, non-negative po-tential. Let H = − ∆ + V be as in (2.6) . Let u = { u n } n ∈ Λ be the landscape function ofH. Assume that u satisfies the scaling condition (1.6) with constant C S , i.e., there is adimensional constant C S > such that (cid:88) n ∈ Q u n ≤ C S (cid:32)(cid:88) n ∈ Q u n + (cid:96) d + (cid:33) (3.24) for every cube Q ⊂ Λ of side length (cid:96) . Here, Q is the tripled cube concentric with Q(see the definition in (2.12) ). Then there exist constants c = c ( d , C s ) , c = c ( d ) > HE LANDSCAPE LAW 25 such that N ( µ ) ≥ c N u ( c µ ) , for all µ > . (3.25)The outline of the proof follows from the non-scaling case in Section 3.2. The con-struction is much simpler with the help of the scaling condition (3.24). Proof.
Let R = (cid:38)(cid:18) C (cid:48)(cid:48) (cid:19) µ − (cid:39) , (3.26)where C (cid:48)(cid:48) ≥ R | K or R (cid:45) K . Similar to the non-scalingcase, the latter can be reduced to the first case by a translation argument. For simplicity,we will only deal with the case that K = K · R for some K ∈ N . Also, we first assumethat R ≥
3, so that the cube of side length R is large enough to construct cut o ff functionsas in (3.10). Otherwise, we start with (cid:101) R = R ≥
3, consider the partition P ( R ; Λ )consisting cubes of side length R as usual. Let F (cid:48)(cid:48) = (cid:26) Q ∈ P ( R ; Λ ) : min n ∈ Q u n ≤ µ (cid:27) and S (cid:48)(cid:48) = span (cid:8) u Q ∈ H : u Qn = u n · χ Qn , Q ∈ F (cid:48)(cid:48) (cid:9) , where χ Q = { χ Qn } is the cut-o ff function as in (A.14) on each Q . The dimension of S (cid:48)(cid:48) equals Card ( F (cid:48)(cid:48) ).Our goal, once again, is to establish estimates as in (3.16), which will allow us tobound the eigenvalues on the subspace S (cid:48)(cid:48) from above. The upper bound for (cid:10) u Q , Hu Q (cid:11) remains the same as we obtained in (3.11): (cid:10) u Q , Hu Q (cid:11) ≤ dR d − · sup Q u n + R d · sup Q u n . (3.27)It remains to obtain the lower bound of (cid:10) u Q , u Q (cid:11) . First, the Moser-Harnack inequality(A.48) implies that: (cid:88) n ∈ Q u n ≥ R d (cid:18) c H · sup Q u n − R (cid:19) . Then we apply the scaling condition (3.24) twice, (cid:88) n ∈ Q u n ≤ C S (cid:32)(cid:88) n ∈ Q u n + R d + (cid:33) ≤ C S (cid:32) C S (cid:32) (cid:88) n ∈ Q / u n + ( R / d + (cid:33) + R d + (cid:33) ≤ C S (cid:88) n ∈ Q / u n + C (cid:48) S R d + . In this step, we need the size of Λ large enough to contain 3 Q and Q /
3, which requiresthat (cid:96) ( Q ) = R ≥
9, and K ≥ C (cid:48)(cid:48) ) − / · µ − / .Therefore, (cid:88) n ∈ Q / u n ≥ C − S c H · R d (cid:18) sup Q u n − C (cid:48)(cid:48) · R (cid:19) , where C (cid:48)(cid:48) depends on d , c H and C S . By the choice of R in (3.26), (cid:18) C (cid:48)(cid:48) (cid:19) µ − ≤ R ≤ (cid:18) C (cid:48)(cid:48) (cid:19) µ − . For Q ∈ F (cid:48)(cid:48) , sup Q u n ≥ µ − . Then we have12 sup Q u n ≥ µ − ≥ · (2 C (cid:48)(cid:48) ) · R = C (cid:48)(cid:48) R . Together with the construction of the cut o ff functions χ Q in (A.14), we have (cid:10) u Q , u Q (cid:11) = (cid:88) n ∈ Q ( u n χ Qn ) ≥ (cid:88) n ∈ Q / u n ≥ C − S c H · R d · (cid:18) sup Q u n − C (cid:48)(cid:48) R (cid:19) ≥ C − S c H · R d · (cid:18)
12 sup Q u n (cid:19) . Putting the upper and lower bounds together, we have that (cid:10) u Q , Hu Q (cid:11) (cid:104) u Q , u Q (cid:105) ≤ d R d − · sup Q u n + R d · sup Q u n R d C − S c H sup Q u n = C (cid:48) R − + C (cid:48) min Q u n ≤ C (cid:48) (4 √ C (cid:48)(cid:48) µ ) + C (cid:48) µ : = C µ , where C = C (cid:48) √ C (cid:48)(cid:48) + C (cid:48) depends only on c H , C S and the dimension d . Therefore,Card (cid:8) eigenvalues λ of H such that λ ≤ C µ (cid:9) ≥ Card (cid:0) F (cid:48)(cid:48) (cid:1) . Notice that in the definition of F (cid:48)(cid:48) , the side length of the cube R is smaller than the sidelength (cid:6) µ − / (cid:7) required as in the definition of N u ( µ ). The box counting using P ( R ; Λ )can be bounded from below by the box counting using P ( (cid:6) µ − / (cid:7) ; Λ ), due to Remark3.1. Also, the above estimates require K (cid:38) µ − / and R ≥
3. These restrictions can be re-moved exactly in the same way as for the non-scaling case, by multiplying a dimensionalconstant (cid:101) c , i.e., N ( C µ ) ≥ (cid:101) cN u ( µ ) . Equivalently, one has that N ( µ ) ≥ (cid:101) cN u ( C − µ ) , for all µ > . HE LANDSCAPE LAW 27 (cid:3)
Lower bound for the periodic potential.
In this part, we prove Corollary 1 fora Z d periodic potential V = { v n } . It is enough to show that the landscape function u associated with the periodic potential satisfies the scaling condition (1.6).Suppose V = { v n } is Z d periodic with period (cid:126) p = ( p , · · · , p d ). Let Γ = (cid:126) , p (cid:127) ×· · · × (cid:126) , p d (cid:127) ⊂ Z d be the fundamental cell of V . Let H = − ∆ + V be as in (2.6),acting on H = (cid:96) ( Λ ). Notice that p i | K , i = · · · , d guarantees that Λ = ( Z / K Z ) d contains finitely many copies of Γ . Let H Γ be the restriction of H on Γ with the periodicboundary condition, and let (cid:101) u = { (cid:101) u n } n ∈ Γ be the landscape function for H Γ , i.e., ( H Γ (cid:101) u ) n = , n ∈ Γ . By the uniqueness of the landscape function (Theorem 5), u = { u n } n ∈ Λ will bethe periodic extension of (cid:101) u = { (cid:101) u n } n ∈ Γ to the entire domain Λ .For s ∈ N and a ∈ Λ , let Q ( s ) = a + (cid:126) , s (cid:127) d be cube in Λ of side length s . Consider Γ + (cid:126) p Z d , which are the disjoint translations (copies) of the fundamental cell by (cid:126) p Z d .Suppose s > p max : = max { p , · · · , p d } . It is easy to verify that Q ( s ) can be packed (frominside) or covered (from outside) by finitely many copies of Γ . And the maximal numberto pack and the minimal number to cover only di ff er by a dimensional constant factor atmost C . In other words, there is a number t ∈ N such that T : = (cid:91) t translation of Γ ⊂ Q ( s ) ⊂ (cid:91) C t translation of Γ . Therefore, 3 Q ( s ) ⊂ (cid:91) C t translation of Γ : = T . Using the periodicity of u = { u n } with respect to all translations of Γ , one has, (cid:88) Q ( s ) u n ≥ (cid:88) T u n = t (cid:88) Γ (cid:101) u n . Then one has (cid:88) Q ( s ) u n ≤ (cid:88) T u n ≤ (3 d C · t ) (cid:88) Γ (cid:101) u n ≤ (3 d C · t ) · t (cid:88) Q ( s ) u n ≤ d C (cid:88) Q ( s ) u n , which shows the scaling condition (1.6) is true for relatively large cubes.It is enough to look at cubes which are about the size of Γ . More precisely, suppose s < p max . Notice that the landscape function u = { u n } satisfies both − ( ∆ u ) n + v n u n ≥ n ∈ Λ and − ( ∆ u ) n ≤ n ∈ Λ . A combination of the Moser-Harnack inequality (A.48) and the Harnack inequality(A.50) implies that for some constant C depending on d and V max , one hassup Q ( s ) u n ≤ C s · inf Q ( s ) u n ≤ C s · inf Q ( s ) / u n ≤ C s · sup Q ( s ) / u n ≤ C s · (cid:32) c − H · ( s / − d (cid:88) Q ( s ) u n + c − H · ( s / (cid:33) . Since C s ≤ C p max , one has (cid:88) Q ( s ) u n ≤ (3 s ) d · sup Q ( s ) u n ≤ (cid:101) C (cid:32)(cid:88) Q ( s ) u n + s d + (cid:33) , where the constant (cid:101) C only depends on the dimension, V max and p max .Therefore, for all cubes Q ( s ), { u n } n ∈ Λ satisfies the scaling condition (1.6). The esti-mates for N in the periodic case then follow directly from Theorem 9.4. L andscape law for the A nderson model In this part, we will concentrate on the Anderson model H = − ∆+ V . For such a poten-tial the values V = { v n } n ∈ Λ are given by independent, identically distributed (i.i.d.) ran-dom variables, with common probability measure P on R . Denote by F ( δ ) = P ( v n ≤ δ )the common cumulative distribution function of v n and bysupp P = { µ ∈ R : P (( µ − ε, µ + ε )) > , ∀ ε > } the support of the measure P . We assume that inf supp P = , sup supp P = V max , andthat supp P contains more than one point. In other words, F ( δ ) = δ < F ( δ ) = δ ≥ V max , and there is a δ ∗ >
0, such that0 < F ( δ ) ≤ F ( δ ∗ ) : = F ∗ < , < δ ≤ δ ∗ . (4.1)We denote by E ( · ) the expectation with respect to the product measure on R | Λ | generatedby P . Let N ( µ ) be the IDS and N u ( µ ) be the landscape box-counting function as definedin (1.2) and (1.3).We will study the following tail estimates for N u first. Theorem 10.
Let V = { v n } n ∈ Λ be an Anderson-type potential as above. Suppose K ≥ .Then there are dimensional constants c , c , γ , K ∗ > such that E ( N u ( µ )) ≥ c µ d / F ( c µ ) γ µ − d / , for all K ∗ / K ≤ µ ≤ . (4.2) Furthermore, there are constants C , C , γ , and µ ∗ > depending on d , δ ∗ , F ∗ only, suchthat E ( N u ( µ )) ≤ C µ d / F ( C µ ) γ µ − d / , for all µ < µ ∗ . (4.3) Remark . All the constants are independent of V max . HE LANDSCAPE LAW 29
We will proceed to prove Theorem 10 first in Section 4.1, 4.2. After we establishthese tail estimates for N u , we will combine them with the deterministic result Theorem1, to prove Theorem 4, and Corollary 2.4.1. Lifshitz tails lower bound.
We will study the lower bound of N u (4.2) in this part.For 0 < µ ≤
1, let r = (cid:108) µ − (cid:109) . Let P ( r ; Λ ) be partition of size r as usual. It is enough toassume that K = K r for some K ∈ N , otherwise the counting can always be boundedfrom below by ignoring the irregular boxes in the last rows / columns of P ( r ; Λ ). For allcubes Q ∈ P = P ( r ; Λ ), let ζ Q = n ∈ Q u n ≤ µ and ζ Q = E ( N u ( µ )) = K d E (cid:18) Card (cid:26) Q ∈ P ( r ) : min n ∈ Q u n ≤ µ (cid:27) (cid:19) = K d E (cid:32) (cid:88) Q ∈P ( r ) ζ Q (cid:33) = K d (cid:88) Q ∈P ( r ) E (cid:0) ζ Q (cid:1) = K d r d (cid:88) Q ∈P ( r ) P (cid:18) min n ∈ Q u n ≤ µ (cid:19) . (4.4)Recall the landscape uncertainty principle (2.11) (cid:104) f , H f (cid:105) = (cid:88) n ∈ Λ (cid:107)∇ f n (cid:107) + v n f n ≥ (cid:88) n ∈ Λ u n f n . For each Q ∈ P , consider translations of Q by a Z d vector k · r · e i for all 1 ≤ i ≤ d directions and | k | ≤ m . m is some large integer that will be specified later. Let B m = B m ( Q ) be the union of these translated cubes, B m = (cid:91) | k |≤ m , ≤ i ≤ d ( Q + kre i ) . Using the similar procedure in (A.14), one can construct a discrete cut-o ff function χ = { χ n } ∈ H (cid:27) R K d , supported on B m , and satisfying χ n = , n ∈ B m , and χ n = , n (cid:60) B m , ≤ χ n ≤ , n ∈ B m \B m , |∇ i χ n | = | χ n + e i − χ n | = , n (cid:60) B m , |∇ i χ n | = | χ n + e i − χ n | < m (cid:96) ( Q ) < m µ , n ∈ B m , ≤ i ≤ d . It follows that min B m u n ≤ |B m | (cid:88) n ∈ Λ u n χ n ≤ |B m | (cid:88) n ∈ Λ ( (cid:107)∇ χ n (cid:107) + v n χ n ) = |B m | (cid:88) B m (cid:107)∇ χ n (cid:107) + |B m | (cid:88) B m v n χ n ≤ |B m | |B m | d (cid:18) m µ (cid:19) + |B m | |B m | max B m v n = (4 m + d (2 m + d d m µ + (4 m + d (2 m + d max B m v n ≤ d dm µ + d max B m v n ≤ µ + d max B m v n , provided m ≥ d + d . Therefore, for all Q ∈ P ( (cid:6) µ − / (cid:7) ; Λ ), P (cid:26) min n ∈B m u n ≤ µ (cid:27) ≥ P (cid:26) max n ∈B m v n ≤ d + µ (cid:27) = ( F ( c µ )) |B m | ≥ ( F ( c µ )) C µ − d / , where c = − d − and C = (4 m + d . On the other hand, notice that all the translations Q + kre i still belong to P ( r ), we can rewrite B m = B m ( Q ) as B m ( Q ) = (cid:91) Q (cid:48) ∈P ( r ) ∩B m ( Q ) Q (cid:48) , which implies that for all Q ∈ P ( r ) (cid:88) Q (cid:48) ∈P ( r ) ∩B m ( Q ) P (cid:26) min n ∈ Q (cid:48) u n ≤ µ (cid:27) ≥ P (cid:26) min n ∈B m ( Q ) u n ≤ µ (cid:27) ≥ ( F ( c µ )) C µ − d / . Summing the l.h.s. of the above inequality w.r.t. all Q ∈ P ( r ), one has (cid:88) Q ∈P ( r ) (cid:88) Q (cid:48) ∈P ( r ) ∩B m ( Q ) P (cid:26) min n ∈ Q (cid:48) u n ≤ µ (cid:27) = (2 m + d (cid:88) Q ∈P ( r ) P (cid:26) min n ∈ Q u n ≤ µ (cid:27) . Combining everything together with (4.4), one has E ( N u ( µ )) ≥ K d r d (cid:88) Q ∈P ( r ) P (cid:18) min n ∈ Q u n ≤ µ (cid:19) ≥ (2 m + − d K d r d (cid:88) Q ∈P ( r ) (cid:32) (cid:88) Q (cid:48) ∈P ( r ) ∩B m ( Q ) P (cid:26) min n ∈ Q (cid:48) u n ≤ µ (cid:27)(cid:33) ≥ (2 m + − d K d r d (cid:88) Q ∈P ( r ) ( F ( c µ )) C µ − d / ≥ (2 m + − d K d r d K d ( F ( c µ )) C µ − d / HE LANDSCAPE LAW 31 ≥ (2 m + − d − d µ d ( F ( c µ )) C µ − d / . We also need to impose the condition on the size of domain so that B m ⊂ Λ , i.e., K ≥ (2 m + r = C (cid:48) · µ − / .4.2. Lifshitz tails upper bound.
