aa r X i v : . [ m a t h . SP ] F e b SCHR ¨ODINGER OPERATORS WITH COMPLEX SPARSEPOTENTIALS
JEAN-CLAUDE CUENIN
Abstract.
We establish quantitative upper and lower bounds for Schr¨odingeroperators with complex potentials that satisfy some weak form of sparsity.Our first result is a quantitative version of an example, due to S. Boegli(Comm. Math. Phys., 2017, 352, 629-639), of a Schr¨odinger operator witheigenvalues accumulating to every point of the essential spectrum. The sec-ond result shows that the eigenvalue bounds of Frank (Bull. Lond. Math.Soc., 2011, 43, 745-750 and Trans. Amer. Math. Soc., 2018, 370, 219-240)can be improved for sparse potentials. The third result generalizes a theoremof Klaus (Ann. Inst. H. Poincar´e Sect. A (N.S.), 1983, 38, 7-13) on the char-acterization of the essential spectrum to the multidimensional non-selfadjointcase. The fourth result shows that, in one dimension, the purely imaginary(non-sparse) step potential has unexpectedly many eigenvalues, comparableto the number of resonances. Our examples show that several known upperbounds are sharp. Introduction and main results
Introduction.
Many examples of Schr¨odinger operators with “strange” spec-tral properties involve sparse potentials. In his seminal work [59] Pearson con-structed examples of real-valued potentials (on the half-line) leading to singularcontinuous spectrum. The potentials consists of an infinite sequence of “bumps”of identical profile, and the separation between these bumps increases rapidly. Thephysical interpretation is that a quantum mechanical particle will ultimately bereflected from a bump. These ideas were further developed in several directions,see e.g. [62] [70], [42], [48], [46], [80], and the references therein. Scattering fromsparse potentials in higher dimesions was studied by Molchanov and Vainberg [53],[54]; see also [52], [37],[38], [65], [45]. The discrete spectrum for multidimensionallattice Schr¨odinger operators was investigated by Rozenblum and Solomyak [63].They constructed examples of sparse potentials whose number of negative eigen-values grows like an arbitrary given polynomial power in the large coupling limit.In the recent work [5] Boegli constructed a complex-valued sparse potential witharbitrary small L q norm ( q > d ) that has infinitely many non-real eigenvaluesaccumulating at every point of the essential spectrum. Since the proof is based oncompactness arguments, there is no quantitative bound on the rate of separationbetween the bumps, and hence no estimate on the pointwise decay of the potentialis possible. Date : February 26, 2021.
A quantitative version of Boegli’s example.
Our first result providesquantitative decay bounds for the example in [5]. We formulate it for the mostinteresting spectral regionΣ = { z ∈ C : | Im z | ≤ ǫ Re z } , (1)where ǫ > Theorem 1.
Let d ≥ , q > d , ǫ , ǫ ∈ (0 , , γ > , and let ( ζ n ) n ⊂ Σ be asequence satisfying (cid:0) X n | ζ n | d | Im ζ n | q − d | log d | Im ζ n /ζ n || (cid:1) q ≤ ǫ . (2) Then there exists a complex sparse potential V such that the following hold: a) For each n there exists a discrete eigenvalue z n of H V = − ∆ + V whichis exponentially close to ζ n , in the sense | z n − ζ n | ≤ exp( −| Im ζ n | − γ ) . b) k V k L q ( R d ) . ǫ . c) V ( x ) decays polynomially.Remark . (i) In particular, for any λ ∈ (0 , ∞ ) there exists a sequence ( ζ n ) n ⊂ Σ satisfying (2) such that lim n →∞ ζ n = λ . In this way one can find a sequenceaccumulating to every point of the essential spectrum. This yields a constructiveproof of the result of Boegli [5].(ii) One can remove the logarithm in (2) at the expense of replacing the L q normof V by the “Davies–Nath norm” (see (28)).(iii) We will give explicit (although non-sharp) bounds on the polynomial decay.(iv) Substituting the trivial lower bound | ζ n | ≥ | Im ζ n | into (2) shows Im ζ n → z n is exponentially close to ζ n .We believe that the pointwise condition is more natural than the L q conditionfor the phenomenon that takes place in Theorem 1. This is because complexanalogues of classical phase space bounds, which motivate the consideration on L q norms in the first place, lack many of the features that make them so usefulfor real potentials (more on that in Subsection 1.5 below). Put simply, the L q norm does not see the separation between the bumps, while the pointwise bounddoes. We will nevertheless work with L q norms since we allow the bumps to havesingularities. In the case where they are bounded the pointwise decay of the wholepotential can easily be estimated by comparing the L ∞ norms of the bumps totheir spatial separation from the origin. In his fundamental work on non-selfadjointSchr¨odinger operators, Pavlov [57], [58] showed that the number of eigenvalues inone dimension is finite if | V ( x ) | . exp( − c | x | / ), and that this exponential rateis best possible. This means that the potential in Theorem 1 cannot decay toofast. The L q bound imposes no decay whatsoever, but we can at least establishpolynomial decay. For recent quantitative improvements of Pavlov’s bound werefer to Borichev–Frank–Volberg [6] and Sodin [72].In Section 6 we construct sparse potentials whose eigenvalues accumulate tothe essential spectrum at a given rate and are arbitrarily close to a given sequenceof spectral points (under some conditions on the sequence, of course). The proofof Theorem 1 in [5] (without the decay estimate) is based on “soft” methods like CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 3 weak convergence, compact embedding and the notion of the limiting essentialspectrum. In contrast, our proof only uses “hard” estimates for the resolventand the Birman-Schwinger operator, combined with tools from complex analysissuch as Rouch´e’s theorem, Jensen’s formula and Cartan type estimates. Rouch´e’stheorem and Jensen’s formula are among the most ubiquitous albeit simple toolsin non-selfadjoint spectral theory, where such machinery as the variational prin-ciple or the spectral theorem is not available. In the present paper Cartan typeestimates are crucial to lower bound a certain Fredholm determinant and get up-per upper bounds on the norm of the resolvent. This opens the way to provingexistence of eigenvalues by means of quasimode construction. We are then in asetting similar to the selfadjoint case where a quasimode of size ǫ guarantees theexistence of a spectral point in an ǫ -neighborhood of the quasi-eigenvalue. Thisfollows from the inequality k ( H V − z ) − k ≤ / d( z, σ ( H V )), where σ ( H V ) is thespectrum. In the non-selfadjoint case the inequality may fail dramatically. Thisphenomenon gives rise to the notion of pseudospectrum, which we will not discusshere (see e.g. the monograph [19]). The upper bounds obtained by Cartan typeestimates generally grow exponentially in 1 / d( z, σ ( H V )). In order to beat this, weare forced to construct exponentially small quasimodes, a challenging task in allbut the simplest models. The strategy is reminiscent of the proof of existence ofresonances close to the real axis due to Tang–Zworski [77] and Stefanov [73] (seealso the recent book by Dyatlov–Zworski [24]). Our method is perhaps closestto that of Dencker–Sj¨ostrand–Zworski [22, Section 6] for non-selfajoint dissipativeSchr¨odinger operators. The difference is that we consider decaying potentials anddo not assume, as these authors do, that the quasi-eigenvalue is real (see [22,Proposition 6.4]). This means that the amplification of the exponential upperbound through the maximum principle (see [22, Proposition 6.2]) is in general notpossible in our case. Another crucial difference is that we need a more quantitativeversion of the Cartan type estimate (Lemma 33) as well as of the conformal trans-formations between the spectral region and the model domain (the unit disk). TheRiemann mapping theorem is notoriously non-quantitative. Instead, we use Cay-ley and Schwarz–Christoffel transformations, which have previously been used inother contexts related to non-selfadjoint spectral theory, especially in connectionwith Lieb–Thirring type inequalities. The combination with Rouch´e’s theoremand the Cartan type bounds is new and leads to results with an inverse problemflavor.1.3. Magnitude bounds.
The second result gives precise bounds on the mag-nitude of eigenvalues of Schr¨odinger operators with complex sparse potentials, ormore generally, potentials of the form V = P Nj =1 V j , where the V j have disjointsupport and separate rapidly from each other. We will call them “separating”.The Schr¨odinger operator H V = − ∆ + V behaves like an almost orthogonal sum,due to the rapid decoupling between the N “channels”. This enables us to im-prove upon the bounds for general complex potentials due to Frank [27], [28]. Forsimplicity we state it here for d ≥
3. The general case along with further refine-ments can be found in Subsection 3.4. We refer to Section 2 for a more in-depthexplanation of the terminology.
J.-C. CUENIN
Theorem 2.
Assume that d ≥ and d/ ≤ q ≤ ( d + 1) / . If V is separating atscale η − , then every eigenvalue z of H V with Im √ z ≥ ( d + 1) η satisfies | z | q − d . sup j ∈ [ N ] k V j k qL q ( R d ) . (3) If q > ( d + 1) / , then | z | d( z, R + ) q − d +12 . sup j ∈ [ N ] k V j k qL q ( R d ) . (4)The bound (3) will follow from (27) by a Birman–Schwinger argument. It couldalso be proved by using the eigenvalue bounds of the author [14], which are inspiredby a method of Davies and Nath [20] in one dimension. For N = 1 the estimates(3), (4) coincide with those of Frank [27], [28], respectively. The difference is thathere V might decay very slowly or not at all. Nevertheless, on the η energy(spectral) scale the estimate is of the same quality as for N = 1.We make a short remark about the connection with the Laptev–Safronov con-jecture [47], which stipulates thatsup V ∈ L q ( R d ) sup z ∈ σ ( − ∆+ V ) \ R + | z | q − d k V k qq < ∞ for all q ∈ [ d/ , d ] . (5)For the range q ∈ [ d/ , ( d + 1) /
2] the conjecture was proven by Frank [27]. Thequestion whether (5) is true for q ∈ (( d + 1) / , d ] is still open. The expectation,based on intuition from counterexamples to Fourier restriction (see e.g. [13], [14]for more explanations) is that the conjecture is false in this range. Incidentally,Theorem 1 clearly implies the necessity of q > d in the conjecture, but this alreadyfollows from B¨ogli’s result (without the pointwise bound). In fact, a single bumpof the sparse potential used in Boegli’s construction (and in Theorem 1) alreadyprovides a counterexample. Since there seems to be some confusion about thisissue we use the results of Section 7 to show necessity of the condition q > d .Indeed, Lemma 22 implies that for small ǫ > V ( ǫ ) andan eigenvalue z ( ǫ ) such that | z ( ǫ ) | q − d / k V ( ǫ ) k qq & ǫ q − d log q (1 /ǫ ). The example (acomplex step potential) is simple but, quite amazingly, generic enough to showoptimality of several estimates in the literature (see [14]). In one dimension, thestep potential can be tuned to essentially saturate any of the known magnitudebounds. For example, the last inequality also shows that the Davies–Nath bound[20] is sharp and, since Im z ( ǫ ) ≍ ǫ , that Frank’s bound [28, Theorem 1.1] is sharpup to a logarithm. In Subsection 1.5 we will show how the complex step potentialalso implies sharpness of another bound in [28].1.4. A generalization of Klaus’ theorem.