We proceed to obtain the upper bound (4.3) of N u .Follow the computation in (4.4), it is enough to bound P (cid:110) min n ∈ Q u n ≤ µ (cid:111) from abovesince E ( N u ( µ )) ≤ (cid:6) µ − / (cid:7) d max Q ∈P ( (cid:100) µ − / (cid:101) ) P (cid:26) min n ∈ Q u n ≤ µ (cid:27) . (4.5)This will be the most delicate part in the entire paper. We need several technicallemmas concerning the growth of the landscape function. Some of these estimates mayhave independent interest in the landscape theory and more for elliptic PDEs.For any r ≥
3, let B ⊂ Λ be cube of side length (cid:96) ( B ) = r and let ˇ B = B / M (large, and only depending on the expectation of the random variable), such that for any µ (small enough, depending on E ( v n )), and any cube B of side length r = (cid:6) (4 M µ ) − / (cid:7) ,the following is true P (cid:26) min n ∈ ˇ B u n ≤ µ (cid:27) ≤ A · M d / · F ( A · M µ ) ( M µ ) − d / / (4.6)for some suitable constants A , A (only depending on the dimension and E ( v n ), andindependent of µ ).The rest of Section 4.2 will be devoted to prove (4.6). We start from the followingdeterministic statement. The lemma states that the landscape function u is forced togrow in a certain rate in some circumstances. Lemma 4.1.
Let u = { u n } be the landscape function given by Theorem 5. Let B ⊂ Λ be a cube of side length r : = (cid:96) ( B ) ≥ , and let ˇ B = B / be the middle third as usual.For any < λ < , there is ε ( d , λ ) > such that for all < ε < ε , there is aC P = C P ( ε, λ, d ) > , a M = M ( ε, λ, d ) > and a r ∗ = r ∗ ( ε, λ, d ) > , such that if Bsatisfies the following conditions: (i): Card (cid:8) j ∈ B : v j ≥ C P r − (cid:9) ≥ λ | B | , (4.7) (ii): there is a ξ ∈ ˇ B such that u ξ ≥ Mr , (4.8) then for all r ≥ r ∗ , there is a ξ (cid:48) ∈ Λ such that | ξ (cid:48) − ξ | ∞ ≤ (cid:106) √ + ε r (cid:107) andu ξ (cid:48) ≥ (1 + ε ) u ξ ≥ M (cid:106) √ + ε r (cid:107) . (4.9) Remark . This lemma holds for any λ , and works for any cube B of side length r . Thechoice of C P , ε and M only depend on λ , and is independent of the choice of B , neitheron its size nor the position. Remark . Note that this Lemma is a completely deterministic result. It has nothingto do with the randomness (structure of v n ). It can be applied to any V and u as long as( Hu ) n = B and its neighborhood). Lemma4.1 will lead to some important probability estimates with application to a random po-tential v n . The small parameter λ will be picked in the very end when we are about toprove (4.6), and the choice will only rely on the tail behavior of F .We need some technical preparations for the average of u n . We will frequently write u ( n ) = u n to make the notations of the sub-index easier to read. For ξ = ( ξ , · · · , ξ d ) ∈ Λ ,and r ∈ Z ≥ , we denote by Q ( r ; ξ ) the box centered at ξ of side length 2 r + Q ( r ; ξ ) = (cid:26) m = ( m , · · · , m d ) ∈ Z d : max ≤ i ≤ d | m i − ξ i | ≤ r (cid:27) . We will omit the center ξ (fixed) and write Q ( r ) = Q ( r ; ξ ) whenever it is clear. We alsodenote by ∂ Q ( r ) ⊂ Q ( r ) the boundary of Q ( r ) and by ∂ ◦ Q ( r ) ⊂ ∂ Q ( r ) the boundaryremoving the “corners”: ∂ Q ( r ) = Q ( r ; ξ ) − Q ( r − ξ ) = (cid:26) m = ( m , · · · , m d ) ∈ Z d : max ≤ i ≤ d | m i − ξ i | = r (cid:27) ,∂ ◦ Q ( r ) = (cid:8) m = ( m , · · · , m d ) ∈ ∂ Q ( r ) : | m i − ξ i | = r for only one 1 ≤ i ≤ d (cid:9) . For r =
0, we have the “degenerate cube” Q (0; ξ ) = ∂ Q (0; ξ ) = { ξ } . Notice that Q ( r ) = ∪ r ρ = ∂ Q ( ρ ). Let a r be the average of u n on ∂ Q ( r ) with respect to the (discrete) Poissonkernel P r ( ξ, n ) (see the precise definition and properties of P r in (A.19) in AppendixA.4): a r = (cid:88) n ∈ ∂ Q ( r ; ξ ) P r ( ξ, n ) u n , r ≥ , a = u ξ . (4.10)Let A r be the average of u n on Q ( r ): A r = | Q ( r ) | r (cid:88) ρ = | ∂ Q ( ρ ) | a ρ : = | Q ( r ) | (cid:88) n ∈ Q ( r ) p n u n , (4.11)where for any n ∈ Q ( r ), and ρ = | n − ξ | ∞ p n = | ∂ Q ( ρ ) | P ρ ( ξ, n ) . (4.12)By the properties of the discrete Poisson kernel, one has (cid:88) n ∈ ∂ Q ( r ; ξ ) P r ( ξ, n ) = = ⇒ (cid:88) n ∈ Q ( r ; ξ ) p n = | Q ( r ) | . (4.13)The first two estimates are lower bounds on a r and A r . HE LANDSCAPE LAW 33
Lemma 4.2.
There is dimensional constant C, such that for any ξ ∈ Λ and r ≥ a r ≥ u ξ − r , (4.14) A r ≥ u ξ − Cr . (4.15) Proof.
Let ˜ u be the landscape function for the free Laplacian on Q ( r ) with zero boundarycondition: (cid:40) − ( ∆ ˜ u ) n = , n ∈ Q ( r − u ) n = , n ∈ ∂ Q ( r ) . Let ˜˜ u n = r − d (cid:80) di = ( n i − ξ i ) . Direct computation shows that (cid:40) − ( ∆ ˜˜ u ) n = , n ∈ Q ( r − u ) n ≥ , n ∈ ∂ Q ( r ) . By the maximum principle (Lemma A.1), one has for all n ∈ Q ( r ), ˜ u n ≤ ˜˜ u n ≤ r . Let w n be the harmonic function on Q ( r ) with boundary condition u n , i.e., (cid:40) − ( ∆ w ) n = , n ∈ Q ( r − w ) n = u n , n ∈ ∂ Q ( r ) . Then by the Poisson integral formula (A.22), w ξ = (cid:80) n ∈ ∂ Q ( r ; ξ ) P r ( ξ, n ) u n = a r . On theother hand, (cid:40) − ( ∆ ( w + ˜ u − u )) n = v n u n ≥ , n ∈ Q ( r − w + ˜ u − u ) n = , n ∈ ∂ Q ( r ) . Therefore, ( w + ˜ u − u ) n ≥ , n ∈ Q ( r ). In particular, u ξ ≤ w ξ + ˜ u ξ ≤ a r + r . Notice the above estimate is true for 1 ≤ ρ ≤ r −
1, which implies | ∂ Q ( ρ ) | u ξ ≤ | ∂ Q ( ρ ) | (cid:88) n ∈ ∂ Q ( ρ ) P ρ ( ξ, n ) u n + | ∂ Q ( ρ ) | ρ ≤ (cid:88) n ∈ ∂ Q ( ρ ) p n u n + C · ρ d + . Summing w.r.t. 1 ≤ ρ ≤ r −
1, one has | Q ( r ) | u ξ = r − (cid:88) ρ = | ∂ Q ( ρ ) | u ξ ≤ r − (cid:88) ρ = (cid:88) n ∈ ∂ Q ( ρ ) p n u n + C · r − (cid:88) ρ = ρ d + ≤ (cid:88) n ∈ Q ( r ; ξ ) p n u n + C · r d + , (4.16)which implies u ξ ≤ A r + C · r . (4.17) (cid:3) Lemma 4.3.
For any < η < / , there is a dimensional constant C > (independentof η ) and a constant C = C ( η, d ) > such that for any cube Q ( r ) = Q ( r ; ξ ) of sidelength r ≥ C /η , there is a subset Q η ( r ) ⊂ Q ( r ) such that | Q ( r ) \ Q η ( r ) | ≤ C η r d andp n ≥ C , for n ∈ Q η ( r ) . (4.18) Remark . The lemma is true for all dimensions. But for d =
1, we actually do notneed to remove any portion of the cube (which is an “interval” in Z ) to obtain (4.18)since the 1-d Poisson kernel P r is rather trivial (constantly 1 / p n . Proof.
Write Q ( r ) = ∪ r ρ = ∂ Q ( ρ ). The estimate follows from the lower bound of P ρ ( ξ, n )on each ∂ Q ( ρ ) as long as n is away from the edges (and the corners). Given 0 < η < c ( ρ, η ) , ρ ( η, d ) such that for all ρ ≥ ρ , P ρ ( ξ, n ) ≥ c ρ − d on ∂ Q ( ρ ) except for C ηρ d − many n ∈ ∂ Q ( ρ ), where C only dependson the dimension. Therefore, p n = | ∂ Q ( ρ ) | P ρ ( ξ, n ) ≥ (cid:101) c > ∪ r ρ = ρ ∂ Q ( ρ ) exceptfor (cid:80) r ρ = ρ C ηρ d − ≤ C η r d many n . For 0 ≤ ρ < ρ , P ρ ( ξ, n ) > c ( ρ, d ) except for those n exactly locate on the edges and corners, whose total number is at most C ρ d − . This againimplies p n ≥ min ≤ ρ<ρ c ( ρ, d ) for all n ∈ ∪ ≤ ρ<ρ ∂ Q ( ρ ) except for (cid:80) ρ ρ = ρ d − (cid:46) r d − (cid:46) η r d many n . Therefore, p n ≥ C for some constant C only depending on η and d , and thetotal exceptional n ∈ Q ( r ) violating this has number at most C η r d . (cid:3) With these two technical lemmas, we are ready to prove of the key lemma. We presentthe most general proof that works in any dimension. Most of the estimates can be signif-icantly simplified for the 1-d case since the 1-d “cube” is just an interval in Z (consec-utive integers) with two endpoints as its boundary. The discrete Poisson and the averageformulas etc for Z are rather trivial which makes the proof way easier. Proof of Lemma 4.1.
Let B and ξ ∈ ˇ B be given as in Lemma 4.1, where (cid:96) ( B ) = r .Clearly, B ⊂ Q ( r ; ξ ). Denote by J the set in condition (4.7): J = { j ∈ B : v j ≥ C P r − } , where C P will be be picked later. Fix 0 < λ <
1, we assume that | J | ≥ λ | B | = λ r d . Let A r , p n be defined in (4.11). Let J = { j ∈ J : u j < A r } and let S ( J ) = (cid:88) n ∈ J p n , S ( J C ) = (cid:88) n ∈ Q ( r ) \ J p n = | Q ( r ) | − S ( J ) , (4.19)where the last equality comes from (4.13).Now we are ready to look for ξ (cid:48) satisfying (4.9) in the following two cases: HE LANDSCAPE LAW 35
Case I: | J | ≥ λ | B | = λ r d . Let C be given in Lemma 4.3, then pick η = min( λ C , /
4) and let C = C ( d , η ) begiven in Lemma 4.3. Then | J | = (cid:88) n ∈ J ∩ Q η ( r ) + (cid:88) n ∈ J \ Q η ( r ) ≤ C − (cid:88) n ∈ J ∩ Q η ( r ) p n + C η r d ≤ C − (cid:88) n ∈ J p n + λ r d . Therefore, S ( J ) = (cid:88) n ∈ J p n ≥ C λ r d : = c ( λ ) r d . Direct computation shows that | Q ( r ) | A r = (cid:88) n ∈ Q ( r ) p n u n = (cid:88) J p n u n + (cid:88) Q ( r ) \ J p n u n ≤ A r · S ( J ) + (cid:88) Q ( r ) \ Q p n u n . By the definition of S ( J C ) and (4.19), this implies that1 S ( J C ) (cid:88) Q ( r ) \ J p n u n ≥ | Q ( r ) | − S ( J ) | Q ( r ) | − S ( J ) A r ≥ (cid:18) + S ( J )2 | Q ( r ) | (cid:19) A r ≥ (1 + c ( d , λ )) A r . Therefore, there is one point ξ (cid:48) ∈ Q ( r ) \ J such that u ξ (cid:48) ≥ (1 + c ) A r . By (4.15) and(4.8), u ξ (cid:48) ≥ (1 + c ) ( u ξ − C d r ) ≥ (1 + c ) (cid:18) − C d M (cid:19) u ξ ≥ (cid:16) + c (cid:17) u ξ , provided M > c ( C d +
1) : = M ( d , λ ) . Therefore, (4.9) holds for ε < c / = ε ( d , λ ). Case II: | J | ≤ λ r d Take R = (cid:106) √ + ε r (cid:107) for some small ε , which gives R − r ≤ ε r . Let a r , a R and G r , G R be the surface average and the Green’s function on Q ( r ) = Q ( r ; ξ ) , Q ( R ) = Q ( R ; ξ ) respectively, as defined in (4.10),(A.20). Applying the discrete Green’s iden-tity (integration by parts formula) (A.22) on Q ( r ) and Q ( R ), one has u ( ξ ) = a R − (cid:88) m ∈ Q ( R − G R ( ξ, m ) ∆ u ( m ) = a r − (cid:88) m ∈ Q ( r − G r ( ξ, m ) ∆ u ( m ) . Then a R − a r = (cid:88) m ∈ Q ( R − \ Q ( r − G R ( ξ, m ) ∆ u ( m ) + (cid:88) m ∈ Q ( r − ( G R ( ξ, m ) − G r ( ξ, m )) ∆ u ( m ) . Recall that u n is the landscape function satisfying − ( ∆ u ) n + v n u n = v n ≥ , u n > , G R ≥
0. Then a R − a r ≥ − (cid:88) m ∈ Q ( R − \ Q ( r − G R ( ξ, m ) − (cid:88) m ∈ Q ( r − G R ( ξ, m ) − G r ( ξ, m ) + (cid:88) m ∈ Q ( r − ( G R ( ξ, m ) − G r ( ξ, m )) v m u m . Notice that ∆ G R ( ξ, · ) = Q ( R − ξ ) \{ ξ } . By the maximum principle Lemma A.3,one has max m ∈ Q ( R − \ Q ( r − G R ( ξ, m ) ≤ max m ∈ ∂ Q ( r ) G R ( ξ, m ) , where we used G R ( ξ, m ) = m ∈ ∂ Q ( R ) and the discussion for an annular regionafter Lemma A.3.On the other hand, ∆ ( G R ( ξ, · ) − G r ( ξ, · )) = Q ( r −
1) and G R ( ξ, n ) − G r ( ξ, n ) = G R ( ξ, n ) for n ∈ ∂ Q ( r ), again by the maximum principle Lemma A.3, for any m ∈ Q ( r − m (cid:48) ∈ ∂ Q ( r ) G R ( ξ, m (cid:48) ) ≤ G R ( ξ, m ) − G r ( ξ, m ) ≤ max m (cid:48) ∈ ∂ Q ( r ) G R ( ξ, m (cid:48) ) . By the choice of r and R , ∂ Q ( r ) is away from ∂ Q ( R ) and the pole ξ . Lemma A.7 impliesthat for r large enough (the largeness only depends on ε ) and all m ∈ ∂ Q ( r ), C r − d ≤ G R ( ξ, m (cid:48) ) ≤ C r − d where C , C only depend on d and ε . Therefore, a R − a r ≥ − C ( R d − r d ) r − d − C r d r − d + r − d (cid:88) m ∈ Q ( r − v m u m ≥ − C ( d , ε ) r + r − d (cid:88) m ∈ J \ J v m u m ≥ − C ( d , ε ) r + C r − d C P r − A r ( | J | − | J | ) ≥ − C ( d , ε ) r + ε A r , provided C P (cid:38) ε/ ( C λ ) . HE LANDSCAPE LAW 37
Finally, by the lower bounds of a r , A r in Lemma 4.2 and the condition (4.8) on u ξ , wehave a R ≥ u ξ − r − C ( d , ε ) r + ε ( u ξ − Cr ) ≥ (1 + ε ) u ξ − (1 + C + C ) r ≥ (1 + ε ) u ξ − (1 + C + C ) 1 M u ξ ≥ (1 + ε ) u ξ , provided M ≥ + C + C ε : = M ( d , ε, λ ) . Recall that a R = (cid:80) ∂ Q ( R ) P R ( ξ, n ) u n and (cid:80) ∂ Q ( R ) P R ( ξ, n ) =
1, therefore, there is one ξ (cid:48) ∈ ∂ Q ( R ; ξ ) such that u ξ (cid:48) ≥ (1 + ε ) u ξ , which completes the proof of Lemma 4.1 in the second case. (cid:3) Notice that Lemma 4.1 is deterministic and requires no randomness of v n . A directconsequence is the following estimate on the probability that u ξ grows for the Andersonmodel. It was first proved for the continuous model in [DFM] (Lemma 3.28 therein).The discrete version can be proved in the same way we did for Lemma 4.1. Lemma 4.4.
Let V ( ω ) = { v n } n ∈ Λ be the Anderson-type potential given as in Theorem 10.Fix < λ < , let ε < ε , C P , M , r ∗ be given as in Lemma 4.1. For any cube Q ⊆ Λ ofside length (cid:96) ( Q ) , define the event Ω ( Q ) : = (cid:8) ω : Card (cid:0) j ∈ Q : v j ≥ C P (cid:96) ( Q ) − (cid:1) ≤ λ | Q | (cid:9) . (4.20) Assume that r ∗ ≥ /ε , otherwise just reset r ∗ to be max { /ε, r ∗ } . For any r ≥ r ∗ ,set r k + = (cid:106) √ + ε r k (cid:107) , k = , , · · · , k max , (4.21) where k max is the largest integer k such that r k < K. Let Ω k = Ω (cid:0) (cid:126) , r k (cid:127) d (cid:1) , k = , , · · · , k max and Ω ∞ = Ω (cid:0) (cid:126) , K (cid:127) d (cid:1) .Then for any cube B ⊂ Λ of side length (cid:96) ( B ) = r , denote by ˇ B = B / its middle thirdpart as in (2.14) , one has P (cid:26) ω : max ξ ∈ ˇ B u ξ ≥ Mr (cid:27) ≤ P ( Ω ∞ ) + C ε − d k max (cid:88) k = P ( Ω k )(4.22) for some dimensional constant C. The idea is to repeatedly use Lemma 4.1 to construct a sequence of growing cubesand stop when the final cube exceeds the size of the entire domain.
Proof.
We start with B = B of side length (cid:96) ( B ) = r . Suppose that max ˇ B u ( ξ ) ≥ Mr ,we pick some ξ ∈ ˇ B such that u ( ξ ) ≥ Mr , which gives (4.8). Suppose E : = Ω fails,then (4.7) hold for B . Lemma 4.1 gives a point ξ , such that ξ ∈ C r : = { n : dist ( n , ˇ B ) ≤ r } , where dist is measured in | · | ∞ for Z d lattice points and u ( ξ ) ≥ (1 + ε ) u ( ξ ) ≥ Mr .Additionally, we can require r ≥ /ε which implies that r > (1 + ε/ r . Clearly,Card (cid:0) C r (cid:1) ≤ (2 r + r ) d ≤ ( ε ) d r d . Therefore, C r can be covered by at most n ∗ = (cid:4) d ε − d (cid:5) + Z d of side length (cid:98) r / (cid:99) , namely, ˇ B (1)1 , ˇ B (2)1 , · · · , ˇ B ( n ∗ )1 . Nowextend all these ˇ B ( j )1 to a cube B ( j )1 such that B ( j )1 has side length r and contains eachˇ B ( j )1 as a middle third part for j = , , · · · , n ∗ . Since we do not know which ˇ B ( j )1 that ξ falls into, in order the induction driven by Lemma 4.1 can continue, we need to excludethe events that (4.7) fails for all B ( j )1 . In other words, we define E = n ∗ (cid:91) j = Ω ( B ( j )1 ) . Assume that E fails, which implies that (4.7) holds for all B ( j )1 . Let B = B ( j )1 be the onethat ξ falls into. Now for ξ ∈ ˇ B ( j )1 (cid:36) B ( j )1 , Lemma 4.1 gives ξ , such that | ξ − ξ | ≤ (cid:106) √ + ε r (cid:107) = r , u ( ξ ) ≥ (1 + ε ) u ( ξ ) ≥ Mr . (4.23)Repeat the construction for r > (1 + ε/ r and ξ ∈ C r : = { n : dist ( n , ˇ B ) ≤ r + r } ,Card (cid:0) C r (cid:1) ≤ (2 r + r + r ) d ≤ (12 /ε ) d r d . Therefore, C r can be covered by at most n ∗ = (cid:4) d ε − d (cid:5) + (cid:98) r / (cid:99) , namely, ˇ B (1)2 , ˇ B (2)2 , · · · , ˇ B ( n ∗ )2 . Extend ˇ B ( j )2 to B ( j )2 in the same way, and define E = n ∗ (cid:91) j = Ω ( B ( j )2 ) . We assume that E fails, and find ξ by Lemma 4.1. Inductively, at step k , we assume allthe previous events, E , E , · · · , E k − fail, we obtain ξ k , r k satisfying (4.9) and then define B r k , cC r k . The same estimates as for r , C r hold for all k such that(1 + ε/ k r ≤ · · · ≤ (1 + ε/ r k − ≤ r k ≤ √ + ε r k − ≤ · · · ≤ (1 + ε ) k / r (4.24) Card (cid:0) C r k (cid:1) ≤ (2 r + r + · · · r k ) d ≤ r k · − (1 + ε/ − < (12 /ε ) d · r dk . (4.25) Note that these B j are not disjoint. But the overlap does not e ff ect our estimate on the probability ofthe events from above. HE LANDSCAPE LAW 39
Then ˇ B ( j ) k , B ( j ) k are defined in the same way as we did for the first two steps. Because of(4.25), for all k , we need the same number n ∗ = (cid:4) d ε − d (cid:5) + B ( j ) k to cover C r k .Then we define the event E k as: E k = n ∗ (cid:91) j = Ω ( B ( j ) k ) . Since v n are i.i.d. r.v., the probability of each Ω ( B ( j ) k ) is translation invariant, and onlydepends on the size of the cube B ( j ) k . In particular, P (cid:16) Ω ( B ( j ) k ) (cid:17) = P (cid:0) Ω ([1 , r k ] d ∩ Z d ) (cid:1) = P ( Ω k ) . Therefore, P ( E k ) ≤ n ∗ P ( Ω k ) ≤ C ε − d P ( Ω k ) , (provided ε < C .We will continue the construction until we reach the k max -th step and obtain ξ k max , r k max , B r k max , ˇ B ( j ) k max , B ( j ) k max and E k max .We need to apply Lemma 4.1 two more times for the final conclusion. But accordingto how we pick k max , the last but one application assuming E k max fails will already resulta cube of side length (cid:106) √ + ε r k max (cid:107) ≥ K which may have already exceeded the maximalsize of the entire domain Λ . To settle down the issue, we need to enlarge the domain atthis point for the last two steps, by making several copies of Λ . We need totally at least p d many copies where p : = (cid:106) √ + ε (cid:107) +
1. Let (cid:101) K = pK so that (cid:101) K = (cid:16)(cid:106) √ + ε (cid:107) + (cid:17) K > (cid:106) √ + ε · K (cid:107) and denote by p Λ : = (cid:126) , (cid:101) K (cid:127) d = (cid:83) j ∈ ( Z mod p Z ) d ( Λ + j K ). We extend the potential V = { v n } n ∈ Λ periodically to (cid:101) V = { (cid:101) v n } n ∈ p Λ , where (cid:101) v m = v n , n ∈ Λ , m = n mod( p Z ) d . Now we consider the landscape equation on p Λ of − ∆ + (cid:101) V with periodic boundarycondition. The enlarged system has a unique solution (cid:101) u by Theorem 5. Suppose u isthe original landscape function on Λ . By the periodicity of (cid:101) V and the uniqueness of thesolution, it is easy to verify that (cid:101) u is the periodic extension of u onto p Λ . More explicitly, (cid:101) u m = u n , n ∈ Λ , m = n mod( p Z ) d . We can do this from the very beginning of the construction, but it will not make any di ff erence untilwe reach the size of K . Now we can return to the construction at the k max th step. Assume the event E k max fails,then (4.7) holds for all possible B k max ⊂ (cid:126) , (cid:101) K (cid:127) d that ξ k max may fall into. Since (cid:101) u ξ k max = u ξ k max ≥ Mr ξ k max . Apply Lemma 4.1 to u Γ on B k max ⊂ (cid:126) , K (cid:127) d , we then obtain a (cid:101) ξ ∈ [1 , (cid:101) K ] d ∩ Z d such that (cid:101) u (cid:101) ξ ≥ (1 + ε ) u ξ k max ≥ M (cid:106) √ + ε r ξ k max (cid:107) ≥ MK , (4.27)where the last inequality follows from the definition of k max . Now let ξ ∞ ∈ (cid:126) , K (cid:127) d bewhere u k attains its maximum. Clearly, (cid:101) u k also attains its maximum at ξ ∞ , u ξ ∞ = max k ∈ Λ u k = max k ∈ p Λ (cid:101) u k = (cid:101) u ξ ∞ . (4.28)Together with (4.27), we have u ξ ∞ = (cid:101) u ξ ∞ ≥ (cid:101) u (cid:101) ξ ≥ MK = M K ) . Now consider ˇ B ∞ = Λ , which is the middle third of B ∞ = Λ ⊂ p Λ . Let E ∞ = (cid:8) Card (cid:0) j ∈ Λ : (cid:101) v j ≥ C P (3 K ) − (cid:1) ≤ λ | Λ | (cid:9) = (cid:8) d Card (cid:0) j ∈ Λ : v j ≥ C P (3 K ) − (cid:1) ≤ λ (3 K ) d (cid:9) = (cid:26) Card (cid:18) j ∈ Λ : v j ≥ C P K − (cid:19) ≤ λ K d (cid:27) . Since Card (cid:0) j ∈ Λ : v j ≥ C P K − (cid:1) ≥ Card (cid:0) j ∈ Λ : v j ≥ C P K − (cid:1) , E ∞ ⊆ (cid:8) Card (cid:0) j ∈ Λ : v j ≥ C P K − (cid:1) ≤ λ K d (cid:9) = Ω ∞ . (4.29)Now if E ∞ fails, apply Lemma 4.1 one last time to (cid:101) u on ˇ B ∞ (cid:40) J ∞ . (4.9) implies thatthere is a ξ (cid:48) ∈ (cid:126) , (cid:101) K (cid:127) d such that (cid:101) u ξ (cid:48) ≥ (1 + ε ) (cid:101) u ξ ∞ > (cid:101) u ξ ∞ . This is a contradiction. Recallthis happens when we start with u ξ ≥ Mr and assume all E j fails. Therefore, at leastone E j must happen to prevent the contradiction. In other words, (cid:26) max ˇ B u ξ ≥ Mr (cid:27) ⊂ E ∞ ∪ k max (cid:91) j = E j = ⇒ P (cid:26) max ˇ B u ξ ≥ Mr (cid:27) ≤ P ( E ∞ ) + k max (cid:88) j = P (cid:0) E j (cid:1) . Together with (4.26) and (4.29), we have that P (cid:26) max ˇ B u ξ ≥ Mr (cid:27) ≤ P ( Ω ∞ ) + C ε − d k max (cid:88) k = P ( Ω k ) , which completes the proof. (cid:3) One also needs to take M M / ≥ M in (4.28) to meet the requirement inLemma 4.1. HE LANDSCAPE LAW 41
The next lemma allows us to estimate the probability of each term on the right handside of (4.22).