The following is a generalization ofa result due to Klaus [43] on the characterization of the essential spectrum. Thegeneralization is twofold: First, we admit complex potentials and second, we proveit for any dimension (whereas Klaus only proved the one-dimensional case). Inthis introduction we again focus on the case d ≥
3, but the statement is valid in d = 1 , q in the range (15). CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 5
Theorem 3 (Klaus’ theorem [43] for complex potentials) . Assume that d ≥ and d/ ≤ q ≤ ( d + 1) / . If V is a separating potential and sup j ∈ [ N ] k V j k L q ( R d ) < ∞ ,then σ e ( H ) = [0 , ∞ ) ∪ S, where S is the set of all z ∈ C \ [0 , ∞ ) such that there exist infinite sequences ( i n ) n , ( z n ) n such that z n ∈ σ ( H V in ) , i n → ∞ and z n → z as n → ∞ . An alternative proof (also in one dimension) of Klaus’ theorem can be foundin [16]. The role of Theorem 3 in this paper will be an auxiliary one, and we willonly use it to argue that the essential spectrum is invariant under the perturbationswe consider in Theorem 1. Although our proof follows the general strategy of [43] itis still worth emphasizing that some parts of it require somewhat novel techniques.1.5.
Weyl’s law and locality.
Recently, Boegli and ˇStampach [4] disproved aconjecture by Demuth, Hansmann and Katriel [21] for one-dimensional Schr¨odingeroperators with complex potentials by establishing a lower bound on a certain Rieszmeans of eigenvalues. More precisely, consider H αV , where V = i [ − , is a purelyimaginary step potential and α is a large semiclassical parameter. Boegli andˇStampach prove that, for any p ≥ α − p X z ∈ σ d ( H αV ) (Im z ) p | z | / ≥ C p log α. (6)The interesting feature of this bound is that it shows a logarithmic violation ofWeyl’s law. To recall Weyl’s law, consider a self-adjoint operator, with a smoothreal-valued potential. Note that if we set h = 1 / √ α , then α − H αV takes the formof a semiclassical Schr¨odinger operator, − h ∂ x + V ( x ). Semiclassical asymptotics(Weyl’s law) yield, for a suitable class of functions f ,Tr f ( H αV ) = √ α π (cid:0) Z f ( α ( ξ + V ( x )))d x d ξ + o (1) (cid:1) (7)as α → ∞ . In particular, if f is homogeneous of degree γ , thenlim α →∞ α − / − γ Tr f ( H αV ) = (2 π ) − Z f ( ξ + V ( x ))d x d ξ. (8)For f ( λ ) := λ γ − and γ ≥ / d = 1) the Lieb-Thirringinequality X λ ∈ σ d ( H αV ) λ γ − ≤ C γ α / γ Z V − ( x ) / γ d x (9)captures the semiclassical behavior (8), but is valid for any α >
0, not onlyasymptotically. Returning to the complex potential V = i [ − , and noticingthat f ( z ) := (Im z ) p | z | / is homogeneous of degree γ = p − /
2, we observe that (6)implies that the formal analogue of (8) cannot hold, i.e. thatlim inf α →∞ α − / − γ X z ∈ σ d ( H αV ) (Im z ) / γ | z | / = ∞ , J.-C. CUENIN hence violating Weyl’s law (8). The comparison with Weyl’s law is formal because f ( H α ) does not make sense in general for a non-normal operator. However, (6)also shows that the complex analogue of the Lieb-Thirring inequality (9), X z ∈ σ d ( H αV ) (Im z ) / γ | z | / ≤ C ′ γ α / γ Z V − ( x ) / γ d x, cannot be true, thus disproving the conjectured bound in [21].For a non-selfadjoint (pseudo)-differential operator with analytic symbol andin one dimension the eigenvalues typically lie on a complex curve, hence violatingWeyl’s law (in terms of complex phase space). Small random perturbations typ-ically restore the Weyl law (in real phase space). The literature on the subjectis vast and we merely refer the interested reader to the book of Sj¨ostrand [71]for an overview of recent developments. In contrast, a classical result of Markusand Macaev [51] implies that Weyl’s law holds for the real part of the eigenvalues,provided the non-selfadjoint perturbation is small.A slightly different view on the same phenomenon (violation of Weyl’s law)is connected with the notion of “locality”. In the semiclassical limit, in the self-adjoint case, each state occupies a volume of (2 π/ √ α ) in phase space T ∗ R . Indeed,with f ( λ ) := ( −∞ , − E ] ( H αV ), (7) yields N ( H αV ; ( −∞ , − E ]) = √ α Vol( S E,V )(1 + o (1)) , where N ( H ; Σ) denotes the number of eigenvalues of an operator H in a set Σ and S E,V := { ( x, ξ ) ∈ T ∗ R : ξ + V ( x ) ≤ − E } is the relevant part of phase space. Ifwe consider a sum of disjoint bumps V = P Nj =1 V j , say with V j ( x ) = W ( x − x j ),then S E,V = N S
E,W , and hence N ( H αV ; ( −∞ , − E ]) N ( H αW ; ( −∞ , − E ]) = N (1 + o (1)) . This means that in the semiclassical limit each bump is responsible for an equalnumber of eigenvalues. In particular, two distinct realizations of V as a sum ofbumps have the same number of eigenvalues, as long as the bumps are disjoint.This feature of locality is also captured by the Lieb-Thirring inequalities since thebound involves the integral linearly.Our third result shows that this kind of locality can be violated in the non-selfadjoint case. We adopt the same notation N ( H ; Σ) for the number of eigen-values of H in Σ as in the self-adjoint case, but emphasize that these are countedaccording to their algebraic multiplicity. We consider one sparse one non-sparse (ornon-separating) realization of V and denote these by V s and V n , respectively. Forsimplicity, we will consider the same potential as in [4], i.e. the purely imaginarystep potential W = i | W | [ − R ,R ] of size | W | and width R . For simplicity we fixthese scales to be of order one. Then V n is the single well of width R = N R , while V s is a sum of N disjoint wells W ( x − x j ) of width R . We will fix the couplingstrength α and consider the limit N → ∞ . For the non-sparse operator this is infact still a semiclassical limit, as can easily be seen by rescaling. CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 7
Theorem 4.
Let d = 1 , N ≫ and consider the rectangular set Σ := { z ∈ C : C − N log N ≤ Re z ≤ C N log N , C − ≤ Im z ≤ C } , where C is a large constant. Then we have N ( H V n ; Σ) & N log N . (10)
Moreover, there exists a sequence ( x j ) j such that N ( H V s ; Σ) . N. (11)The estimates in [4] would be sufficient to prove a lower bound of size N − ǫ in (10). Their argument proceeds by approximating the characteristic eigenvalueequation and finding the roots of this equation in an asymptotic regime. They didnot prove that the original equation has nearby roots. This can be done e.g. by acontraction mapping argument [75]. We will give a proof using Rouch´e’s theorem.Note that, by power counting (dimensional analysis), the constants in (10), (11)only depend on the dimensionless quantity | W | / R .In the selfadjoint case, i.e. when W is replaced by | W | [ − R ,R ] , the numberof negative eigenvalues of V n is of order N , in agreement with semiclassics. Alsonote that, since in one dimension each H V j has at least one negative eigenvalue, wehave N ( H V s ; R − ) ≍ N in this case. Hence the left hand sides of (10) and (11) areequal in magnitude, which may be seen as a manifestation of locality. However,this locality is violated if one takes into account not only eigenvalues but also res-onances. Zworski [81] proved that, for a compactly supported, bounded, complexpotential, the number n ( r ) of resonances λ j in a disk | λ j | ≤ r is asymptoticallysatisfies n ( r ) = 2 | ch supp(V) | π r (1 + o V (1)) (12)as r → ∞ . Moreover, for any ǫ >
0, the number of resonances in | λ j | ≤ r butoutside the sector Σ (see (1)) is o ( r ). This and the fact that eigenvalue boundsoutside Σ are “trivial” (in the sense that they can be proved by the same standardestimates as for real potentials, with the provisio that the constants blow up asthe implicit constant in (1) becomes small) motivates us to often restrict attentionto the spectral set Σ . The result (12) was obtained earlier by Regge [61] in somespecial cases. Different proofs were given by Froese [31] and Simon [68]. Froese’sproof also works for complex potentials. Note that eigenvalues are included in thedefinition of resonances. Formula (12) can be seen as a Weyl law for resonancesbut is nonlocal as it includes the convex hull of the support of the potential. Anobvious corollary of Zworski’s formula is that the right hand side of (12) is anupper bound for the number of eigenvalues λ j in the disk | λ j | ≤ r (resonances inthe upper half plane). For the potential V n , taking r ≍ N/ log N and observingthat ch supp(V n ) ≍ N , we find that the leading term in (12) is of order N / log N ,in agreement with the lower bound (10). Note, however, that the asymptoticsare not uniform, i.e. the error may depend on N (recall that V n depends on N ). J.-C. CUENIN
Korotyaev [44, Theorem 1.1] proved an upper bound which yields n ( r ) ≤ Nπ log 2 r + O ( N log N )for r ≍ N/ log N . Hence, (10) shows that, at the scale considered here, a sub-stantial fraction of resonances are actual eigenvalues . In the special case of a steppotential Stepanenko [76] proved that the total number of eigenvalues is boundedby N / log N . The lower bound (10) shows that this is sharp up to constants; thiswas observed independently by Stepanenko [75].In dimension d ≥ H V j , and thus also H V s , have no eigenvalues at all if | W | / R is small (by the CLR bound), while V n has of the order N d eigenvalues. This is not the kind of phenomenon that takesplace in Theorem 4. Indeed, there we allow | W | / R to be of unit size. We alsonote that in higher dimensions the results on the asymptotics of the resonancecounting function are weaker, and there are example of complex potentials withno resonances at all (see Christiansen [8]). On the other hand, Christiansen [7]and Christiansen–Hislop [9], [10] proved that n ( r ) has maximal growth rate r d for“most” potentials in certain families.A final comment regarding the implications of Theorem 4, also related to lo-cality (or the lack thereof), concerns a general observation on Lieb–Thirring typeinequalities for complex potentials. It is a fact that all known upper bounds foreigenvalue sums have a superlinear dependence on N . For example, in d = 1, [28,Theorem 1.3] yields the bound X j d( z j , R + ) a . (cid:18)Z R | V | b d x (cid:19) c (13)with ( a, b, c ) = (1 , , c cannot bedecreased, while preserving the homogeneity condition − a = ( − b + 1) c . Indeed, X j d( z j , R + ) a ≥ X z j ∈ Σ d( z j , R + ) a & | W | a N log N , while (cid:18)Z R | V n | b d x (cid:19) c = (2 N R | W | b ) c , so that the ratio of the first to the second is bounded below by ( | W | / R ) − c N − c log N ,which tends to infinity as N → ∞ , for every c <
2. On the other hand, for sufficientrapid separation of the bumps in V s , we can prove Frank’s bound (13) with a lineardependence on N , at least locally in the spectrum. Theorem 5.
Let V ∈ ℓ L ( R ) and N ≫ . For any η > there exists a sequence ( x j ) Nj =1 , x j = x j ( η, k V k ℓ L ) , such that X Im √ z j > ( d +1) η d( z j , R + ) . N N X j =1 (cid:18)Z R | V j | d x (cid:19) Our methods could be used to prove similar generalizations of the higher-dimensional results of [28], but we will not pursue this.
Notation.
If not stated otherwise we choose the branch of the square root on C \ [0 , ∞ ) with Im √· >
0. We write σ ( H ), ρ ( H ) for the spectrum, respectivelythe resolvent set of a closed linear operator H . The free and the perturbed re-solvent operators are denoted by R ( z ) = ( − ∆ − z ) − and R V ( z ) = ( H V − z ) − ,respectively. We denote by S p the Schatten spaces of order p over the Hilbert space L ( R d ) and by k · k p the corresponding Schatten norms. We also write k · k = k · k ∞ for the operator norm. To distinguish it from L p norms of functions we denote thelatter by k · k L p . We will use the notation V / = V / | V | / and h x i := 2 + | x | . Thestatement a . b means that | a | ≤ C | b | for some absolute constant C . We write a ≍ b if a . b . a . If the estimate depends on a list of parameters τ , we indicatethis by writing a . τ b . The dependence on the dimension d and the Lebesgueexponents p, q is usually suppressed. We write a ≪ b if | a | ≤ c | b | with a smallabsolute constant c , independent of any parameters. By an absolute constant wealways understand a dimensionless constant C = C ( d, p, q ). Here we choose unitsof length l such that position, momentum and energy have dimensions l , l − and l − , respectively. We chose the branch of the square root √· on C \ [0 , ∞ ) suchthat √ z ∈ H , where H = { κ ∈ C : Im κ > } denotes the upper half plane. Theopen unit disk in C is denoted by D . Acknowledgements.
The author gratefully acknowledges correspondence withSabine Boegli and comments of Rupert Frank, who pointed out the failure ofWeyl’s law and the connection with nonlocality. Many thanks also go to St´ephaneNonnenmacher for useful discussions on resonances and to Alexei Stepanenko forexplaining his recent preprint. Special thanks go to Tanya Christiansen for manyhelpful remarks on a preliminary version of the introduction.2.
Definitions and preliminaries
Separating and sparse potentials.
We consider sparse potentials of theform V ( x ) = V j ( x ) , x ∈ Ω j , (14)where j ∈ [ N ] = { , , . . . , N } , N ∈ N ∪ {∞} , and Ω j ⊂ R d are mutually disjoint(not necessarily bounded) sets. We then set L j := d(Ω j , ∪ i ∈ [ N ] Ω i \ Ω j ) . We assume that V j ∈ ℓ p L q , where the norms are defined by k V k ℓ p L q := (cid:0) X j ∈ [ N ] k V j k pL q ( R d ) (cid:1) /p for p ∈ [1 , ∞ ) and k V k ℓ ∞ L q := sup j ∈ [ N ] k V j k L q ( R d ) . Here q will be in the range q ∈ [1 , ∞ ] if d = 1 , q ∈ (1 , ∞ ] if d = 2 , q ∈ [ d , ∞ ] if d ≥ . (15) In particular, we have k V k L q = k V k ℓ q L q . We sometimes write V = V ( L ) or V = V ( L, Ω) to emphasize the dependence of V on the sequences L = ( L j ) Nj =1 orΩ = (Ω) Nj =1 . Definition 1.
We say that V = V ( L ) is separating ifsep( L, η ) := X j ∈ [ N ] exp( − ηL j ) < ∞ for every η >
0. We call V separating at scale η − if sep( L, η ) ≤
1. We say that V is strongly separating if sep( L, δη ) . δ sep( L, η ) for every δ, η >
L, η ) only depends on L and not on V itself. We will some-times abuse terminology and call the sequence L separating. Note that since ηL j is dimensionless, η has the dimension of inverse length. We shall always assumethat the sequence L is increasing. Definition 2.
We say that V = V ( L, Ω) is sparse if it is separating, the supportsΩ n are bounded, and lim n →∞ diam(Ω n ) /L n = 0.Most of our results hold for separating potentials. The strong separation condi-tion is convenient and facilitates some of the proofs. Typical examples of stronglyseparating sequences are ( η ≪ η ):a) It η L k & k α for α >
0, then sep(
L, η ) . ( η/η ) − /α .b) If η L k & exp( k ), then sep( L, η ) . log( η /η ).c) If η L k & exp(exp( k )), then sep( L, η ) . log log( η /η ).See Subsection 8.2 for a proof. The explicit example used to prove Theorem 1turns out to be sparse. Note that, by the disjoint supports, the definition (14) isequivalent to V = P Nj =1 V j .2.2. Comparison with a direct sum.
In Section 5 we will compare the twooperators H V = − ∆ + V, H diag = M j ∈ [ N ] ( − ∆ + V j ) . Note that the point spectrum (eigenvalues) of H diag , σ p ( H diag ) = N [ j =1 σ p ( H V j ) = N [ j =1 σ p ( H W j ) , is independent of the sequence L . We will consider σ p ( H W j ) as part of the dataand seek to prove lower bounds on L j or | x i − x j | that imply that σ p ( H V ) is closeto σ p ( H diag ). In fact, we will consider a subset of the point spectrum, the discretespectrum. We will consider the V j as given only up to translations , i.e. we stipulatethat V j ( x ) = W j ( x − x j ) , CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 11 where W j ∈ ℓ p L q contains the origin in its support. Using the triangle inequality,it is easy to see that | x i − x j | ≥ L j for i = j , and thereforesup i ∈ [ N ] X j ∈ [ N ] \{ i } exp( − η | x i − x j | ) ≤ sep( L, η ) . Straightforward arguments also show that L i ≥ | x i − x j | − diam(Ω i ) − diam(Ω j )for all j ∈ [ N ] \ { i } . In particular, for sparse potentials, L i (1 − o (1)) ≥ sup j
For technical reaons, it will turn out to be convenient to havea bit more flexibility in the choice of operators being compared to each other. For n ∈ [ N ] we define V ( n ) := X j ∈ [ n ] V j , H ( n ) := − ∆ + V ( n ) . (17)2.4. Abstract Birman-Schwinger principle.
We mostly disregard operatortheoretic discussions here and refer e.g. to [28] for the (standard) definition of H V j as m -sectorial operators. The rigorous definition of H V is a bit more subtle since V need not be decaying. However, a classical construction of Kato [40] producesa closed extension H V of − ∆ + V via a Birman-Schwinger type argument. Thisapproach works as soon as one can find point z in the resolvent set of H = − ∆at which the Birman-Schwinger operator BS V ( z ) := | V | / ( − ∆ − z ) − V / (18)has norm less than one. Such bounds are provided by Lemma 6, but to avoidtechnicalities it is useful to think of the potential as being bounded by a largecutoff (and all of the bounds will be independent of that cutoff). By iterating thesecond resolvent identity, R V ( z ) = R ( z ) − R ( z ) V R ( z ) , it is then easy to see that R V ( z ) − R ( z ) = − R ( z ) V / ( I + BS V ( z )) − | V | / R ( z ) . (19)For more background on the abstract Birman-Schwinger principle in a nonselfad-joint setting we refer to [32], [28], [3], [36]. The essential and discrete spectrum.
We briefly recall some facts aboutthe essential and discrete spectrum of a closed operator H . There are severalinequivalent definitions of essential spectrum for non-selfadjoint operators (butthese all coincide for Schr¨odinger operators with decaying potentials [28, AppendixB]). We use the following standard definition. σ e ( H ) := { z ∈ C : H − z is not a Fredholm operator } . The discrete spectrum is defined as σ d ( H ) := { z ∈ C : z is an isolated eigenvalue of H of finite multiplicity } . Note that, if H is not selfadjoint, then, in general, σ ( H ) is not the disjoint unionof σ e ( H ) and σ d ( H ). However, by [33, Theorem XVII.2.1], if every connectedcomponent of C \ σ e ( H ) contains points of ρ ( H ), then σ ( H ) \ σ e ( H ) = σ d ( H ) . (20)In the situations we consider here (20) will always be true for H = H V and H diag . In fact, Corollary 11 tells us that σ e ( H ) = [0 , ∞ ), just as for decayingpotentials. 3. Universal bounds for separating potentials
In this section we consider H V as a perturbation of − ∆. We will thus onlymake assumptions about V and not about H diag .3.1. Birman–Schwinger analysis.
Since the V j have mutually disjoint sup-ports, we can write the Birman-Schwinger operator (18) as BS V ( z ) = N X i =1 BS ii ( z ) + X i = j BS ij ( z ) , where BS ij ( z ) = | V i | / R ( z ) V / j . The first term is unitarily equivalent to the orthogonal sum BS diag ( z ) := N M i =1 BS ii ( z ) . on the Hilbert space H ≃ L Ni =1 H i , where H = L ( R d ), H i = L (Ω i ). By abuseof notation we will always identify these two Hilbert spaces and the correspondingoperators. The off-diagonal contribution is BS off ( z ) := X i = j BS ij ( z ) . In the following we will use the notation ω q ( z ) := ( | z | d q − if q ≤ q d , | z | − q d( z, R + ) qdq − if q ≥ q d , (21) CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 13 where q d = ( d + 1) /
2. Note that for z ∈ Σ (see (1)) we have d( z, R + ) = | Im z | .We use the abbreviation s ( L, z ) := sep( L, Im √ z/ ( d + 1)) (22)and set α ( q ) := 2 max( q, q d ). Lemma 6.