Lemma 4.5.
Let V = { v j } j ∈ Λ be the Anderson potential as in Theorem 10. Let F and δ ∗ be as in (4.1) . For any B ⊂ Λ and < λ < , if δ > is such that − λ − F ( δ ) > , then P (cid:8) Card (cid:0) j ∈ B : v j ≥ δ (cid:1) ≤ λ | B | (cid:9) ≤ e − D (1 − λ (cid:107) F ) ·| B | , (4.30) where D ( x (cid:107) y ) = x log xy + (1 − x ) log 1 − x − y (4.31) is the Kullback–Leibler divergence between Bernoulli distributed random variables withparameters x and y respectively.As a consequence, for any r ∈ N , P (cid:8) Card (cid:0) j ∈ (cid:126) , r (cid:127) d : v j ≥ δ (cid:1) ≤ λ r d (cid:9) ≤ (cid:0) C ( λ ) F ( δ ) − λ (cid:1) r d (4.32) where C ( λ ) = (1 − λ ) − λ − λ .Furthermore, there is a λ ∗ > , which only depends on F ( δ ∗ ) , such that for all < δ ≤ δ ∗ , < λ ≤ λ ∗ and any r ∈ N , one has P (cid:8) Card (cid:0) j ∈ (cid:126) , r (cid:127) d : v j ≥ δ (cid:1) ≤ λ r d (cid:9) ≤ ( F ( δ )) r d / . (4.33) Remark . The λ ∗ can be taken as of order (1 − F ( δ ∗ )) . See (4.39) in the end of theproof. Proof.
Let ζ j be the characteristic function for the event v j ≤ δ , i.e., ζ j = (cid:40) v j ≤ δ v j > δ . Since {{ v j } are i.i.d. random variables, all ζ j are i.i.d. Bernoulli random variables, takingvalues in { , } , with common expectation E (cid:0) ζ j (cid:1) = P (cid:0) v j ≤ δ (cid:1) = F ( δ ). Let S B : = (cid:88) j ∈ B ζ j . By the Cherno ff –Hoe ff ding Theorem, [Ho] (see Lemma B.1 in the appendix), P { S B ≥ (1 − λ ) | B | } ≤ e − D (1 − λ (cid:107) F ) | B | , (4.34)where F = F ( δ ) and D ( x (cid:107) y ) is as in (4.31). Then (4.30) follows directly from (4.34)since | B | − Card (cid:0) j ∈ B : v j > δ (cid:1) = Card (cid:0) j ∈ B : v j ≤ δ (cid:1) = (cid:88) j ∈ B ζ j = S B . For { , } Bernoulli distribution with p = P ( X = < λ ∗ only depends on p . Examining the the Kullback–Leibler divergence with parameter 1 − λ and F , one has D (1 − λ (cid:107) F ) = (1 − λ ) log 1 − λ F + λ log λ − F ≥ log (cid:0) (1 − λ ) − λ λ λ (cid:1) − log F − λ ≥ log (cid:0) (1 − λ ) λ λ (cid:1) − log F − λ , where we used 1 − F < − λ ) − λ ≥ − λ . Therefore, P (cid:8) Card (cid:0) j ∈ [1 , r ] d ∩ Z d : v j ≥ δ (cid:1) ≤ λ r d (cid:9) ≤ e − D (1 − λ (cid:107) F ) r d ≤ (cid:0) (1 − λ ) − λ − λ F − λ (cid:1) r d (4.35)which is (4.32).Let q = − F ( δ ∗ ) ∈ (0 , < δ ≤ δ ∗ , F ( δ ) ≤ F ( δ ∗ ) = − q < = ⇒ log F ( δ ) ≤ log(1 − q ) < . (4.36)On the other hand, it is easy to check thatlim λ → log (cid:0) (1 − λ ) λ λ (cid:1) / − λ = . (4.37)Then there is a λ ∗ ≤ / < λ ≤ λ ∗ , one haslog (cid:0) (1 − λ ) λ λ (cid:1) / − λ > log(1 − q ) ≥ log F ( δ ) = ⇒ (1 − λ ) λ λ > ( F ( δ )) / − λ = ⇒ (1 − λ ) − λ − λ < ( F ( δ )) λ − / . (4.38)Together with (4.35), one has P (cid:8) Card (cid:0) j ∈ (cid:126) , r (cid:127) d : v j ≥ δ (cid:1) ≤ λ r d (cid:9) ≤ ( F ( δ )) r d / , which completes the proof of Lemma 4.5. One can be more specific on the exact orderof λ ∗ . For 0 < δ ≤ δ ∗ , log F ( δ ) ≤ log(1 − q ) < − min( q ,
12 )If λ < /
4, then log(1 − λ ) > − λ > − √ λ, and λ log λ > − √ λ. Let λ ∗ = (cid:18)
18 min( q ,
12 ) (cid:19) . (4.39)For λ < λ ∗ ≤ /
4, one has0 > log (cid:0) (1 − λ ) λ λ (cid:1) / − λ ≥ (cid:0) log((1 − λ ) + λ log λ (cid:1) ≥ − √ λ HE LANDSCAPE LAW 43 ≥ − ·
18 min( q ,
12 ) ≥ log F ( δ ) , which gives (4.38) in the same way. (cid:3) Combining Lemma 4.4 and Lemma 4.5, one immediately has:
Lemma 4.6.
Let δ ∗ and λ ∗ be as in Lemma 4.5. Fix λ ≤ λ ∗ , take ε < ε ( λ, d ) andC P , M , r ∗ as in Lemma 4.4. Then for any cube B (cid:40) Λ of side length (cid:96) ( B ) = r ≥ r ∗ , andits middle third part ˇ B, one has P (cid:26) max ξ ∈ ˇ B u ξ ≥ Mr (cid:27) ≤ C ε − d − F ( δ ∗ ) · (cid:0) F ( C P r − ) (cid:1) r d / (4.40) for some dimensional constant C > .Remark . The exponent r d / r d , by taking λ to besmaller. And it will also result a large factor 1 /λ in front of F . Proof.
Let r = r and define the sequence r k as in Lemma 4.4. We also write r ∞ = K forsimplicity. Let δ k = C P r − k , k = , · · · , k max and δ ∞ = C P K − . By the construction of r k and (4.24), one has r dk ≥ (1 + ε/ dk r d ≥ (1 + kd ε/ r d ≥ r d + k , k = , · · · , k max . It is also clear that for k = , · · · , k max and k = ∞ δ k ≤ δ = C P r − ≤ δ ∗ provided r ≥ √ C P /δ ∗ . Notice that in the proof of Lemma 4.5, by the choice of δ ∗ in(4.1), F ( δ ) < F ( δ ∗ ) : = − q . Therefore, F ( δ k ) ≤ − q < , k = , · · · , k max , and ∞ since the distribution F is non-decreasing. Now apply Lemma 4.5 to all r k , combing(4.33) with (4.22), one has P (cid:26) max ξ ∈ ˇ B u ξ ≥ Mr (cid:27) ≤ F ( δ ∞ ) K d / + C ε − d k max (cid:88) k = F ( δ k ) r dk / ≤ F ( δ ) r d / + C ε − d k max (cid:88) k = F ( δ ) ( r d + k ) / ≤ F ( δ ) r d / + C ε − d F ( δ ) r d / · − F ( δ ) ≤ F ( δ ) r d / (cid:18) + C ε − d q (cid:19) : = C ( d , ε, δ ∗ ) F ( δ ) r d / , which is the desired bound. (cid:3) Now we are ready to complete:
Proof of (4.3) in Theorem 10.
Let δ ∗ and λ ∗ be given as in Lemma 4.6. Fix λ ≤ λ ∗ , take ε < ε ( λ, d ) and C P , M , r ∗ as in Lemma 4.5.For any µ ≤ / (4 M ), let r = (cid:6) (4 M µ ) − / (cid:7) such that µ − / < Mr ≤ µ − . To applyLemma 4.6, one also needs to ensure that r ≥ r ∗ , which requires µ to be taken in therange µ ≤ µ ∗ = / ( Mr ∗ ).Now for any cube B of side length r = (cid:6) (4 M µ ) − / (cid:7) , and its middle third part ˇ B .Lemma 4.6 implies that P (cid:26) max ξ ∈ ˇ B u ξ ≥ µ − (cid:27) ≤ P (cid:26) max ξ ∈ ˇ B u ξ ≥ Mr (cid:27) ≤ C · (cid:0) F ( C P r − ) (cid:1) r d / , where C > d , ε and F ( δ ∗ ). Then P (cid:26) min n ∈ ˇ B u n ≤ µ (cid:27) ≤ C · (cid:0) F ( C P r − ) (cid:1) r d / ≤ C · ( F (4 C P M µ )) ( M µ ) − d / / . (4.41)Notice that (cid:96) ( ˇ B ) ≥ r / ≥ (4 M µ ) − / /
6. Recall the cubes used in the definition of N u has side length (cid:6) µ − / (cid:7) . For any Q ∈ P ( (cid:6) µ − / (cid:7) ; Λ ), Q can be covered by at most (cid:16) (cid:100) µ − / (cid:101) (4 M µ ) − / / (cid:17) d + ≤ C (cid:48) · M d / disjoint cubes of side length (cid:96) ( ˇ B ) = (cid:6)(cid:6) (4 M µ ) − / (cid:7) / (cid:7) , forsome C (cid:48) only depends on M , d . Notice also that the estimate (4.41) is independent of theposition of ˇ B , and can be applied to all cubes ˇ B of the same size. Therefore, for any Q ,one has P (cid:26) min n ∈ Q u n ≤ µ (cid:27) ≤ C (cid:48) · M d / P (cid:26) min n ∈ ˇ B u n ≤ µ (cid:27) ≤ C (cid:48) M d / · C · ( F (4 C P M µ )) ( M µ ) − d / / ≤ C (cid:48)(cid:48) · ( F ( C µ )) γ µ − d / . Together with (4.5), we obtain the desired upper bound E ( N u ( µ )) ≤ (cid:6) µ − / (cid:7) d max Q ∈P ( (cid:100) µ − / (cid:101) ) P (cid:26) min n ∈ Q u n ≤ µ (cid:27) ≤ C µ d / · ( F ( C µ )) γ µ − d / for all µ ≤ µ ∗ . The constants C , C , γ > d , M and C P , which eventu-ally only depend on d and F ∗ = F ( δ ∗ ). (cid:3) In comparison with the IDS.
Putting together the general upper / lower boundsin Theorem 1 for the deterministic case, and the Lifshitz tails in Theorem 10 for theAnderson model, we have Theorem 11.
Let C be as in Theorem 1 and δ ∗ be as in Theorem 10. Then there areconstants c , c > depending on d, δ ∗ and V max such thatc E N u ( c µ ) ≤ E N ( µ ) ≤ E N u ( C µ ) , for all µ > . (4.42) Let µ ∗ , K ∗ be as in Theorem 10, which only depends on d and δ ∗ . Then for all µ < µ ∗ , (4.42) holds with constants c , c also independent of V max . HE LANDSCAPE LAW 45
If, in addition, K ∗ / K < µ < µ ∗ / C , then there are constants ¯ C , ¯ c , ¯ C , ¯ c , ¯ γ , ¯ γ depending only on δ ∗ such that ¯ c µ d / F (¯ c µ ) ¯ γ µ − d / ≤ E N ( µ ) ≤ ¯ C µ d / F ( ¯ C µ ) ¯ γ µ − d / . (4.43) Proof.
The upper bound in (4.42) is the average of the upper bound in Theorem 1. Weonly need to study the lower bound with the help of Theorem 8 and Theorem 10. Let c , c , C and α < α <
1, be as in Theorem 8. For µ ≥ d + V max , N ( µ ) = µ < d + V max , let us call µ = c α d + µ and µ = α µ = c α d + µ , Theorem 8 implies that E N ( µ ) ≥ c α d E N u ( µ ) − C E N u ( µ ) . (4.44)Next let c , c , C , C , γ , γ > µ ∗ be given as in Theorem 10, then by (4.2) and(4.3), if µ ≤ µ ≤ µ ∗ , one has E N u ( µ ) ≥ c µ d / F ( c µ ) γ µ − d / and E ( N u ( µ )) ≤ C µ d / F ( C · µ ) γ µ − d / . (4.45)Therefore, E N ( µ ) ≥ c α d µ d / F ( c µ ) γ µ − d / − C µ d / F ( C µ ) γ µ − d / (4.46) = c µ d / · (cid:16) F ( c µ ) γ µ − d / − C F ( C α µ ) γ µ − d / (cid:17) (4.47)provided for µ ≤ µ ∗ , µ ≤ , µ < d + V max . This requires that α ≤ α and α ≤ α , where α : = ( c (4 d + V max )) − / ( d + , and α : = µ / ( d + ∗ · ( c (4 d + V max )) − / ( d + . If we assume in addition that α ≤ α and α ≤ α , where α : = (cid:112) c / C , and α : = δ / ( d + ∗ · ( c · c · (4 d + V max )) − / ( d + , then for all µ < d + V max , one has C µ < c µ ≤ δ ∗ . Therefore, 0 < F ≤ F ≤ F ∗ < F = F ( C µ ) = F ( C α µ ) , F = F ( c µ ) , F ∗ = F ( δ ∗ ) . The di ff erence term in (4.47) is then bounded from below by F γ µ − d / − C · F γ µ − d / ≥ F γ µ − d / − C · F γ µ − d / . We want to pick α small enough (independent of µ ) such that, F γ µ − d / − C · F γ µ − d / ≥ F γ µ − d / (4.48) ⇐⇒ F γ µ − d / ≥ C · F γ µ − d / ⇐⇒ (2 C ) − ≥ F γ µ − d / − γ µ − d / = F µ − d / · ( γ α − d − γ )4 . (4.49) Notice that µ < = ⇒ µ − d / > µ − d / ( γ α − d − γ ) > γ α − d − γ > α < ( γ /γ ) / d : = α . Then F ≤ F ∗ < F µ − d / ( γ α − d − γ )4 ≤ F µ − d / ( γ α − d − γ ) ∗ ≤ F γ α − d − γ ∗ . Then we solve 1 ≥ (2 C ) − ≥ F γ α − d − γ ∗ for α , and obtain that γ α − d − γ ≥ log(2 C )log F ∗ ⇐⇒ α ≤ (cid:18) γ − log(2 C )log F ∗ + γ − γ (cid:19) − / d : = α under which (4.49) will be true. Putting everything together, we see that if we set α = α ∗ : = min { α , α , · · · , α } , then for all µ < d + V max , (4.48), (4.47) and (4.45) imply that E N ( µ ) ≥ c µ d / · (cid:16) F γ µ − d / − C F γ µ − d / (cid:17) ≥ c µ d / · (cid:18) F γ µ − d / (cid:19) ≥ c · (cid:18) · C − · E N u ( µ ) (cid:19) = c C − · E N u ( c α d + ∗ · µ ): = c E N u ( c · µ ) , which completes the proof for the l.h.s. of (4.42).It is also easy to verify that if we are only interested in small µ , then all the α i can bepicked independent of V max . Therefore, the final constants c , c are also independent of V max if we require µ to be small in (4.42). In particular, let δ ∗ , µ ∗ be as in Theorem 10,then for all c (cid:48)∗ µ < C µ < µ ∗ E N ( µ ) ≤ E N u ( C µ ) ≤ C ( C µ ) d / F ( C C µ ) γ · ( C µ ) − d / : = ¯ C µ d / F ( ¯ C µ ) ¯ γ µ − d / and E N ( µ ) ≥ c E N u ( C µ ) ≥ c C ( c µ ) d / F ( c c µ ) γ · ( C µ ) − d / : = ¯ c µ d / F (¯ c µ ) ¯ γ µ − d / where are the constants ¯ c , ¯ c , ¯ C , ¯ C , ¯ γ , ¯ γ only depend on d and δ ∗ , µ ∗ , and are inde-pendent of V max . (cid:3) Dual landscape and Energy near the top.