Assume q is in the range (15) and p ∈ [ α ( q ) , ∞ ] . Then the followinghold. (i) For any i ∈ [ N ] , k BS ii ( z ) k α ( q ) . ω q ( z ) k V i k L q . (23)(ii) For any i, j ∈ [ N ] , i = j , k BS ij ( z ) k α ( q ) . exp( − d +1 Im √ z d(Ω i , Ω j )) ω q ( z ) k V i k / L q k V j k / L q . (24)(iii) The diagonal part satisfies k BS diag ( z ) k p . ω q ( z ) k V k ℓ p L q . (25)(iv) The off-diagonal part satisfies k BS off ( z ) k α ( q ) . s ( L, z ) ω q ( z ) k V k ℓ ∞ L q , (26)(v) The full Birman–Schwinger operator satisfies k BS V ( z ) k p . ω q ( z ) h s ( L, z ) i k V k ℓ p L q . (27) Proof.
It follows from k BS diag ( z ) k p = (cid:0) X i ∈ [ N ] k BS ii ( z ) k pp (cid:1) /p that (iii) is a consequence of (i). In view of the embeddings S p ⊂ S p , ℓ p ⊂ ℓ p for p ≤ p , (v) follows from (iii) and (iv). Moreover, (iv) follows from (ii) and thetriangle inequality, using the estimate d(Ω i , Ω j ) ≥ L i + L j to sum the doubleseries. The estimate (i) is the same as for the N = 1 case and follows from knownresults in the literature: For q ≤ q d , from [30, Theorem 12] for d ≥
3, from [12,Theorem 4.1] for d = 2 and from [1, Theorem 4] for d = 1, for q ≥ q d and d ≥ q ≤ q d and in [28, Proposition 2.1] for q ≥ q d . Theonly difference is that we include the (second) exponential in the pointwise bound | ( − ∆ − z ) − ( a +i t ) ( x − y ) | ≤ C e C t e − Im √ z | x − y | | x − y | − d +12 + a for a ∈ [( d − / , ( d + 1) /
2] and d ≥
2, see e.g. [49, (2.5)]. For d = 1 one can usethe explicit formula for the resolvent kernel. (cid:3) Remark . Using the results of [14] (or [20] in one dimension) in the proof ofLemma 6 we could replace the bounds (23), (24) by the following. For q ≤ q d and i, j ∈ [ N ], k BS ij ( z ) k α ( q ) . exp( − d +1 Im √ z d(Ω i , Ω j )) | z | d q − F / V i ,q (Im √ z ) F / V j ,q (Im √ z ) , where F V,q ( s ) is the “Davies–Nath norm” F V,q ( s ) := sup y ∈ R d Z R d | V ( x ) | q exp( − s | x − y | )d x ! q . (28)This implies the bounds k BS diag ( z ) k p . | z | d q − sup i ∈ [ N ] F V i ,q , k BS off ( z ) k α ( q ) . s ( L, z ) | z | d q − sup i ∈ [ N ] F V i ,q , k BS V ( z ) k α ( q ) . h s ( L, z ) i | z | d q − sup i ∈ [ N ] F V i ,q . Norm resolvent convergence.Lemma 7.
Under the assumptions of Lemma 6 we have k| V | / R ( z ) k p . | Im z | − ω q ( z ) h s ( L, z ) i k V k / ℓ p L q . Proof.
The claim readily follows from (27), the identity R ( z ) R ( z ) = 12Im z ( R ( z ) − R ( z ))and a T T ∗ argument. (cid:3) Proposition 8.
Under the assumptions of Lemma 6 the Schr¨odinger operators H V ( n ) with truncated potentials converge in norm resolvent sense to H V .Proof. We first note that all the bounds of Lemma 6 also hold for H V ( n ) , uniformlyin n . Since V ( n ) converges to V in ℓ p L q , it follows that the Birman-Schwingeroperator associated to H V ( n ) converges to BS V ( z ). Moreover, by Lemma 7, theoperators | V ( n ) | / R ( z ) converge in S p -norm. Note that the square root of V is trivial to compute, owing to the disjointness of supports of the V j . We choose z as in the proof of Corollary 11, i.e. z = i t , t ≫
1. For such z the norm ofthe Birman-Schwinger operator is <
1, hence a Neumann series argument and theresolvent identity (19) yield the claim. (cid:3)
Proposition 9. As s ( L, z ) → for fixed z , the Schr¨odinger operators H V (de-pending on L ) converge in norm resolvent sense to H diag .Proof. By (26), it follows that BS off ( z ) → s ( L, z ) →
0. The remain-der of the proof is similar to that of Proposition 8. (cid:3)
Proof of Klaus’ theorem.
The proof is an adaptation of Klaus’ original ar-gument, which is based on the Birman-Schwinger principle. In the non-selfadjointsetting one may use e.g. [19, Lemma 11.2.1] as a substitute for Weyl sequences.This yields an easy proof of the inclusion σ e ( H V ) ⊃ [0 , ∞ ) ∪ S . For the converseinclusion we prove an analogue of [43, Proposition 2.3]. Proposition 10.
For V ∈ ℓ ∞ L q and z ∈ C \ [0 , ∞ ) we have σ ( BS diag ( z )) = [ i σ ( BS ii ( z )) . CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 15
Proof.
Only the inclusion ⊂ is nontrivial. The resolvent set of a direct sum ofbounded operators A = L i A i is known to be ρ ( A ) = { λ ∈ \ i ρ ( A i ) : sup i k ( A i − λ ) − k < ∞} . We set A i = BS ii ( z ), whence A = BS diag ( z ). Assume that λ ∈ C \ S i σ ( BS ii ( z )).This clearly implies λ ∈ T i ρ ( A i ). It remains to prove that sup i k ( A i − λ ) − k < ∞ .By assumption, there exists δ > λ, σ ( A i )) ≥ δ . By [2, Theorem 4.1], k ( A i − λ ) − k ≤ δ − exp( a k A i k αα δ − α + b ) , where α = α ( q ) and a = a ( q ), b = b ( q ) are constants. Then (25) yields that k A i k α . k V k ℓ ∞ L q and hence sup i k ( A i − λ ) − k < ∞ . (cid:3) With Proposition 10 in hand is immediate that [43, Lemma 2.4] holds in thegenerality needed here ( K i = BS ii ( z ) in our notation). The Schatten bound (26)provides a substitute for the compactness arguments [43, Proposition 2.1-2.2]. Forthe remainder of the proof one can follow the arguments in [43] verbatim. Corollary 11. If V ∈ ℓ p L q with q in the range (15) and p < ∞ , then we have S = ∅ , i.e. σ e ( H V ) = [0 , ∞ ) . The same holds for H diag .Proof. By (27), we have that lim t →∞ k BS V (i t ) k = 0, which means that the inverseexists as a bounded operator for z = i t and t ≫
1. Corollary 7 implies that R ( z ) V / is compact, whence the resolvent difference (19) is compact. The claimnow follows by Weyl’s theorem [25, Theorem IX.2.4]. (cid:3) Magnitude bounds.
The following universal bounds generalize those ofTheorem 2 in the introduction. They are an immediate consequence of (27),Remark 2 and the Birman–Schwinger principle.
Theorem 12.
Let q be in the range (15) . If V is separating, then every eigenvalue z of H V satisfies ω q ( z ) − h s ( L, z ) i − . k V k ℓ ∞ L q (29) as well as | z | q − d h s ( L, z ) i − . sup j ∈ [ N ] F V j ,q (Im √ z ) . (30) Remark . ( i ) In the case of a single “bump” ( N = 1) the bound (29) was provedby Frank in [29] for q ≤ q d and in [28] for q > q d . In the latter case it was observedthat the inequality implies Im z → z → + ∞ for eigenvalues z of H V . Moreprecisely, if we fix the norm of the potential, then | Im z | − qdq . (Re z ) − q . In the case N = 1 (29) implies that the above holds with an additional factor h s ( L, z ) i on the right. If L grows at least polynomially, η L k & k α , then weobtain | Im z | − qdq + α . (Re z ) − q + α η − α , see Example a) after Definition 2. Hence, for sufficiently large α (depending on q and d ) the exponent of | Im z | remains positive, while that of Re z remains negative,and we still get the conclusion that Im z → z → + ∞ .( ii ) The N = 1 case of (30) was proved in one dimension by Davies and Nath[20] and in higher dimensions by the author [14]. The inequality is similar to (29)for q > q d . Both are relevant for “long-range” potentials. In the case of the steppotential (30) is sharp, while (29) (both for N = 1) loses a logarithm (see (56)and (57)). 4. Determinant bounds
Assumption 1.
Let q be in the range (15), p ∈ [2 max( q, q d ) , ∞ ), V = V ( L )strongly separating and k V k ℓ p L q . k V k ℓ p L q . Upper bounds.
We collect some useful estimates that we will repeatedlyuse (these follows from [69, Theorem 9.2]): | f ( z ) | ≤ exp( O (1) hk BS ( z ) k p i p ) , (31) | f diag ( z ) − f V ( z ) | ≤ k BS off ( z ) k p exp( O (1) hk BS diag ( z ) k p + k BS V ( z ) k p i p ) , (32)where f = f diag or f V and BS ( z ) = BS diag ( z ) or BS V ( z ). Formulas (31), (32),together with the bounds of Lemma 6 motivates the following definitions, ψ p ( t ) := exp( O p (1) h t i p ) , ϕ p ( t ) := t exp( O p (1) h t i p ) , t ≥ , and for z ∈ C \ [0 , ∞ ), Ψ p,q ( z ) := ψ p ( ω q ( z ) k V k ℓ p L q ) , Φ p,q ( z ) := ϕ p ( ω q ( z ) k V k ℓ p L q ) , Φ p,q ( L, z ) := ϕ p ( s ( L, z ) ω q ( z ) k V k ℓ p L q ) , (33)where we suppressed the dependence on V . We recall that ω q ( z ) and s ( L, z )were defined in (21) and (22). The constant O p (1) is allowed to vary from oneoccurence to another. Thus, for example, the inequality ϕ p ( t ) ψ p ( t ) . ϕ p ( t ) holds,but ψ p ( t ) . Lower bounds away from zeros.
The following lemma can be consideredone of the key technical results. To state it we introduce the notation M p,q ( z ) := h z i| Im z | (cid:18) h z i| z | (cid:19) p ( qdq − − +5 h ω q ( z ) i p , (34) M p,q ( L, z ) := M p,q ( z ) h s ( L, (cid:18) | z |h z i (cid:19) z ) i p , Moreover, we set δ H ( z ) := min( , d( z, σ ( H ))) . (35)In the following H denotes either H diag or H V . We then write δ diag ( z ) := δ H diag ( z )and δ V ( z ) := δ H V ( z ). CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 17
Lemma 13.
Suppose Assumption 1 holds. Then for all z ∈ Σ , | f diag ( z ) | − ≤ exp( O (1) M p,q ( z ) log 1 δ diag ( z ) ) , (36) | f V ( z ) | − ≤ exp( O (1) M p,q ( L, z ) log 1 δ V ( z ) ) . (37) Proof.
In the following, f denotes either f diag or f V . We are going to applyLemma 34 with parameters (in the notation of that lemma) r = c | z | , r = c | z | , r = c | z | ,R = c − | z | , R = c − h z i , ϕ = ǫ | arg( z ) | , ϕ = ǫ | arg( z ) | , ϕ = cǫ | arg( z ) | ,θ = π − ϕ , θ = π − ϕ , θ = π − ϕ , where c ≪ ǫ = ǫ ( z ) = c (cid:18) | z |h z i (cid:19) πθ − ϕ +1 . c (cid:18) | z |h z i (cid:19) . Note that | arg( z ) | ≪ z ∈ Σ and thus sin(2 ϕ j ) ≍ ϕ j ≍ tan(2 ϕ j ), whichwill be used repeatedly in the proof. The first condition in (86) is satisfied bydefinition. Using that, for j = 1 , ∂U j , ∂U ) = sin(2 ϕ j ) r j ≍ ( cǫ ) j | Im z | , we find that the second and third condition in (86) are also satisfied. We will showthat, for a suitable choice of z ∈ U ,max w ∈ U log | f ( w ) | − log | f ( z ) | . c ( h s ( L, ǫ Im √ z ) i ν ǫ − qdq − − h ω q ( z ) i ) p , (38)where ν = 0 if f = f diag and ν = 1 if f = f V . Since in the present case R d( ∂U , ∂U ) (cid:18) R r (cid:19) πθ − ϕ . c (cid:18) h z i| Im z | (cid:19) (cid:18) h z i| z | (cid:19) , we see that (86) holds. Lemma 34 thus implies that (36), (37) hold. To prove(38) we first observe that, by the maximum principle, | f | attains its maximumon the boundary of the wedge U . We estimate this on the boundary com-ponent corresponding to the ray wρ e ϕ , ρ > r , the estimates for the othertwo boundary components being similar. By (31) and the bounds of Lemma 6,this maximum is bounded by the right hand side of (38), where we used thatsup ρ>r ω q ( ρ e ϕ ) . c ǫ − qdq − − ω q ( z ) and that L is strongly separating. Thisproves (38) for the first term on the left. The other part follows by selecting, forinstance, z = i4 R and using the estimates (similar to (32), see [69, Theorem 9.2]) | f diag ( z ) − | . Φ p,q ( z ) , | f V ( z ) − | . Φ p,q ( L, z ) , where we once again used Lemma 6 and ω q ( z ) . ω q ( z ). Note that in the case of f V we can absorb the factor h s ( L, z ) i in the definition of Φ p,q ( L, z ) into the O (1)term since Im √ z & L is strongly separating. Since ω q ( z ) ≪ R ≫ c ≪ | f ( z ) | ≥ / c sufficientlysmall. This finishes the proof of (38). (cid:3) Upper bound on the resolvent away from eigenvalues.
As a conse-quence of Lemma 13 we also obtain an upper bound for the resolvent of H V awayfrom the spectrum. The idea is to use the following infinite-dimensional analogueof Caramer’s rule (see [67, (7.10)]), k ( I + BS ( z )) − k ≤ exp( O (1) k BS ( z ) k pp ) | f ( z ) | . (39) Proposition 14.
Suppose Assumption 1 holds. Then for all z ∈ Σ , k ( H diag − z ) − k ≤ exp( O (1) M p,q ( z ) log 1 δ diag ( z ) ) , (40) k ( H V − z ) − k ≤ exp( O (1) M p,q ( L, z ) log 1 δ V ( z ) ) , (41) where M q ( z ) , M q ( L, z ) are given by (34) .Proof. In view of the trivial bound k ( − ∆ − z ) − k ≤ | Im z | − , the claim follows from Lemma 15, the resolvent identity (19), Corollary 7 and thefact that | Im z | − . M p,q ( z ). (cid:3) Lemma 15.
The operator norms of ( I + BS diag ( z )) − and ( I + BS V ( z )) − arebounded by the right hand side of (40) and (41) , respectively.Proof. We only prove the claim for BS diag ( z ); the other part is similar. By (39),(27) and (33), we have k ( I + BS diag ( z )) − k . | f diag ( z ) | − Ψ p,q ( z ). Since Ψ p,q ( z ) . M p,q ( z ), Lemma 13 implies that the latter is bounded by (40). (cid:3) Comparison between H diag and H V Ghershgorin type upper bounds.
We record the following Ghershgorintype bound. We temporarily restore the norm of the potential and define ω q,i ( z ) := ω q ( z ) k V i k L q , and M p,q,i ( z ) is defined by (34) with ω q ( z ) replaced by ω q,i ( z ). Proposition 16.
Under Assumption 1 the discrete spectrum of σ ( H V ) in Σ iscontained in the set [ i ∈ [ N ] { z ∈ C : Im √ zL i − log h s ( L, z ) i . − M p,q,i ( z ) log δ H Vi ( z ) + log ω q,i ( z ) } . (42) Proof.
Assume first that
N < ∞ , and consider the Hilbert spaces H n = L (Ω n ), H = L n ∈ [ N ] H n , with operators A ij = δ ij I H + BS ij ( z ) and A = ( A ij ) Ni,j =1 .Applying the Gershgorin theorem for bounded block operator matrices due toSalas [64] (see also [78, Theorem 1.13.1]) yields σ ( A ) ⊂ N [ i =1 { λ ∈ C : k ( A ii − λ ) − k − ≤ X j ∈ [ N ] \{ i } k A ij k} . CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 19
Note that here we are using the convention that k ( A ii − λ ) − k = ∞ if λ ∈ σ ( A ii ).By the Birman–Schwinger principle, this implies that σ ( H V ) ⊂ N [ i =1 { z ∈ C : k ( I H + BS ii ( z )) − k − ≤ X j ∈ [ N ] \{ i } k BS ij ( z ) k} . Again, we include the spectrum of A ii in the set on the right. By Lemma 15, wehave k ( I H + BS ii ( z )) − k ≤ exp( O (1) M p,q,i ( z ) log 1 δ H Vi ( z ) ) . On the other hand, by (24) and the strong separation property, X j ∈ [ N ] \{ i } k BS ij ( z ) k . s ( L, z ) exp( − d +1 Im √ zL i ) ω q ( z )The claim for N < ∞ follows. Similarly, it follows for the truncated operators H ( n ) . Since the set (42) is independent of n the claim for the case N = ∞ thenfollows from the norm resolvent convergence of the truncated operators (Proposi-tion 8). (cid:3) Lower bounds: Qualitative results.
In the following we establish criteriaon the sequence ( L j ) j that guarantee proximity of σ d ( H V ) to σ d ( H diag ) in variousregions of the spectral plane. Let us fist discuss some standard facts from pertur-bation theory ([41, Chaper IV]). It is well knows that the spectrum of a closedoperator H is upper semicontinuous under perturbations, and the same is true foreach separated part of the spectrum [41, Theorem 3.16]. Moreover, a finite system of eigenvalues { ζ , . . . , ζ n } changes continuously, just as in the finite-dimensionalcase. This follows from the fact that the Riesz projection12 π i Z Γ ( ζ − H ) − d ζ is continuous in H in the uniform topology (in the sense of generalized conver-gence of operators). Here Γ is a closed contour (a piecewise smooth curve) in theresolvent set of H and encircling the eigenvalues ζ , . . . , ζ n (and no other point ofthe spectrum) once in the counterclockwise direction. Then12 π i Tr Z Γ ( ζ − H ) − d ζ = n. Here we consider a finite system of eigenvalues of H diag in some compact subsetΣ ⋐ C \ [0 , ∞ ). By Corollary 11, Assumption 1 implies that each point in Σis either in the resolvent set or a discrete eigenvalue of H diag . By compactness,Σ ∩ σ ( H diag ) is a finite set. We then have δ (Σ) := min { d( ζ, σ ( H diag ) \ { ζ } ) : ζ ∈ Σ ∩ σ ( H diag ) } > . (43)For δ ∈ (0 , min(1 , δ (Σ))) we set U δ := [ ζ ∈ Σ ∩ σ ( H diag ) D ( ζ, δ ) , Γ δ := ∂U δ . (44) In general it is hard to determine δ (Σ), but we still have Γ δ ⊂ ρ ( H diag ) forgeneric δ . This is all that is needed for a lower bound on the number of eigenval-ues in U δ . The norm resolvent convergence (Proposition 9) implies the followingproposition. In Subsection 5.4 we will give an alternative proof using the argumentprinciple.Let us first state our assumptions for the remainder of this section. Assumption 2.
Let Σ ⊂ Σ ∩ C \ [0 , ∞ ) be a compact subset, let δ (Σ) , U δ , Γ δ be defined by (43), (44) and let δ ∈ (0 , δ (Σ)). Proposition 17.
Suppose Assumptions 1, 2 hold. Then for any δ ∈ (0 , δ (Σ)) there exists a constant C = C ( δ, Σ) such that, if s ( L, ζ ) ≤ /C , then N ( H V ; U δ ) = N ( H diag ; U δ ) . Argument principle.
The argument in the previous subsection involvedcompactness and continuity and is obviously non-quantitative. The issue is ofcourse the need for a quantitative estimate of the resolvent on the curve Γ δ . Wewill prove such estimates in Proposition 14. Here we argue in a slightly different(albeit closely related) manner. We will use the regularized Fredholm determinants(see for instance [34, IV.2], [69, Chapter 9] or [23, XI.9.21]) f diag ( z ) := det p ( I + BS diag ( z )) ,f V ( z ) := det p ( I + BS V ( z )) , where p ∈ [2 max( q, q d ) , ∞ ) is assumed to be an integer. The main property thatwe will use is that the f diag , f V are analytic functions in C \ [0 , ∞ ) and have zeros(counted with multiplicity) exactly at the eigenvalues of H diag , H V , respectively.Moreover, by the generalized argument principle (see e.g. [28, Theorem 3.2] or [3,Theorem 6.7]), N ( H ; U δ ) = 12 π i Z Γ δ dd z log f ( ζ )d ζ, where H = H diag or H V and f = f diag or f V . This suggests a comparison between H diag and H V via Rouch´e’s theorem (see e.g. [66] for related ideas). We set r δ := sup z ∈ Γ δ | f diag ( z ) − f V ( z ) || f diag ( z ) | . (45)We will show that r δ < j ∈ [ n ] s ( L, ζ j ) is sufficiently small. Rouch´e’s theoremthen asserts that f diag and f have the same number of zeros in U δ .5.4. Alternative proof of Proposition 17.