Let H = − ∆ + V be as in (1.1) actingon H = (cid:96) ( Λ ) , Λ = ( Z / K Z ) d . In this part, we will briefly discuss the so-called duallandscape and see how it is applied to the eigenvalue-counting for high energy modes.We refer readers to Section 2.4 in [WZ] for more discussion about the dual landscape.For ϕ ∈ H = (cid:96) ( Λ ), we define a dual vector (cid:101) ϕ (cid:101) ϕ n = ( − s ( n ) ϕ n , n ∈ Λ , (4.50) HE LANDSCAPE LAW 47 where s ( n ) = (cid:80) dj = n j for n = ( n , n , · · · , n d ) ∈ Z d . We assume in addition that K is aneven number so that (cid:101) ϕ ∈ (cid:96) ( Λ ). Now suppose ( µ, ϕ ) is an eigenpair of H = − ∆ + V in H = (cid:96) ( Λ ), direct computation shows that( − ∆ + V max − V ) (cid:101) ϕ = (cid:101) µ (cid:101) ϕ, (4.51)where V max − V stands for the non-negative potential { V max − v n } n ∈ Λ acting on H in thesame way as V does, and (cid:101) µ = d + V max − µ . (4.52)In other words, ( µ, ϕ ) is an eigenpair of H i ff ( (cid:101) µ, (cid:101) ϕ ) is an eigenpair of a dual operator (cid:101) H : = − ∆ + V max − V , defined by the left hand side of (4.51). This dual operator (cid:101) H is the same type of discrete Schr¨odinger operator as H only with a di ff erent potential(also taking value in [0 , V max ]). We can define a dual landscape function (cid:101) u satisfying( (cid:101) H (cid:101) u ) n =
1, and a dual box-counting function N (cid:101) u ( µ ; (cid:101) H ) as in (1.3) for (cid:101) H .It is easy to check thatCard (cid:8) eigenvalues of H ≤ µ (cid:9) = Card { Λ } − Card (cid:8) eigenvalues of H > µ (cid:9) = Card { Λ } − Card (cid:110) eigenvalues of (cid:101) H < (cid:101) µ (cid:111) . (4.53)Therefore, N ( µ ; H ) = − N − ( (cid:101) µ ; (cid:101) H ) , (4.54)where N ( · ; H ) and N − ( · ; (cid:101) H ) are the finite volume IDS for H and (cid:101) H respectively. Note thatthe counting for N − ( · ; (cid:101) H ) in (4.53) is defined for eigenvalues strictly less than µ , which isdi ff erent from the definition of N ( · ; H ) in (1.2). For the Anderson-type potential V withcommon distribution F ( δ ) = P ( v n ≤ δ ), V max − V is also an Anderson-type potential,with common distribution P ( V max − v n ≤ δ ). We denote by (cid:101) F ( δ ) = P ( V max − v n < δ ) = − F ( V max − δ ). We now want to apply Theorems 1, 4 and 10 to the dual operator (cid:101) H and the dual counting function N ( (cid:101) µ ; (cid:101) H ) for (cid:101) µ near 0. All the estimates still hold if wereplace N , F by N − and (cid:101) F . In particular, the second part of Theorem 4 implies that thereare constants a i , µ ∗ only depending d and the tail behavior of (cid:101) F such that for (cid:101) µ ≤ µ ∗ a E N (cid:101) u ( a (cid:101) µ ; (cid:101) H ) ≤ E N − ( (cid:101) µ ; (cid:101) H ) ≤ a E N (cid:101) u ( a (cid:101) µ ; (cid:101) H ) . (4.55)Therefore, by (4.52) and (4.54), one has for µ ≥ d + V max − µ ∗ : = (cid:101) µ ∗ ,1 − a E N (cid:101) u ( a (cid:101) µ ; (cid:101) H ) ≤ E N ( µ ; H ) ≤ − a E N (cid:101) u ( a (cid:101) µ ; (cid:101) H ) , which is (1.13) in Corollary 3. In addition, there are constants b i only depending d andthe tail behavior of (cid:101) F such that for (cid:101) µ ≤ µ ∗ , b (cid:101) µ d / (cid:101) F ( b (cid:101) µ ) b (cid:101) µ − d / ≤ E N − ( (cid:101) µ ; (cid:101) H ) ≤ b (cid:101) µ d / (cid:101) F ( b (cid:101) µ ) b (cid:101) µ − d / . (4.56) Again by (4.52) and (4.54), one has for µ ≥ d + V max − µ ∗ : = (cid:101) µ ∗ ,1 − b (cid:101) µ d / (cid:2) − F ( V max − b (cid:101) µ ) (cid:3) b (cid:101) µ − d / ≤ E N ( µ ; H ) ≤ − b (cid:101) µ d / (cid:2) − F ( V max − b (cid:101) µ ) (cid:3) b (cid:101) µ − d / , (4.57)which completes (1.14) in Corollary 3.A ppendix A. D iscrete L aplacian and harmonic functions A.1.
Maximum principles for discrete sub-solutions.Lemma A.1 (The maximum principle for subharmonic functions) . Let Q = [ a , b ] ×· · · × [ a d , b d ] ⊂ Z d be a box in Z d and let ∂ Q = { n ∈ Q : n + e i (cid:60) Q or n − e i (cid:60) Q for some e i , ≤ i ≤ d } (A.1) be the (inner) boundary of Q. LetE ( Q ) = { n ∈ ∂ Q : n + e i (cid:60) Q or n − e i (cid:60) Q for more than one e i , ≤ i ≤ d } be the edges (including the corners) of Q, and let ∂ ◦ Q = ∂ Q − E ( Q ) be the (flat) boundary removing all the “edges” and “corners”. Let V = { v n } n ∈ Q bea non-negative potential on Q, i.e., v n ≥ , n ∈ Q. A vector f = { f n } n ∈ Q is called asub-solution, w.r.t. − ∆ + V, on (the interior of) Q if − ( ∆ f ) n + v n f n ≥ , n ∈ Q \ ∂ Q . If f is a sub-solution, then the minimum of f n in Q \ E ( Q ) must be attained on ∂ ◦ Q. Moreprecisely, min n ∈ Q \ ∂ Q f n ≥ min n ∈ ∂ ◦ Q f n . (A.2) Proof.
Let m = min n ∈ ∂ ◦ Q f n . It is enough to prove (A.2) for m =
0, otherwise just let g n = f n − m .Now suppose f n ≥ n ∈ ∂ ◦ Q , we want to show that f n ≥ n ∈ Q \ ∂ Q .Suppose not, then − a : = min n ∈ Q \ ∂ Q f n < . Let j ∈ Q \ ∂ Q be such that the minimum is attained, i.e., f j = − a < f j ± e i ≥ f j , ≤ i ≤ d . − ( ∆ f ) j + v j f j ≥ d f j + v j f j ≥ (cid:88) ≤ i ≤ d ( f j + e i + f j − e i ) ≥ d · ( − a )Therefore, v j = HE LANDSCAPE LAW 49 and f j ± e i = f j = − a , ≤ i ≤ d . If any one of these 2 d nearest neighborhood j ± e i belongs to the flat boundary ∂ ◦ Q , thenit is a contradiction. Suppose not, pick any one of them, say, j + e ∈ Q \ ∂ Q . Notice that f j + e = − a is also the (global) minimum, then we can repeat what we did for f j (and onlystop if we reach the flat boundary). It is easy to check that after finitely many steps, wewill push this negative minimum to the flat boundary. More precisely, there is a j (cid:48) ∈ ∂ ◦ Q such that f j (cid:48) = − a <
0, which is a contraction. (cid:3)
There will be several direct corollaries of the above maximum principle as. We willsimply list them as independent lemmas and omit most of the details for the proof.
Lemma A.2 (Positivity for Periodic boundary conditions) . Let
Λ = ( Z / K Z ) d be a pe-riodic lattice. If V = { v n } n ∈ Λ , b n ≥ is a non-negative potential, and not constantlyzero. Let H = − ∆ + V be the discrete Schr ¨odinger operator on this periodic lattice. If ( H f ) n ≥ for all n ∈ Λ , then f n ≥ for all n ∈ Λ . The periodic boundary condition case has been proved previously in [WZ] (see Lemma2.12 therein). The argument is similar to Lemma A.1. Notice that there is actually noboundary value given for the periodic case, the contradiction comes from the potential v n being not constantly zero, see (A.3). We omit the details here.Another important application is for (discrete) harmonic functions, we have controlfor both the maximum and the minimum. Lemma A.3 (Maximum principles for discrete harmonic functions) . Let Q = [ a , b ] ×· · · × [ a d , b d ] ⊂ Z d be a box in Z d and let ∂ Q , E ( Q ) , ∂ ◦ Q be the same as in Lemma A.1.Suppose f = { f n } n ∈ Q is discrete harmonic on (the interior of) Q, i.e., ( ∆ f ) n = , n ∈ Q \ ∂ Q , then for all n ∈ Q \ ∂ Q min m ∈ ∂ Q f m ≤ min m ∈ ∂ ◦ Q f m ≤ f n ≤ max m ∈ ∂ ◦ Q f m ≤ max m ∈ ∂ Q f m . (A.4)This is a direct application of Lemma A.1, to f and − f . We only state maximum prin-ciples as above for the boxes in Z d for simplicity, it is not hard to believe that the sameconclusion would hold for more general domains in Z d , as long as they are “connected”w.r.t. the discrete Laplacian operator in a suitable sense. In particular, it definitely worksfor “annular” domain given by the di ff erence of two cubes: Q (cid:40) Q ⊂ Z d are two Z d boxes and A = Q − Q . The boundary of A is defined in the same as in (A.1), i.e., ∂ A = { n ∈ A : n + e i (cid:60) A or n − e i (cid:60) A for some e i } . Then if ( ∆ f ) n = n ∈ A \ ∂ A ,then for all n ∈ A \ ∂ A min m ∈ ∂ A f m ≤ f n ≤ max m ∈ ∂ A f m . (A.5) A.2.
The discrete Poincar´e inequality.
The result essentially can be generalized toany “connected” region in Z d . We only need the version on a rectangular domain. Lemma A.4.
Let Q = I × · · · × I d be a a rectangular domain in Z d , where I i = [ a i , b i ] ∩ Z for some a i < b i ∈ Z and (cid:96) i = Card ( I i ) ∈ N , i = · · · , d. For any (real-valued) sequence { f n } n ∈ Q , let | Q | = Card ( Q ) = (cid:96) × · · · × (cid:96) d and ¯ f Q = | Q | (cid:80) n ∈ Q f n . Then we have (cid:88) n ∈ Q ( f n − ¯ f Q ) ≤ d (cid:96) (cid:88) n ∈ Q (cid:107)∇ f n (cid:107) . (A.6) In particular, if Q ⊂ Z d is a cube of side length (cid:96) ( Q ) , then (cid:88) n ∈ Q ( f n − ¯ f Q ) ≤ d (cid:96) ( Q ) (cid:88) n ∈ Q (cid:107)∇ f n (cid:107) . (A.7) Proof.
It is enough to prove (A.6) for ¯ f Q =
0. It is easy to check that (cid:88) m ∈ Q (cid:88) n ∈ Q ( f m − f n ) = | Q | (cid:88) n ∈ Q f n . (A.8)For m = ( m , · · · , m d ) ∈ Q , n = ( n , · · · , n d ) ∈ Q , let Γ ( n , m ) = { γ → γ → · · · → γ d + } be a discrete path in Z d connecting n and m , defined as follows. All vertices γ i ∈ Q ,and is given explicitly γ = n , γ i + = γ i + t i e i , i = · · · , d , γ d + = m and the all edges E i connecting the consecutive vertices are parallel to e i where { e i } is the canonical basis of Z d . Then ( f m − f n ) = (cid:32) (cid:88) ≤ i ≤ d ( f γ i + − f γ i ) (cid:33) ≤ d (cid:88) ≤ i ≤ d ( f γ i + − f γ i ) . (A.9)For each i = , · · · , d , we claim that (cid:88) m , n ∈ Q ( f γ i + − f γ i ) ≤ (cid:96) × | Q | × (cid:88) n ∈ Q |∇ i f n | , (A.10)where (cid:96) max = max j (cid:96) j . Then (A.6) follows from (A.8)-(A.10).For n i , m i ∈ I i = [ a i , b i ] ∩ Z , suppose t i = m i − n i >
0, then the edge E i = { γ i , γ i + e i , · · · , γ i + t i e i } . Notice that for all n (cid:48) = ( n (cid:48) , n (cid:48) , · · · , n (cid:48) d ) in E i , only the i-th coordinatesare di ff erent, and the rest d − n and m . We denote by E iQ the extension of E i to the boundary of Q ( f γ i + − f γ i ) = (cid:32) (cid:88) ≤ k ≤ t i − f γ i + ( k + e i − f γ i + ke i (cid:33) ≤ | t i | (cid:88) n (cid:48) ∈ E i ( ∇ i f n (cid:48) ) ≤ (cid:96) max (cid:88) n (cid:48) ∈ E iQ ( ∇ i f n (cid:48) ) . (A.11) HE LANDSCAPE LAW 51
It is easy to verify that the estimate for t i = m i − n i < d − n and m , except the i-thone. Therefore, (cid:88) m , n ∈ Q ( f γ i + − f γ i ) ≤ (cid:88) rest ( d + m j , n j ∈ I j (cid:88) ( d − m j , n j ∈ I j in E iQ ( f γ i + − f γ i ) ≤ (cid:88) rest ( d + m j , n j ∈ I j (cid:88) ( d − m j , n j ∈ I j in E iQ (cid:96) max (cid:88) n (cid:48) ∈ E iQ ( ∇ i f n (cid:48) ) ≤ (cid:96) max (cid:88) ( d + m j , n j ∈ I j not appear in E iQ (cid:32)(cid:88) n (cid:48) ∈ Q ( ∇ i f n (cid:48) ) (cid:33) ≤ (cid:96) max × ( (cid:96) i × | Q | ) × (cid:88) n (cid:48) ∈ Q ( ∇ i f n (cid:48) ) ≤ (cid:96) × | Q | × (cid:88) n (cid:48) ∈ Q ( ∇ i f n (cid:48) ) , which completes the proof of (A.10) and Lemma A.6. (cid:3) A.3.