Without loss of generality we mayassume that Σ contains exactly one eigenvalue ζ of H diag . We are going to useLemma 32. For this purpose we set U = U δ and let U ⊂ C \ [0 , ∞ ) be a pre-compact simply connected neighborhood of U containing a point ζ / ∈ σ ( H diag ).This is possible by (3) applied to V j (i.e. with N = 1) since we can take ζ = − A ,where A ≫
1. By (32) we find thatsup z ∈ Γ δ | f diag ( z ) − f V ( z ) | ≤ C s ( L, ζ ) , (46) CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 21 where C = C ( δ, Σ). We take A so large thatΦ p,q ( z ) ≤ . (47)This is possible since lim A →∞ ω q ( − A ) = 0 by (25). By Lemma 32 there exists aconstant C = C ( δ, Σ) such thatlog | f diag ( z ) | ≥ − C for all z ∈ Γ δ . (48)Here we used that log | f diag ( z ) | ≥ − log 2, which follows from (47), (25) and (31).Combining (46) and (48), we infer that r δ < s ( L, ζ ) is sufficiently small.5.5.
Lower bounds: Quantitative results.
In the following we establish quan-titative versions of Proposition 17.We return to estimating the quantity (45) featuring in Rouch´e’s theorem.
Lemma 18.
Suppose Assumptions 1, 2 hold. Then r δ ≤ max ζ ∈ Σ s ( L, ζ ) exp (cid:0) O (1) h s ( L, ζ ) i M p,q ( ζ ) log 1 δ (cid:1) , (49) where δ H diag ( ζ ) , s ( L, ζ ) , M p,q ( ζ ) are given by (35) , (22) , (34) .Proof. Again we may assume that Σ contains exactly one eigenvalue ζ of H diag , sothat U δ = D ( ζ, δ ) and Γ δ = ∂D ( ζ, δ ). We clearly have δ ( z ) = δ and ω q ( z ) ≍ ω q ( ζ )for z ∈ Γ δ . It is easy to see that (32) and Lemma 6 implysup z ∈ Γ δ | f diag ( z ) − f V ( z ) | . Φ p,q ( L, ζ )Ψ p,q ( ζ ) . We have also used that L is strongly separating, hence s ( L, z ) . s ( L, ζ ). In orderto estimate (45) it remains to bound | f diag ( z ) | from below using (36). (cid:3) As an immediate corollary we obtain an improvement of Proposition 17.
Proposition 19.
Suppose Assumptions 1, 2 hold. Then N ( H V ; U δ ) = m ( H diag ; U δ ) , provided that L is so large that r δ < in (49) . From quasimodes to eigenvalues
Existence of a single eigenvalue.
We record a useful corollary of Propo-sition 14.
Corollary 20.
Suppose Assumption 1 holds. Assume that there is a normalized ψ ∈ L ( R d ) such that k ( H V − ζ ) ψ k ≤ ǫ, where ǫ − is larger than the right hand side of (41) at z = ζ . Then σ d ( H V ) ∩ D ( ζ, δ )) = ∅ . Existence of a sequence of eigenvalues.
If instead of a single quasi-eigenvalue we consider a sequence ( ζ j ) j with lim j →∞ Im p ζ j = 0, an across-the-board assumption like the one in Corollary 20 is not feasible since the right handside of (41) at z = ζ n tends to infinity as n → ∞ . One possible solution wouldbe to modify the previous arguments and select L as a function of the sequence( ζ j ) j . We will follow a similar, albeit slightly different strategy which we find moreintuitive. It is also closer in spirit to the inductive argument in [5], which is basedon strong resolvent convergence. Once more, the approach we will outline can beviewed as a quantitative version of that method.The strategy will be to first construct quasimodes of H V in a direct way (Lemma21) and then use Corollary 20 to obtain existence of eigenvalues. The quasi-eigenvalues and quasimodes will be actual eigenvalues and eigenfunctions of H diag .We introduce a sequence of scales ε j and a j , where ε j has dimension of energy and a j has dimension of length, and assume that the eigenfunctions ψ j correspondingto ζ j decay exponentially away from Ω j in such a way that k V i ψ j k ≤ C q a − d/ e qj exp( − c Im p ζ j d (Ω i , Ω j )) k V i k L e q , (50)where e q ≥ c >
0. In the following applications we can take c = 1. We willthen choose L such thatIm p ζ n L n ≥ C log n log h n i sup j ∈ [ n ] ε − j a − d/ e qj sup i ∈ [ n ] k V i k L e q ! , (51)where C = C ( d, e q ) is a large constant. Lemma 21.
Assume that V ∈ ℓ ∞ L e q for some e q ≥ and that ζ j are eigenvaluesof H V j with normalized eigenfunctions ψ j satisfying (50) . Then there exists anabsolute constant C = C ( d, q ) such that if V ( L ) is separating and satisfies (51) ,then H V has a sequence of normalized quasimodes ψ j , k ( H V − ζ j ) ψ j k ≤ ε j . Remark . Lemma 21 could be seen as a quantitative version of Lemma 2 in [5].
Proof.
Note first that the assumption that ζ j is an eigenvalue of H W j is translation-invariant, i.e. is equivalent to ζ j being an eigenvalue of H V j . Let ψ j be the eigen-functions of H V j corresponding to ζ j , i.e. ( H V j − ζ j ) ψ j = 0. For n ∈ [ N ] we set H ( n ) := H V ( n ) (recall (17) for the definition) and make the following (stronger)induction hypothesis P ( n ): k ( H ( n ) − ζ j ) ψ j k ≤ ε j (cid:0) − h n i (cid:1) , j ∈ [ n ] . (52)The base case n = 1 is true by assumption. Assume now that P ( n −
1) holds. Bythe exponential decay (50), k ( H ( n ) − ζ n ) ψ n k ≤ n − X j =1 k V j ψ n k ≤ nC q a − d/ e qn exp( − c η n L n ) sup j ∈ [ n ] k V j k L e q , CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 23 where we have set η n := Im √ ζ n . Moreover, by induction hypothesis, for j ∈ [ n − k ( H ( n ) − ζ j ) ψ j k ≤ k ( H ( n − − ζ j ) ψ j k + k V n ψ j k≤ ε j (cid:0) − h n − i (cid:1) + C q a − d/ e qj exp( − c η j L n ) k V n k L e q . Hence P ( n ) would hold if L n satisfied the estimates nC q a − d/ e qn exp( − c η n L n ) sup j ∈ [ n ] k V j k L e q ≤ ε n , (53) C q a − d/ e qj exp( − c η j L n ) k V n k L e q ≤ ε j (cid:0) h n − i − h n i (cid:1) , (54)for j ∈ [ n ]. By the mean value theorem (cid:0) h n − i − h n i (cid:1) & n log h n i , and it is easy to check that (53)–(54) are satisfied for the choice (51). This com-pletes the induction step. For N < ∞ the claim now follows from (52) with n = N .Now consider the case N = ∞ . Since L is separating,lim n →∞ k ( H V − H ( n ) ) ψ j k = lim n →∞ k ( V − V ( n ) ) ψ j k = lim n →∞ ∞ X k = n +1 k V k ψ j k ≤ a − d/ e qj k V k ℓ ∞ L e q lim n →∞ ∞ X k = n +1 exp( − η j L k ) = 0 . Together with (52) this yields the claim for N = ∞ . (cid:3) Remark . The factor n log n in (51) comes from the induction hypothesis andshould not be taken too seriously. One could of course replace log h n i by any otherslowly varying sequence tending to infinity. However, this would not change thebound (51) significantly.6.3. Quasimode construction.
We now construct the potential W j having ζ j as an eigenvalue. Lemma 22.
Given ζ ∈ Σ and x ∈ R d there exists a potential W = W ( ζ, x ) ∈ L ∞ comp ( R d ) such that the following hold. (1) H W has eigenvalue ζ ; (2) supp W ⊂ B ( x , R ) , where R = R ( ζ ) ≍ | ζ | | Im ζ | − | log | Im ζ | / | ζ || . (55)(3) For any ≤ q ≤ ∞k W k L q ( R d ) ≍ | ζ | d q | Im ζ | − dq | log dq | Im ζ/ζ || . (56)(4) For ≤ q ≤ q d , F W,q (Im p ζ ) . | ζ | d q | Im ζ | − dq (57) (5) The normalized eigenfunction ψ = ψ ( ζ, x ) of H W corresponding to ζ satisfies the exponential decay estimate | ψ ( x ) | ≤ C | ζ | / | x | − d − exp( − Im p ζ d ( x, supp W )) . (58) Proof.
By scaling it suffices to prove this for | ζ | ≍
1. In view of the results ofSection 7 (and ζ = E in the notation of that section) we can then simply choose W as a shifted step potential. The shift of course does not affect the eigenvalues northe L q norms. The latter are trivial to compute using the size bound | W | = O ( ǫ )and the formula (55) for the width of the step. The estimate (57) follows from adirect computation. The exponential decay follows from Lemma 23 or the explicitform of the wavefunction for the step potential. (cid:3) Remark . Similar results involving complex step potentials are contained in [5],[14], [15], albeit in a less quantitative form. A technical detail that distinguishesour proof from these is that we first pick the eigenvalue, then find the potential.This avoids the use of Rouch´e’s theorem in [14], [15].6.4.
Exponential decay.
We prove that the exponential decay bound (50) holdsfor a class of compactly supported potentials that will be relevant in the nextsection. The important point here is that the constant C in (59) is independentof W . Lemma 23.
Assume that supp W ⊂ B (0 , R ) and κ ∈ H , Im √ ζ ≤ R − log R , | κ | ≥ KR − for a large absolute constant K . Assume that ψ is a normalizedeigenfunction of H W with eigenvalue ζ . Then there exists an absolute constant C = C ( d ) such that for | x | > R , | ψ ( x ) | ≤ C | ζ | / | x | − d − exp( − Im p ζ | x | ) . (59) Proof.
Since ψ is normalized in L it has units l − d/ . By homogeneity, we maythus assume that | κ | = 1. Since ψ solves the Helmholtz equation − ∆ ψ ( x ) = κ ψ ( x )for | x | > R and κ = ζ , we have (see e.g. [79, Chapter 1, Section 2]) ψ ( x ) = A | x | − ν H (1) ν ( κ | x | )in this region, where H (1) ν is the Hankel function, ν = ( d − / A = A ( d, W ) isa normalization constant. By the well-known asymptotics of the Hankel functionat infinity, ψ ( x ) = Ac d | x | − d − exp( − Im κ | x | )(1 + O ( | x | − )) . This would imply (59) if we could show that A has an upper bound independentof W . Since ψ is normalized, A c d Z | x | >R | x | − ( d − exp( − κ | x | )(1 + O ( | x | − ))d x ≤ k ψ k = 1 . For sufficiently large K we estimate the integral from below by(1 − O ( K − )) Z RR exp( − κr )d r ≥ (1 − O ( K − )) R exp( − log R ) ≥ , CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 25 which proves that A ≤ /c d . (cid:3) Corollary 24.
Assume that V j ( x ) = W j ( x − x j ) and that the assumptions ofLemma 23 are satisfied for W j , ζ, ψ j , R j . Then (50) holds for any e q ≥ and with a − d/ e qj . | ζ j | / (cid:16) L j Im p ζ j (cid:17) − d − ( − e q ) . (60) Proof.
Let e q + r = . By H¨older, k V i ψ j k ≤ k V i k L e q k ψ j k L r (Ω i ) and by (59), k ψ j k L r (Ω i ) . | ζ j | / (cid:0) Z Ω i ( | x − x j | − d − exp( − Im p ζ j | x − x j | )) r d x (cid:1) /r . Since | x − x j | ≥ d( x, Ω j ), k ψ j k L r (Ω i ) . | ζ j | / (cid:16) L j Im p ζ j (cid:17) − d − r . The claim follows. (cid:3)
A quantitative version of Boegli’s example.
In view of Corollary 20,given ζ j ∈ σ H Vj and δ j > ε j = ǫ j ( ζ j , δ j , L ) as ε − j = exp( O (1) M p,q ( L, ζ j ) log 1 δ j ) (61)and require that (51) holds with a j as in (60). This gives a sufficient condition onthe sequence L ensuring that d( ζ j , σ ( H V )) ≤ δ j . The following proposition followsimmediately from Corollary 20, Lemma 21 and Corollary 24. Proposition 25.
Suppose Assumption 1 holds, V j ∈ ℓ ∞ L e q for some e q ≥ and that supp V j ( · + x j ) ⊂ B (0 , R j ) for some positive R j . Let ζ j , δ j be sequences satisfying Im p ζ j ≤ R − j log R j , | ζ j | / ≥ KR − j for some large absolute constant K and δ j ∈ (0 , / . Assume that L satisfies (51) with a j as in (60) and ǫ j as in (61) .If ζ j ∈ Σ is an eigenvalue of H V j of multiplicity m j , then D ( ζ j , δ j ) contains atleast m j eigenvalues of H V , counted with multiplicity. In the following we apply Proposition 25 with V j = W ( ζ j , x j ), where W is thecomplex step potential in Lemma 21. Clearly, V j ∈ L q ( R d ) for every q ∈ [1 , ∞ ],with k V k ℓ p L q . (cid:0) X n (cid:0) | ζ n | d q | Im ζ n | − dq | log dq | Im ζ n /ζ n || (cid:1) p (cid:1) p , (62)sup j ∈ [ n ] k V j k L ∞ . sup j ∈ [ n ] | Im ζ n | . (63)We will also take e q = ∞ , so that a − d/ e qj = 1. For the remainder of this section weassume the following. Assumption 3.
Let q > d , and assume that 2 holds. Without loss of generalitywe may also assume, as we will for the remainder of this section, that Im ζ n ismonotonically decreasing. Lemma 26.
Under Assumption 3 the following hold. (1) k V k ℓ p L q ≤ ǫ , k V k ℓ ∞ L ∞ . . (2) | Im ζ n | . . (3) h ζ n i . | Im ζ n | − q − d ) d . (4) M p,q ( ζ n ) & | Im ζ n | p ( ( q − d ) dq + qdq − . (5) M q ( ζ n ) . | Im ζ n | − − q − d ) d − p ( qdd − − p ( ( q − d ) dq + qdq − .Proof. Condition 2 states that the right hand side of (62) with p = q is boundedby ǫ . Since p > q and the embedding ℓ q ⊂ ℓ p is contractive, the first claim in (1)follows. Since | Im ζ n | ≤ | ζ n | and | log | Im ζ n /ζ || ≥ ζ n ∈ Σ , Condition 2 alsoimplies | Im ζ n | − d q ≤ ǫ , | ζ n | d ≤ ǫ | Im ζ n | q − d . (64)Since q > d the first bound implies (2) and thus the second claim in (1) followsfrom (63). The claim (3) follows from the second bound in (64) and (2). Using (3),we find ω q ( ζ n ) = | ζ n | − q | Im ζ n | qdq − & | Im ζ n | ( q − d ) dq + qdq − . (65)This yields (4) since M p,q ( ζ n ) ≥ ω q ( ζ n ) p . It also follows from (2) that | ζ n |h ζ n i & | Im ζ n | . (66)Combining (3), (65) and (66) with the trivial lower bound | ζ n | ≥ | Im ζ n | in (34)yields (5). (cid:3) Remark . From the first equality in (65) and the definition of M p,q ( z ) in (34) itis easy to see that for | ζ n | ≍
1, we have better bounds | Im ζ n | p ( qdq − . M p,q ( ζ n ) . | Im ζ n | p ( qdq − − . Lemma 27.
Suppose that L k & k α . Then, under Assumption 3, s ( L, (cid:18) | ζ n |h ζ n i (cid:19) ζ n ) i . | Im ζ n | − α ( + ( q − d ) d ) ,M p,q ( L, ζ n ) . | Im ζ n | − − q − d ) d − p ( qdd − − p ( ( q − d ) dq + qdq − − pα ( + ( q − d ) d ) . Proof.
Combining (66) with the estimateIm p ζ n ≍ | Im ζ n || ζ n | / & | Im ζ n | ( q − d ) d , (67)where the first bound holds since ζ n ∈ Σ and the second bound follows fromLemma 26 (3), we obtain s ( L, (cid:18) | ζ n |h ζ n i (cid:19) ζ n ) i . sep( L, | Im ζ n | + ( q − d ) d ) . The claim thus follows from Proposition 35 and Example a) following it. (cid:3)
CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 27
Remark . For | ζ n | ≍
1, we again have better bounds s ( L, (cid:18) | ζ n |h ζ n i (cid:19) ζ n ) i . | Im ζ n | − α ,M p,q ( L, ζ n ) . | Im ζ n | p ( qdq − − − pα . We will write the second estimate in Lemma 27 as M p,q ( L, ζ n ) . | Im ζ n | − κ α .We also assume now that δ n ≥ exp( −| Im ζ n | − γ ) (68)for some γ >
0. This lower bound is motivated from the corresponding upperbound that results from the Ghershgorin estimate (42) and a posteriori by (70).
Lemma 28.
Fix a compact set Σ ⊂ Σ ∩ C \ [0 , ∞ ) . The there exists c = c (Σ) such that σ ( H V ) ∩ Σ ⊂ { z : δ H Vn ( z ) ≤ exp( − cL n ) } . The following lemma is obvious.
Lemma 29. If ǫ n is defined by (61) , L k & k α and δ n satisfies (68) , then underAssumption 3, log ǫ − n . | Im ζ n | − κ α − γ . Lemma (26) (1) and Lemma 29 imply that the right hand side of (51) (with e q = ∞ ) is bounded by | Im ζ n | − κ α − γ log h n i . We will show that h n i ≤ | Im ζ n | − d − q +1 , (69)for all but finitely many n ∈ N , which will then give a sufficient condition for thechoice of L in Proposition 25, namely L n ≥ C | Im ζ n | − κ α − γ − − ( q − d ) d log( | Im ζ n | − ) . Here we have used (67) to estimate Im √ ζ n from below. In order to be consistentwith our assumption L k & k α we actually choose L n = C | Im ζ n | − e κ , (70)where, in view of (69), it suffices to take e κ := max( κ α + γ + 2 + ( q − d ) d , α ( d q − . (71)The exact choice of α is not important for us and we choose α = 1 for convenience. Lemma 30.
Under Assumption 3 we have (69) for all but finitely many n ∈ N .Proof. Suppose the claim is false. Then there exists a subsequence, again denotedby ( ζ n ) n , such that h n i > | Im ζ n | − d − q +1 . Since | ζ n | ≥ | Im ζ n | , (2) implies X n h n i − < X n | Im ζ n | d + q − . , a contradiction. (cid:3) Proof of Theorem 1.
We now specialize Proposition 25 to the step potential V j = W ( ζ j , x j ) and the explicit choice (70), which will prove Theorem 1. Since wealready know that the exponential decay bound is true for these potentials (see(58)) we do not need to check the conditions Im p ζ j ≤ R − j log R j , | ζ j | / ≥ KR − j , but it is easy to see from (55) that they do hold. Proposition 31.
Suppose Assumption 3 holds, δ n > satisfies (68) for some γ > , and let V = V ( L ) be the potential whose bumps V n = W ( ζ n , x n ) areseparated by L n in (70) . Then D ( ζ n , δ n ) contains an eigenvalues of H V . Moreover, k V k L q ≤ ǫ and V decays polynomially, | V ( x ) | . h x i − e κ , where e κ is given by (71) for some arbitrary α > .Proof. By (56), we have the bound | V ( x ) | . | Im ζ n | for | x − x n | ≤ R ( ζ n ) and zeroelsewhere. Since | Im ζ n | . V is bounded. Since e κ ≥
2, a comparison between L n and | Ω n | = R ( ζ n ) in (55) shows that V is sparse.Therefore, by (16), we have L n & | x n | . Hence, (70) yields | V ( x n ) | . | Im ζ n | . L − e κ n . x − e κ n , from which the decay bound follows. (cid:3) Complex step potential
In this section we will establish precise estimates for eigenvalues of the spericallysymmetric complex step potential V = V B (0 ,R ) , where V ∈ C and R >
0. Thebound state problem for V < d = 1 , χ = √ E, κ = p χ − V . (72)Here E ∈ C is the eigenvalue parameter, i.e. we consider the stationary Schr¨odingerequation − ∆ ψ + ( V − E ) ψ = 0 , (73)which becomes − ∆ ψ − κ ψ = 0 inside the step and − ∆ ψ − χ ψ = 0 outside thestep.7.1. One dimension.