Discrete cut-o ff functions. Let Q = [1 , R ] d ∩ Z d , j max = (cid:100) R / (cid:101) . Let ∂ Q and Q / ∂ Q = { n ∈ Q : n + e i (cid:60) Q or n − e i (cid:60) Q for some 1 ≤ i ≤ d } (A.12)and Q / = { n ∈ Q : dist ( n , ∂ Q ) ≥ j max } . (A.13)Let I ( j ) be the level set in Q which is j -th layer starting from ∂ Q (from inside), i.e., I ( j ) : = { n ∈ Q | dist ( n , ∂ ) = j − } , ≤ j ≤ j max where ∂ Q = I (0).It is easy to check that Q / I ( j ) are pairwise disjoint and Q = Q / ∪ j max (cid:91) j = I ( j )for j = , · · · , j max , i.e., I ( j ) : = { n ∈ Q \I (1) \I (1) · · · \I ( j − | | m − n | = m ∈ (cid:96) ( j − } . The side length of Q / (cid:96) ( Q / = R − (cid:100) R / (cid:101) satisfies R / − < (cid:96) ( Q / ≤ R / . Now we can define the step function χ = { χ n } as χ n = n ∈ Q / R ( j − , n ∈ I ( j ) , j = , · · · , j max , j (cid:60) Q . (A.14)For any n , n + e i which are both in Q \ ( Q / n , n + e i belongto the same I ( j ) or n , n + e i belong to I ( j ) , I ( j +
1) one each for some j = , · · · , j max .Therefore, either χ n − χ n + e i = χ n − χ n + e i = R in the case when both n , n + e i ∈ Q \ ( Q / n , n + e i belong to I ( j max ) and Q / | χ n − χ n + e i | = − R ( j max − = − R ( (cid:6) R (cid:7) − R / ≤ (cid:6) R (cid:7) < R / + < | χ n − χ n + e i | ≤ R .Two remaining cases are either n , n + e i both belong to Q /
3, or one of them does notbelong to Q . It is easy to check χ n − χ n + e i = ≤ i ≤ d , | χ n + e i − χ n | (cid:40) ≤ R n , n + e i ∈ Q = , n , or n + e i (cid:60) Q . Clearly, the construction is independent of where the cube Q locates, i.e., works forany translation of the cube, Q n : = Q + n , n ∈ Z d .A.4. Dirichlet problem on a cube.
We study the Dirichlet problem for the discreteLaplacian on a cube in Z d . We restate some of the notations and definitions that we haveintroduced previously here for self-consistency. For ξ = ( ξ , · · · , ξ d ) ∈ Z d , and r ∈ Z ≥ ,We denote by Q ( r ; ξ ) the box centered at ξ of side length 2 r + Q ( r ) = Q ( r ; ξ ) = (cid:26) m = ( m , · · · , m d ) ∈ Z d : max ≤ i ≤ d | m i − ξ i | ≤ r (cid:27) . We may frequently omit the center ξ , whenever it is clear. We also denote by ∂ Q ( r ) ⊂ Q ( r ) the boundary of Q ( r ) and by ∂ ◦ Q ( r ) ⊂ ∂ Q ( r ) the boundary removing the “corners”:for r ≥ ∂ Q ( r ) = Q ( r ) − Q ( r − = (cid:26) m = ( m , · · · , m d ) ∈ Z d : max ≤ i ≤ d | m i − ξ i | = r (cid:27) ,∂ ◦ Q ( r ) = (cid:8) m = ( m , · · · , m d ) ∈ ∂ Q ( r ) : | m i − ξ i | = r for only one 1 ≤ i ≤ d (cid:9) and ∂ Q (0) = Q = { ξ } .For a Z d sequence f = { f n } n ∈ Z d , its directional (right) derivatives and the gradient on Z d are defined as ∇ i f n = f n + e i − f n , i = , · · · , d , and its gradient ∇ f : M (cid:55)→ R d is ∇ f ( n ) = ( ∇ f n , ∇ f n , · · · , ∇ d f n ) . HE LANDSCAPE LAW 53
We also denote the dot product and the induced norm of ∇ f , ∇ g , as R d vectors, by( ∇ g · ∇ f )( n ) = d (cid:88) i = ∇ i g n · ∇ i f n , and |∇ f | ( n ) : = (cid:112) ( ∇ f · ∇ f )( n ) . Lemma A.5 (Green’s formula) . For any f , g ∈ H = (cid:96) ( Λ ) , (cid:88) n ∈ Q ( r − g n · ( ∆ f ) n = − (cid:88) n , n + e i ∈ Q ( r ) ( ∇ g ) n · ( ∇ f ) n + (cid:88) n ∈ ∂ ◦ Q ( r ) g n · ∂ f ∂ N ( n )(A.15) = − (cid:88) n , m ∈ Q ( r ) | m − n | = ( g m − g n )( f m − f n ) + (cid:88) n ∈ ∂ ◦ Q ( r ) , m ∈ ∂ Q ( r − | m − n | = g n · ( f n − f m ) . (A.16) As a consequence, (cid:88) n ∈ Q ( r − g n ( ∆ f ) n − (cid:88) n ∈ Q ( r − f n ( ∆ g ) n = − (cid:88) n ∈ ∂ ◦ Q ( r ) , m ∈ ∂ Q ( r − | m − n | = f n ( g n − g m ) + (cid:88) n ∈ ∂ ◦ Q ( r ) , m ∈ ∂ Q ( r − | m − n | = g n ( f n − f m ) . (A.17) Remark
A.1 . One can find the general discrete version of Green’s formula on graphs,for example in [Ch, Gu]. We sketch the proof for cubes in Z d lattices for the reader’sconvenience. Proof.
For n = ( n , · · · , n d ) ∈ Z d , let ˇ n i : = ( n , · · · , n i − , n i + , · · · , n d ) , i = · · · , d . Forfixed ˇ n ir − (cid:88) n i = − r ( g n + e i − g n ) · ( f n + e i − f n ) = r (cid:88) n i = − r + ( g n − e i − g n ) · ( f n − e i − f n ) = r − (cid:88) n i = − r + g n · (2 f n − f n + e i − f n − e i ) + g | n i = r ( f | n i = r − f | n i = r − ) + g | n i = − r ( f | n i = − r − f | n i = − r + ) . Summing over all ˇ n i and 1 ≤ i ≤ d , one gets (cid:88) n , m ∈ Q ( r ) | m − n | = ( g m − g n )( f m − f n ) = − · (cid:88) n ∈ Q ( r − g n · ( ∆ f ) n + · (cid:88) n ∈ ∂ ◦ Q ( r ) , m ∈ ∂ Q ( r − | m − n | = g n · ( f n − f m )which implies (A.15). (cid:3) Given { f n } n ∈ Q ( r − and { h n } n ∈ ∂ Q ( r ) , we proceed to solve the linear system on Q ( r ) (cid:40) − ( ∆ u ) n = f n , n ∈ Q ( r − u n = h n , n ∈ ∂ Q ( r ) . (A.18)The problem can be decomposed into the following two systems, which give us thediscrete Poisson Kernel and the discrete Green’s function.Fix any m ∈ ∂ Q ( r ), the discrete Poisson kernel P r ( n ; m ) : Q ( r ) × ∂ Q ( r ) → [0 ,
1] is theunique solution to the boundary problem on Q ( r ). (cid:40) ∆ P r ( n , m ) = , n ∈ Q ( r − P r ( n , m ) = δ m ( n ) , n ∈ ∂ Q ( r ) . (A.19)Fix any m ∈ Q ( r − m , G r ( n , m ) : Q ( r ) → [0 ,
1] is the unique solution to the system. (cid:40) − ∆ G r ( n , m ) = δ m , n ∈ Q ( r − G r ( n , m ) = , n ∈ ∂ Q ( r ) . (A.20)Consider − ∆ with zero boundary condition as an invertible matrix of size | Q ( r − | ×| Q ( r − | . Clearly, for n , m ∈ Q ( r − G ( n , m ) = ( ∆ ) − ( n , m ) = G ( m , n ) since ∆ isself-adjoint.Moreover, for fixed m ∈ ∂ Q ( r ) and m (cid:48) ∈ Q ( r ), if we apply the Green’s formula (A.17)to g n = P r ( n , m ) and f n = G r ( n , m (cid:48) ), then (cid:88) n ∈ Q ( r − P r ( n , m ) ( − δ m (cid:48) ( n )) = (cid:88) n ∈ ∂ ◦ Q ( r ) , n (cid:48) ∈ ∂ Q ( r − | n (cid:48) − n | = δ m ( n )( G r ( n , m (cid:48) ) − G r ( n (cid:48) , m (cid:48) )) , which implies for any m ∈ ∂ Q ( r ) and m (cid:48) ∈ Q ( r ), P r ( m (cid:48) , m ) = G r ( n (cid:48) , m (cid:48) ) = G r ( m (cid:48) , n (cid:48) ) , n (cid:48) ∈ ∂ Q ( r − , | n (cid:48) − m | = . (A.21)Notice that this can be considered as the (negative) normal derivative of G r ( m (cid:48) , · ) in thedirection of the outward pointing to the surface of Q ( r ).Back to the system (A.18), using P r and G r , we can solve the system u n = (cid:88) m ∈ ∂ Q ( r ) P r ( n , m ) h m + (cid:88) m (cid:48) ∈ Q ( r − G r ( n , m (cid:48) ) f m (cid:48) . In particular, we have the following integration by parts formula (Green’s identity) forany { u n } n ∈ ∂ ◦ Q ( r ) at the center of the box Q ( r ; ξ ): u ξ = (cid:88) m ∈ ∂ Q ( r ) P r ( ξ, m ) u m − (cid:88) m (cid:48) ∈ Q ( r − G r ( ξ, m (cid:48) ) · ( ∆ u ) m (cid:48) . (A.22) HE LANDSCAPE LAW 55
With an application to a constant solution (constantly 1) to (A.19), we have that forany ξ and r , (cid:88) m ∈ ∂ Q ( r ) P r ( ξ, m ) = . (A.23)A.5. Estimates on the Green’s function and the Poisson kernel.
Retain the defini-tions in the previous section, for R ∈ N , let Q ( R ) = Q ( R ; ξ ) ⊂ Z d be the discrete cubecentered at ξ ∈ Z d of side length 2 R +
1, and let G R ( ξ, n ) be the discrete Green’s functionas defined in (A.20). In this part, we study the behavior of the discrete Green’s functionaway both from the pole ξ and the boundary ∂ Q ( R ). We will approximate the discreteGreen’s function by a continuous one to obtain the desired estimates. Let us also recallsome of the definitions for the di ff erential case. Fix ξ ∈ Z d , let Q = ξ + [ − , d be acube in R d centered at ξ of side length 2. Let G ( ξ, · ) be the continuous Green’s functionon the cube Q with zero boundary condition: (cid:40) − ∆ c G ( ξ, x ) = δ c ξ ( x ) G ( ξ, x ) = , x ∈ ∂ Q , (A.24)where ∆ c = (cid:80) di = ∂ ∂ x i is the standard (di ff erential) Laplacian on R d , and δ c ξ ( x ) is the Diracdelta function at ξ in the distribution sense. For any R ∈ N , consider a square mesh ofsize h = R on Q . Denote the collection of all the mesh points by Ω h = { τ n = ξ + n · h , n ∈ [ − R , R ] d ∩ Z d } . (A.25)We see that Ω h is exactly indexed (one-to-one correspondence) by Q ( R ). For τ n ∈ Ω h ,let G h ( ξ, τ n ) : = h d − G R ( ξ, n ) . (A.26)It is easy to verify that the equation of G R in (A.20) implies that (cid:40) − h − (cid:80) | n − m | = G h ( ξ, τ m ) + h − · d · G h ( ξ, τ n ) = h − d δ ξ ( n ) , n ∈ Q ( R ; ξ ) G h ( ξ, τ n ) = , n ∈ Q ( R ; ξ ) \ ∂ Q ( R ; ξ ) . For readers who are familiar with finite di ff erence method, it is not hard to find that G h is exactly the finite di ff erence approximation to the continuous problem (A.24). We aregoing to prove that Lemma A.6.
There are positive dimensional constants C , h such that if h ≤ h , then forall τ n ∈ Ω h ∩ (cid:0) Q \Q / (cid:1) |G h ( ξ, τ n ) − G ( ξ, τ n ) | ≤ C · | log h | d + h . (A.27) Remark
A.2 . Such approximation are proved in a rectangle domain in R in [La]. It waslater generalized to the interior of a domain in any dimension with smooth boundaryby Schatz and Wahlbin, see Theorem 6.1 [SW], using finite element approach. The method in [SW] potentially can be generalized to any convex polyhedral domains, up tothe boundary. Here we present a direct proof using the series expansion of G h and G .Similar estimate also holds for G h ( y , x ) − G ( y , x ) where the pole y is not far away fromthe center ξ . We will only deal with the case y = ξ which will be enough for our use. Proof.
W.L.G we assume that ξ = . In this case, the mesh points τ n = nh and Q = R Q ( R ). Due to the symmetry of the problem, it is also enough to consider the upper halfcube Q + = { x ∈ Q : x d ≥ } . We deal with a partial eigenfunction representation of G (0 , x ) on Q + , w.r.t. the first d − k = ( k , · · · , k d ) ∈ Z d − + , considernormalized eigenfunctions f k ( (cid:101) x ) = (cid:81) d − i = sin( k i π ( x i + (cid:101) x = ( x , · · · , x d − ). Let α k = (cid:118)(cid:117)(cid:117)(cid:116) d − (cid:88) i = (cid:18) k i π (cid:19) : = π (cid:107) k (cid:107) . (A.28)Then sinh( α k ( x d + , sinh( α k (1 − x d )) solves g (cid:48)(cid:48) − α k g = − δ for − < x d < g ( − = , g (1) = α k sinh(2 α k ) (constant). By separation of variables, it is easy to checkthat for x ∈ Q + G (0 , x ) = (cid:88) k ∈ Z d − + α k sinh(2 α k ) sinh( α k ) · sinh( α k (1 − x d )) · f k (0) · f k ( (cid:101) x ) , < x d < = (cid:88) k ∈ Z d − + sinh( α k (1 − x d ))2 α k cosh( α k ) · (cid:32) d − (cid:89) i = sin( k i π · sin( k i π x i + (cid:33) . (A.30)The same idea can be used to derive the formula for G h (0 , τ n ) on the finite dimensionalspace. Let h = R and k = ( k , · · · , k d − ) ∈ [1 , R − d − ∩ Z d : = T R . Since2 R − (cid:88) k i = sin k i π ( n i h + k i π ( m i h + = R − (cid:88) k i = cos( k i θ ) − R − (cid:88) k i = cos( k i ϕ ) , θ = π h ( n i − m i )2 , ϕ = π ( hn i + hm i + = (2 R − + − sin(2 R − / ϕ ϕ , if n i = m i , ϕ = π ( hn i + = R − + / − sin 2 π ( n + R ) · cos( ϕ/ − cos 2 π ( n + R ) · sin( ϕ/ ϕ = R − + / − + / = R . HE LANDSCAPE LAW 57 If n i (cid:44) m i , 2 R θ = π ( n i − m i ) , R ϕ = π ( n i + m i ) + π R ∈ π Z , then(A.31) = − + sin(2 R − / θ θ + − sin(2 R − / ϕ ϕ = sin(2 R θ ) · cos( θ/ − cos(2 R θ ) · sin( θ/ θ − sin(2 R ϕ ) · cos( ϕ/ − cos(2 R ϕ ) · sin( ϕ/ ϕ = − / · cos π ( n i − m i ) + / · cos π ( n i + m i ) = − sin( π n i ) sin( π m i ) = . Therefore, (cid:80) R − k i = √ h sin k i π ( n i h + · √ h sin k i π ( m i h + = δ ( n i − m i ) form an orthogonal basisfor R R − . We can consider normalized basis on ( R R − ) d − given by f hk ( (cid:101) n ) = √ h d − d − (cid:89) i = sin( k i π n i h + , (cid:101) n = ( n , · · · , n d − ) ∈ [ − R , R ] d ∩ Z d . Write the discrete Laplacian ∆ = (cid:101) ∆ + ∆ d , where (cid:101) ∆ is the second order di ff erence w.r.t.the first d − (cid:101) n = ( n , · · · , n d − ) and ∆ d is the second order di ff erence w.r.t. thelast variable n d . Direct computation shows that (cid:101) ∆ f hk ( (cid:101) n ) = f hk ( (cid:101) n ) · d − (cid:88) i = (cid:18) k i π h − (cid:19) . On the other hand, for H ( n d ) = sinh( β k n d ), ( ∆ d F )( n d ) = H ( n d ) · (2 cosh β k − n d , expand G h (0 , n ) as G h (0 , nh ) = (cid:80) k H ( n d ) f hk ( (cid:101) n ). Notice that δ ( (cid:101) n , n d ) = (cid:80) k δ ( n d ) · f hk (0) · f hk ( (cid:101) n ). Then ∆ G h (0 , · ) = − h − d δ ( · ) will be reduced to a di ff erenceequation only on the last variable: F k ( n d ) · d − (cid:88) i = (cid:18) k i π h − (cid:19) + ( ∆ d F k )( n d ) = − h − d · f hk (0) · δ ( n d ) . (A.32)Define β k = β k ( h ) to be the positive solution of the equation2 cosh β k − + d − (cid:88) i = (cid:18) k i π h − (cid:19) = ⇐⇒ cosh β k = d − d − (cid:88) i = cos π k j h . (A.33)Then sinh( β k ( R − n d )) , sinh( β k ( R + n d )) solve the homogeneous part of (A.32) withboundary conditions v ( R ) = = v ( − R ). Let H ( n d ) = (cid:40) sinh( β k ( R + n d )) , − R ≤ n d ≤ β k ( R − n d )) , ≤ n d ≤ R . Direct computation shows that H (0) (2 − β k ) + H (1) + H ( − − H (0) = − β k R ) sinh( β k ) . Therefore, F k ( n d ) = h − d · f hk (0)2 cosh( β k R ) sinh( β k ) H ( n d ) . solves the inhomogeneous equation (A.32) with boundary condition F k ( ± R ) = G h at the grid points on the upper halfcube n d ≥ G (0 , nh ) = (cid:88) k ∈ T R F k ( n d ) · f hk ( (cid:101) n ) , < n d < R (A.34) = (cid:88) k ∈ T R sinh( β k ( R − n d ))2 R sinh( β k ) · cosh( β k R ) · (cid:32) d − (cid:89) i = sin( k i π k i π n i h + (cid:33) . (A.35)It was proved in [La] that for d =
2, and (cid:107) n (cid:107) d : = (cid:112) n + · · · + n d > |G (0 , nh ) −G h (0 , nh ) | ≤ Ch | n | − . The method can be extended to higher dimensions. We will notbrother to give the full generalization of the exact singularity of order 1 / | n | . We onlyneed the version for R ≥ n d ≥ R / β k ( h ) in (A.33) as h → β k is given by (A.33), then either β k ≥ R · β k ≥ (cid:107) k (cid:107) d for all R = h and 1 ≤ k i ≤ R −
1. We sketch the proof forreader’s convenience. If β k ≤
1, then1 + β k ≥ cosh β k = d − d − (cid:88) i = cos π k i h ≥ d − d − (cid:88) i = (cid:18) − ( k i h (cid:19) (A.36) = + (cid:107) k (cid:107) h , (A.37)which gives β k ≥ (cid:107) k (cid:107) h /
2. In (A.36), we used the elementary inequalities 1 + x ≥ cosh x ,for x ∈ [0 ,
1] and 1 − cos π x ≥ x for x ∈ [0 , x d = n d / R ∈ [1 / , ffi cients of G h decays exponentially (cid:12)(cid:12)(cid:12)(cid:12) sinh( β k ( R − n d ))2 R sinh( β k ) · cosh( β k R ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) k (cid:107) e −(cid:107) k (cid:107) / , or , (cid:12)(cid:12)(cid:12)(cid:12) sinh( β k ( R − n d ))2 R sinh( β k ) · cosh( β k R ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) k (cid:107) e − R / . On the other hand, for any fixed k ∈ T R , we want to expand β k ( h ) in h explicitly. Let α k = π (cid:107) k (cid:107) be as in (A.28), one has β k = cosh − (cid:32) d − d − (cid:88) i = cos π k i h (cid:33) = cosh − (cid:32) d − d − (cid:88) i = (1 − π k i h + O ( k i h )) (cid:33) HE LANDSCAPE LAW 59 = cosh − (cid:18) + π (cid:107) k (cid:107) h + O ( (cid:107) k (cid:107) h ) (cid:19) = log + π (cid:107) k (cid:107) h + O ( (cid:107) k (cid:107) h ) + (cid:115)(cid:18) + π (cid:107) k (cid:107) h + O ( (cid:107) k (cid:107) h ) (cid:19) − = log (cid:32) + π (cid:107) k (cid:107) h + O ( (cid:107) k (cid:107) h ) + (cid:114) π (cid:107) k (cid:107) h + O ( (cid:107) k (cid:107) h ) (cid:33) = log (cid:18) + π (cid:107) k (cid:107) h + O ( (cid:107) k (cid:107) h ) + π (cid:107) k (cid:107) h + O ( (cid:107) k (cid:107) h ) (cid:19) = π (cid:107) k (cid:107) h + π (cid:107) k (cid:107) h + O ( (cid:107) k (cid:107) h ) − (cid:18) π (cid:107) k (cid:107) h + π (cid:107) k (cid:107) h + O ( (cid:107) k (cid:107) h ) (cid:19) = π (cid:107) k (cid:107) h + O ( (cid:107) k (cid:107) h ) = α k · h + O ( (cid:107) k (cid:107) h ) . Therefore, R · β k = α k + O ( (cid:107) log h (cid:107) h ) for (cid:107) k (cid:107) ≤ C d | log h | (with any constant C d onlydepending on the dimension). For any x d = n d / R ∈ [1 / ,
1] and t = − x d ∈ [0 , / ffi cients of G and G h in (A.30) and (A.35) up to (cid:107) k (cid:107) ≤ C d | log h | .Let f ( x ) = sinh( xt )cosh( x ) . Then 0 ≤ f ( x ) ≤ , | f (cid:48) ( x ) | ≤ x ≥
0. Therefore, f ( β k R ) = f ( α k ) + O ( (cid:107) log h (cid:107) h ). And 1 / ( R sinh( xh )) = x (1 + O ( x h ) implies that1 R sinh( β k ) = β k R (1 + O ( β k h )) = α k (1 + O ( (cid:107) log h (cid:107) h )) . Putting all these together, we have thatsinh( β k ( R − n d ))2 R sinh( β k ) · cosh( β k R ) = sinh( α k (1 − x d ))2 α k cosh( α k ) + O ( | log h | h ) . Therefore, we break the series of G (0 , nh ) − G h (0 , nh ) into low frequency and highfrequency part for some a ∼ | log h | ∼ R , |G (0 , nh ) − G h (0 , nh ) | ≤ (cid:88) (cid:107) k (cid:107)≤ a ( · ) + (cid:88) (cid:107) k (cid:107)≥ a ( · ) ≤ (cid:88) (cid:107) k (cid:107)≤ a O ( | log h | h ) + (cid:88) (cid:96) ≥ a (cid:88) (cid:96) ≤(cid:107) k (cid:107) <(cid:96) + (cid:96) e − (cid:96)/ + (2 R − d − e − R ≤ a d O ( | log h | h ) + a d − e − a / + (2 R − d − e − R ≤ C d | log h | d + h , for su ffi ciently large R ≥ R d . (cid:3) For any 0 < ε < /
4, by the positivity and smoothness (away from the pole) ofthe the continuous Green’s function G ( ξ, x ), there are c ( ε, d ) > , c ( ε, d ) > c ≤ G ( ξ, x ) ≤ c / Q − ε/ \Q / . Combine this with the approximation in (A.27),we have that for h < (cid:101) h ( ε, d ) and τ n = ξ + nh ∈ Ω h ∩ (cid:0) Q − ε/ \Q / (cid:1) , c ≤ G h ( ξ, τ n ) ≤ c . Then by (A.26), one has for R = / h ≥ / (cid:101) h , c ≤ R d − G R ( ξ, n ) ≤ c . Notice that τ n = ξ + nh ∈ Ω h ∩ (cid:0) Q − ε/ \Q / (cid:1) is equivalent to n ∈ Q ( R ; ξ ) , and R / ≤ | n − ξ | ∞ < (1 − ε/ R . (A.38)For any 0 < ε < / r ∈ /ε , if we set R = (cid:4) (1 + ε ) / r (cid:5) , then it is easy to verifythat (1 + ε/ r ≤ R ≤ (1 + ε/ r , which implies that R / ≤ r ≤ (1 + ε/ − R < (1 − ε/ · R . In other words, if n ∈ ∂ Q ( r ; ξ ), then n will satisfy (A.38). In conclusion, we have showedthat Lemma A.7.
For any < ε < / , and r ∈ N , let R = (cid:4) (1 + ε ) / r (cid:5) . There are constants (cid:101) c , c , r depending on d and ε such that if r ≥ r , then for all m ∈ Q ( r ; ξ ) c r − d ≤ G R ( ξ, m ) ≤ c r − d . On the other hand, we are also interested in the behavior of G and G r near the bound-ary. By the comparison principle of harmonic functions, G r ( ξ, x ) can be bounded frombelow by the distance of x to the boundary of Q as long as x is away from all the edges(and corners). Lemma A.8.
Let x c = ξ + (0 , · · · , , ∈ ∂ Q be the center of the top surface of Q . Forany < η < / , let T − η = { x ∈ Q : | x − x c | R d ≤ − η } . There is a constant c ( η, d ) such that for all x = ( x , · · · , x d ) ∈ T − η G ( ξ, x ) ≥ c ( η, d ) | ξ d + − x d | = c ( η, d ) · dist( x , ∂ Q ) . (A.39) Proof.
Consider a slightly larger semi-sphere T − η/ such that T − η (cid:40) T − η/ (cid:40) T . Let h ( x ) = ξ d + − x d . HE LANDSCAPE LAW 61
Clearly, G ( ξ, x ) and h ( x ) are two strictly positive harmonic functions on T − η/ . By thethe comparison principle of harmonic functions, see e.g. [Da, Ke], G r ( ξ, x ), there is aconstant c only depends on d and η such that for all x , y ∈ T − η G ( ξ, x ) h ( x ) ≥ c · G ( ξ, y ) h ( y ) . In particular, take y = (0 , · · · , , ξ d + η ) ∈ T − η . h ( y ) = − η and use the well-knownsingularity of G near the pole, G ( ξ, y ) ≥ (cid:40) C log | y − ξ | − , d = C | y − ξ | − ( d − , d (cid:44) ≥ C ( η, d ) . Therefore, G ( ξ, x ) ≥ c · h ( x ) = C · ( ξ d + − x d ) , where the constant c only depends on the dimension and η . (cid:3) Now we consider the approximation of (A.27) for the mesh size h = r , we see thatfor r ≥ / h , and n = ( n , · · · , n d − , r − ∈ ∂ Q ( r − τ n = ξ + n r ∈ Ω h ∩ (cid:0) Q \Q / (cid:1) .(A.27) implies that |G ( ξ, τ n ) − G / r ( ξ, τ n ) | ≤ C (log r ) r . Next suppose | n i | < (1 − η ) r , ≤ i ≤ d −
1, for h ≤ h ( η ), | τ n − x c | ≤ − η , i.e., τ n ∈ T − η . Then (A.39) implies that G ( ξ, τ n ) ≥ c · dist( τ n , ∂ Q ) = c · r . Therefore, for r ≥ r ∗ ( d , η ) G / r ( ξ, τ n ) ≥ c · r − C (log r ) r ≥ (cid:101) c r . Finally, by (A.26) (for h = / r ), we have that for n = ( n , · · · , n d − , r − ∈ ∂ Q ( r − | n i | < (1 − η ) r , i = , · · · , d − G r ( ξ, n ) = h d − G / r ( ξ, τ n ) ≥ c r d − . Clearly, the argument works for any side of the surface of Q and Q ( r ), as long as n ∈ ∂ Q ( r −
1) and | n i | < (1 − η ) r (except the largest one).Together with the relation (A.21) between the Poisson kernel P r ( ξ, n ) and the Green’sfunction G r ( ξ, n ), we actually proved that Lemma A.9.
Let P r ( ξ, m ) be given as in (A.19) . For any < η < / , and m = ( m , · · · , m d ) ∈ ∂ Q ( r ; ξ ) such that n i ∗ = max i n i = ± r and | m j | ≤ (1 − η ) r , j (cid:44) i ∗ . There are constants c = c ( η, d ) , r ( η, d ) only depends on the dimension and η such thatfor all r ≥ r P r ( ξ, m ) ≥ c r d − . (A.40)A.6. Harnack type inequalities.
Let Q ( r ) = Q ( r ; ξ ) and ∂ Q ( r ) be as in Section A.4.We prove the sub-mean value property and the Moser-Harnack inequality for a discretesub-harmonic function first. Lemma A.10.
Suppose f = { f n } is a discrete subharmonic function on Q ( r ; ξ ) in thesense that: − ( ∆ f ) n = − (cid:88) ≤ i ≤ d ( f n + e i + f n − e i ) + d f n ≤ , n ∈ Q ( r − ξ ) . There is a dimensional constant C = C ( d ) > , such that if additionally f n ≥ , thenf ξ ≤ Cr − d (cid:88) n ∈ ∂ Q ( r ; ξ ) f n , (A.41) f ξ ≤ Cr − d (cid:88) n ∈ Q ( r ; ξ ) f n . (A.42) As a consequence, for any cube Ω ⊂ Z d , if f n is non-negative and subharmonic on adomain containing the tripled cube Ω , then | (cid:96) ( Ω ) | d sup ξ ∈ Ω f ξ ≤ C (cid:88) n ∈ Ω f n . (A.43) Proof.
Fix ξ ∈ Z d , let (cid:101) f be the discrete harmonic function on Q ( r ; ξ ) which coincides f on ∂ Q ( r ; ξ ), i.e., ( ∆ (cid:101) f ) = , n ∈ Q ( r − ξ ) , (cid:101) f n = f n , n ∈ ∂ Q ( r ; ξ ) . By the integration by parts formula (A.22), (cid:101) f ξ = (cid:88) m ∈ ∂ Q ( r ) P r ( ξ, m ) f m , where P r ( n , m ) is the discrete Poisson kernel on Q ( r ) as given by (A.19). It was showedin [Gu, GuMa] that there is a dimensional constant C > m ∈ ∂ Q ( r ; ξ ) P r ( ξ, m ) ≤ Cr − d . Let w n = (cid:101) f n − f n . Clearly, − ( ∆ w ) n = ( ∆ f ) n ≥ n ∈ Q ( r − ξ ) and w n = n ∈ ∂ Q ( r ; ξ ). By the maximum principle Lemma A.1, one has w n = (cid:101) f n − f n ≥ , n ∈ HE LANDSCAPE LAW 63 Q ( r − ξ ). In particular, for all 0 ≤ r (cid:48) ≤ r , f ξ ≤ (cid:101) f ξ = (cid:88) m ∈ ∂ Q ( r (cid:48) ; ξ ) P r (cid:48) ( ξ, m ) f m ≤ Cr (cid:48) − d (cid:88) m ∈ ∂ Q ( r (cid:48) ; ξ ) f m (A.44)since f n ≥
0. We multiply (A.44) by r (cid:48) d − , and then sum for all 1 ≤ r (cid:48) ≤ r to obtain f ξ r (cid:88) r (cid:48) = r (cid:48) d − ≤ C r (cid:88) r (cid:48) = (cid:88) n ∈ ∂ Q ( r (cid:48) ; ξ ) f n ≤ C (cid:88) n ∈ Q ( r ; ξ ) f n . Therefore, f ξ ≤ (cid:101) Cr − d (cid:88) n ∈ Q ( r ; ξ ) f n , (A.45)which completes (A.42). (A.45) and H¨older inequality imply that r d f ξ ≤ C (cid:32) (cid:88) n ∈ Q ( r ; ξ ) (cid:33) (cid:32) (cid:88) n ∈ Q ( r ; ξ ) f n (cid:33) ≤ (cid:101) Cr d (cid:88) n ∈ Q ( r ; ξ ) f n . Then for any ξ , and r , r d f ξ ≤ (cid:101) C (cid:88) n ∈ Q ( r ; ξ ) f n . (A.46)In particular, let Ω be a cube in Z d of side length (cid:96) ( Ω ) and let 3 Ω be the tripled cubeas usual. If ξ ∈ Ω , then for r = (cid:96) ( Ω ), Q ( r ; ξ ) ⊂ Ω . (A.46) implies that | (cid:96) ( Q ) | d f ξ ≤ (cid:101) C (cid:88) n ∈ Q ( r ; ξ ) f n ≤ (cid:101) C (cid:88) n ∈ Ω f n . Therefore, | (cid:96) ( Q ) | d sup ξ ∈ Ω f ξ ≤ (cid:101) C (cid:88) n ∈ Ω f n . (A.47) (cid:3) As a direct consequence of the Moser-Harnack inequality for subharmonic functions,we have
Lemma A.11 (Moser-Harnack inequality for sub-solutions) . Let Ω ⊂ Z d be a cube ofside length (cid:96) ( Ω ) , let Ω be the tripled cube defined as in (2.12) of side length (cid:96) ( Ω ) .Suppose g n is a non-negative, sub-solution to an inhomogeneous equation on a domaincontaining Ω − ( ∆ g ) n ≤ , g n ≥ , n ∈ Ω , then there is a dimensional constant c H > such that (cid:88) n ∈ Ω g n ≥ | (cid:96) ( Ω ) | d · (cid:18) c H · sup Ω g n − | (cid:96) ( Ω ) | (cid:19) . (A.48) Proof.
Let g n be the non-negative sub-solution as given. Suppose 3 Ω = [ a , b ] × · · · × [ a d , b d ], b i − a i + = (cid:96) ( Ω ) , i = , · · · , d . And denote by | Ω | = d (cid:96) d ( Ω ) its cardinalityas usual. For n = ( n , · · · , n d ) ∈ Ω , let h n = d d (cid:88) i = ( n i − a i )( b i − n i ) . Direct computations show that − ( ∆ h ) n = − d (cid:88) i = (cid:0) h n + e i + h n − e i − h n (cid:1) = − ≤ h n ≤ | (cid:96) ( Ω ) | , n ∈ Ω . Therefore, let f n = h n + g n , one has − ( ∆ f ) n = − ( ∆ h ) n − ( ∆ g ) n ≤ − + = f n ≥ g n ≥
0. We can apply Lemma A.10 to the non-negative subharmonic function f n .(A.43) implies that | (cid:96) ( Ω ) | d sup n ∈ Ω g n ≤ | (cid:96) ( Ω ) | d sup n ∈ Ω f n ≤ C (cid:88) n ∈ Ω f n ≤ C (cid:88) n ∈ Ω (cid:0) h n + g n (cid:1) ≤ C · | (cid:96) ( Ω ) | · | Ω | + C (cid:88) n ∈ Ω g n ≤ C (cid:32) | (cid:96) ( Ω ) | + d + (cid:88) n ∈ Ω g n (cid:33) . Let c H = C − and rewrite the inequality, we have (cid:88) n ∈ Ω g n + | (cid:96) ( Ω ) | + d ≥ c H | (cid:96) ( Ω ) | d sup n ∈ Ω g n = ⇒ (cid:88) n ∈ Ω g n ≥ | (cid:96) ( Ω ) | d · (cid:18) c H · sup n ∈ Ω g n − | (cid:96) ( Ω ) | (cid:19) . (cid:3) Next, we study the discrete Harnack inequality on cubes in Z d for discrete sup-solutions to a homogeneous Schr¨odinger equation with a bounded potential Lemma A.12.
Suppose v n ≤ V max is a bounded potential. Let f n be a non-negativesuper-solution on a cube Ω ⊂ Z d such that − ( ∆ f ) n + v n f n ≥ , f n ≥ , n ∈ Ω . (A.49) There is a constants C > depending on d and V max such that for any cube Q ⊂ Ω ofside length (cid:96) ( Q ) , sup n ∈ Q f n ≤ C (cid:96) ( Q ) · inf n ∈ Q f n . (A.50) HE LANDSCAPE LAW 65
Proof.
Assume that the finite dimensional vector { f n } n ∈ Q attains its minimum and max-imum at m , n ∈ Q respectively. Connect m , n by a discrete path { γ j } sj = in Q , where γ = m , γ s = n and γ j + = γ j ± e i for some i . It is easy to check that the minimum stepsto reach n from m is s = | m − n | ≤ d · (cid:96) ( Q ). The upper bound for v n and (A.49) implythat for all k ∈ Ω (2 d + V max ) f k ≥ (cid:88) | k (cid:48) − k | = f k (cid:48) = ⇒ f n ≥ d + V max (cid:88) | k (cid:48) − k | = f k (cid:48) . By the non-negativity of f n , one has inductively f m ≥ d + V max (cid:88) | m (cid:48) − m | = f m (cid:48) ≥ d + V max f γ ≥ d + V max ) f γ ≥ · · · d + V max ) s f γ s . Therefore,inf n ∈ Q f n ≥ (2 d + V max ) − d · (cid:96) ( Q ) · sup n ∈ Q f n = ⇒ sup n ∈ Q f n ≤ (2 d + V max ) d · (cid:96) ( Q ) · inf n ∈ Q f n . (cid:3) A ppendix B. C hernoff bound
Lemma B.1 (Cherno ff –Hoe ff ding Theorem, [Ho]) . Suppose B ⊂ Z d has cardinality | B | . Suppose { ζ j } j ∈ B are i.i.d. Bernoulli random variables, taking values in { , } , withcommon expectation p = E (cid:0) ζ j (cid:1) ∈ (0 , . Then for any < λ < − p, P (cid:40) (cid:88) j ∈ B ζ j ≥ (1 − λ ) | B | (cid:41) ≤ e − D (1 − λ (cid:107) p ) | B | (B.1) where D ( x (cid:107) y ) = x log xy + (1 − x ) log 1 − x − y (B.2) is the Kullback–Leibler divergence between Bernoulli distributed random variables withparameters x and y respectively. We sketch the proof, following the arguments used in [DFM] (which is also close tothe original proof of Hoe ff ding), for readers’ convenience. Proof.
Let S = (cid:80) j ∈ B ζ j . For any t > P { S ≥ (1 − λ ) | B | } = P (cid:0) e tS ≥ e t (1 − λ ) | B | (cid:1) ≤ e − t (1 − λ ) | B | E (cid:0) e tS (cid:1) = e − t (1 − λ ) | B | (cid:89) j ∈ B E (cid:0) e t ζ j (cid:1) (by the independence of ζ j ) = e − t (1 − λ ) | B | (cid:0) E (cid:0) e t ζ (cid:1)(cid:1) | B | . Clearly, E (cid:0) e t ζ (cid:1) = e t p + − p . Therefore, for all t > P { S ≥ (1 − λ ) | B | } ≤ − t (1 − λ ) | B | + | B | log (cid:0) e t p + − p (cid:1) : = −| B | f ( t ) , (B.3)where f ( t ) = t (1 − λ ) − log (cid:0) e t p + − p (cid:1) . It is enough to optimize f ( t ) in t . As computed in [DFM], under the condition of 1 − λ − p > f ( t ) attains its only local maximum at t ∗ near 0, where t ∗ = log (cid:18) − λλ − pp (cid:19) (B.4)and f ( t ∗ ) = t ∗ (1 − λ ) − log (cid:0) e t ∗ p + − p (cid:1) = (1 − λ ) log (cid:18) − λλ − pp (cid:19) − log (cid:18) − λλ − pp p + − p (cid:19) = (1 − λ ) (cid:18) log 1 − λ p + log 1 − p λ (cid:19) − log 1 − p λ = (1 − λ ) log 1 − λ p − λ log 1 − p λ = (1 − λ ) log 1 − λ p + λ log λ − p = D (1 − λ (cid:107) p )where D ( x (cid:107) y ) is given in (B.2). Returning to (B.3), we have that P { S ≥ (1 − λ ) | B | } ≤ inf t e −| B | f ( t ) = e −| B | f ( t ∗ ) = e − D (1 − λ (cid:107) p ) | B | which is (B.1). (cid:3) R eferences [AW] Aizenman, M., and Warzel, S., Random Operators: Disorder E ff ects on QuantumSpectra and Dynamics. Vol. 168. American Mathematical Soc., 2015. 11, 12[ADFJM1] Arnold, D. N., David, G., Jerison, D., Mayboroda, S., and Filoche, M., E ff ective confining potential of quantum states in disordered media. Phys. Rev.Lett. 116.5 (2016): 056602. 2[ADFJM2] Arnold, D. N., David, G., Filoche, M., Jerison, D., and Mayboroda, S.,
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To appear in Ann.Henri Poincar´e. 8, 13, 14, 46, 49————————————–
D. Arnold, S chool of M athematics , U niversity of M innesota , 206 C hurch S t SE, M inneapolis , MN55455 USA
E-mail address : [email protected] M. Filoche, P hysique de la M ati ` ere C ondens ´ ee , E cole P olytechnique , CNRS, I nstitut P olytechniquede P aris , P alaiseau , F rance E-mail address : [email protected] S. Mayboroda, S chool of M athematics , U niversity of M innesota , 206 C hurch S t SE, M inneapolis ,MN 55455 USA
E-mail address : [email protected] W. Wang, S chool of M athematics , U niversity of M innesota , 206 C hurch S t SE, M inneapolis , MN55455 USA
E-mail address : [email protected] S. Zhang, S chool of M athematics , U niversity of M innesota , 206 C hurch S t SE, M inneapolis , MN55455 USA
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