We start with one-dimensional case. The solution space to(73) then splits into even and odd functions, while in higher dimensions it splitsinto functions with definite angular momentum ℓ . We consider odd functions asthese also provide a solution for the case d = 3 and ℓ = 0 ( s -waves). The standardprocedure to solving the square well problem reduces the task to finding zeros ofthe nonlinear scalar function F ( V , κ ) := i χ − κ cot( κR ), where χ = √ κ + V by(72). A complete study of all the complex poles of this equation was initiated byNussenzveig [56] for V ∈ R \{ } . Subsequent articles in the physics literature [39],[18], [17], [35] investigated the case of complex potentials. The solution κ = κ ( V )is not single-valued as there are branch points where ∂F/∂χ = 0. The viewpointendorsed by [35] is to regard the equation F ( V , κ ) = 0 as the definition of a CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 29
Riemann surface. This approach treats the complex variables κ and V on equalfooting. In fact, it is easy to see that one can always use κ as a coordinate, i.e.one can solve for V , V = − κ sec ( κR ) . (74)For the purpose of the construction of the sparse potential in Subsection 6.5 wedo not need to solve for κ . Instead, we pick κ first and then define V by (74). Toget an eigenvalue (i.e. a resonance on the physical sheet) we simply need to takecare of the condition Im χ >
0, i.e.Re ( κR cot( κR )) > . (75)We are only interested in complex eigenvalues E with | E | ≍ V in (74) small, i.e. we postulatethat V = ǫ e V , where ǫ > e V ∈ C is of unit size. By(72) this implies that | κ | ≍
1, and (74) then reveals that | sin ( κR ) | ≍ ǫ − , whichmeans that e κR ≍ ǫ − , e − κR ≍ ǫ. (76)Since we are free to choose κ , set κ = ± ǫσ with σ = 0, which is will yield aneigenvalue with Re E = 1 + O ( ǫ ) and | Im E | = O ( ǫ ). Going back to (76) we seethat we must have R = 12 | σ | ǫ log 1 ǫ (1 + o (1)) . (77)It is quickly checked that this is consistent with the bound of Abramov et al. [1]since k V k L & log ǫ and | E | ≍
1. In view of (76) we may write e κR = ǫu , where u = C e κR . Using the Taylor approximationsec ( κR ) = − ǫu (1 + 2 ǫu + O ( ǫ )) (78)we obtain from (74) that V = 4 ǫu (Re κ ) + O ( ǫ ) , (79)which provides the desired smallness | V | = O ( ǫ ). As already mentioned, we needto make sure that (75) hold. Using the Taylor approximationcot( κR ) = i(1 + 2 ǫu + O ( ǫ )) , we find that (75) holds if − (Re κ )(Im u ) − (Im κ )(Re u ) > u ∈ i R + , we find that (80) forces us to choose Re κ = −
1. Itis easy to check that in this way we get an eigenvalue with Re E = 1 + O ( ǫ ) andIm E = ± σǫ + O ( ǫ ) as desired. By simple scaling arguments this proves theone-dimensional case of Lemma 22. Before we conclude the one-dimensional wenote that the same result could have been obtained with an even wavefunction,in which case sec replaced by csc in (74) and cot is replaced by tan in (75). TheTaylor approximationscsc ( κR ) = 4 ǫu (1 + O ( ǫ )) , tan( κR ) = i(1 − ǫu + O ( ǫ )) , and the freedom to choose the signs and the imaginary part of u yields a proof ofLemma 22 using odd solutions.7.2. Higher dimensions.
By symmetry reductions we are led to consider theradial Schr¨odinger equation( − ∂ r − d − r ∂ r + ℓ ( ℓ + d − r + V ( r ) − E ) ψ ℓ ( r ) = 0 . It can be shown (see e.g. [10, (5.12)]) that an eigenvalue E corresponds to a zeroof the function (Wronskian) F ( V , κ ) := κJ ′ ν ( κR ) H (1) ν ( χR ) − χJ ν ( κR ) H (1) ′ ν ( χR ) , where ν = ℓ + d − . We recall that χ, κ, E, V are related by (72). Computationsof resonances for spherically symmetric potentials can be found in [55], [82], [74],[10]. The last three papers use uniform asymptotic expansion of Bessel functionsfor large order. Here we only consider s -waves, i.e. ℓ = 0. Then we have theasymptotics J ν ( z ) = (cid:0) πz (cid:1) / cos( z − πν − π O ( | z | − )) ,H (1) ν ( z ) = (cid:0) πz (cid:1) / exp(i z − i πν − i π O ( | z | − )) ,J ′ ν ( z ) = − (cid:0) πz (cid:1) / sin( z − πν − π O ( | z | − )) ,H (1) ′ ν ( z ) = i (cid:0) πz (cid:1) / exp(i z − i πν − i π O ( | z | − ) . With the same choice of κ as in the one-dimensional case and with u ∈ C suchthat e κR − πν − π ) = ǫu , we then obtain that the zeros of F ( V , κ ) coincide withthe zeros of a function κ sin( ω ( κ )) − i χ cos( ω ( κ )) + O ( R − ) , where ω ( κ ) = κR − πν − π and the κ -derivative of the error term is O (1). Recallthat χ = χ ( V , κ ) is given by (72). The zeros of the function without the errorterm are found exactly as in the one-dimensional case and can be parametrized by κ , v.i.z. V = − κ csc ( ω ( κ )) . (81)This follows by dividing the above expression by cos( ω ( κ )) which has no zerossince Im κ >
0. Since | cos( ω ( κ )) − | = O ( ǫ / ) we get from (77) that the errorafter dividing is O ( ǫ / ), i.e. we are looking for the zeros of a function e F ( V , κ ) = V + κ csc ( ω ( κ )) + O ( ǫ / ) , where the derivative of the error is O ( ǫ / ). The implicit function theorem thusyields ∂ e F ( V , κ ) /∂V = 1 + O ( ǫ / ), which means that we can solve e F ( V , κ ) for V , and the solution satisfies (81) up to errors O ( ǫ / ). Hence we obtain that | V | = O ( ǫ ) as before. CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 31
Proof of Theorem 4.
We return to one dimension. We first prove the upperbound (11). Since p | V | R is of order one, the bound in [29] yields that the totalnumber of eigenvalues of H V j is also of order one. By Proposition 17 (it is clearthat the assumption on the norm of V can be dropped), given N ≫
1, we canfind L = L ( N ) such that H s has the same number of eigenvalues in Σ = Σ( N ) as H diag , which is just the N -fold orthogonal sum of the H V j and hence has less than O ( N ) eigenvalues by the first part of the argument.To prove the lower bound (10) we return to the formulas (74), (75), but wenow fix V = i. We also set R = N R and R ≍
1, so that the dimensionlessparameter p | V | R is of size N . We first solve an approximate equation and thenuse Rouch´e’s theorem to show that the exact equation (74) has solutions close tothe approximate ones. Finally, we use (75) to check that we have found a pole onthe physical plane (i.e. an eigenvalue). The approximate equation is G ( κ ) = 0,where G ( κ ) := V − κ e κR , and the approximation will be valid in the regime Im κR ≫
1. Since G can befactored, G ( κ ) = ( p V − κ e i κR )( p V + 2 κ e i κR ) , we only look for zeros of the first factor. These zeros κ n are expressed by meansof the Lambert W function, κ n R = − i W n (i p V R/ , where n ∈ Z and W n are the branches of the Lambert W function. According toin [11, (4.19)] the asymptotic expansion of W n ( z ) as | z | → ∞ is W n ( z ) = log( z ) + 2 π i n − log(log( z ) + 2 π i n ) + O ( log(log( z ) + 2 π i n )log( z ) + 2 π i n ) , where log is the principal branch of the logarithm on the slit plane with the negativereal axis as branch cut. For z = i √ V R/ κ n R = 2 πn − i log(i p V R/
2) + i log(log(i p V R/
2) + 2 π i n ) + E n ( V , R ) , where, for N ≫
1, the error satisfies | E n ( V , R ) | . log(log N + | n | ) / (log N + | n | ),where we recalled that p | V | R ≍ N . For the assumption Im κR ≫ κ n we requireRe log (cid:18) log(i √ V R/
2) + 2 π i n i √ V R/ (cid:19) ≫ ⇐⇒ (cid:12)(cid:12)(cid:12)(cid:12) log(i √ V R/
2) + 2 π i n i √ V R/ (cid:12)(cid:12)(cid:12)(cid:12) ≫ . Since N ≫ | n | ≫ N , which we will assume henceforth. This gives us the errorbound | E n ( V , R ) | . log | n | / | n | , which implies thatIm κ n R & log | n | N , in agreement with the assumption Im κR ≫
1. We also obtain the more preciseformulas Re κ n R = 2 πn + O (log | n | / | n | ) , Im κ n R = log | n | N + O (1) , (82)where we Taylor expanded the logarithm and used | n | ≫ N ≫
1. Having foundthe large zeros of G ( κ ) we proceed to find those of G ( κ ) := V + κ sec ( κR ) , which determines the eigenvalues of the step potential (see the beginning of Sub-section 7.1). We define ǫ n := exp( − κ n R ), so that e κ n R = ǫ n u n for some u n on the unit circle. Note that, by (82), ǫ n = O (1) (cid:0) N | n | (cid:1) . Using (78) with ǫ = ǫ n , u = u n , we estimate, for e ǫ n ≪ | κ − κ n | = e ǫ n | G ( κ ) − G ( κ ) | = O ( n ǫ n ) . (83)Moreover, for | κ − κ n | = e ǫ n we have | G ′ ( κ ) | & | κ | R e − κR & N n ǫ n , | G ′′ ( κ ) | . | κ | R e − κR . N n ǫ n . Using G ( κ n ) = 0 and Taylor expanding, it follows that | G ( κ ) | & N n ǫ n e ǫ n + O ( N n ǫ n e ǫ n ) . For this to be meaningful we must of course assume e ǫ n ≪ /N , which we do. Thenwe have | G ( κ ) | & N n ǫ n e ǫ n . Comparing this with (83) we see thatsup | κ − κ n | = e ǫ n | G ( κ ) | − | G ( κ ) − G ( κ ) | < , provided e ǫ n ≫ ǫ n /N . Adopting the choice e ǫ n = C Nn , where C is a large constant,we see that there exists a zero e κ n ∈ D ( κ n , C Nn ) of G . By the smallness of N/n ,it follows that e κ n also satisfies (82). We drop the tilde, i.e. we now denote thezeros of G by κ n . Summarizing what we have done so far, we have found infinitelymany resonances κ n , | n | ≫ N , of of the step potential satisfying (82). The laststep is to check which of the resonances lie on the physical sheet, i.e. are actualeigenvalues. For this we need to check condition (75). Writing κ n R = α n + i β n ,we find that Re ( κ n R cot( κ n R )) = α n sin(2 α n ) + β n sinh(2 β n )cosh(2 β n ) − cos(2 α n ) (84)Using (82) we see that the denominator is cosh(2 β n )(1 + O ( N /n ). Hence (75)is satisfied if the nominator is positive. The latter has the expansion2 πn (1 + O (1 / log | n | )) + O (1) n N log | n | N , where the O (1) term is positive (since it is the exponential of a real number).We now recall from the discussion at the end of Subsection 7.1 that (79) and(80), together with (82) and the assumption V = i made at the beginning of thissubsection, imply that n must be negative. Hence the condition 7.3 is nonvoidand is satisfied whenever | n | N log | n | N = O (1), which holds for | n | = O ( N / log N ). CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 33
Thus the expression (84) is positive for κ n satisfying (82) with N ≪ | n | ≪ N log N and n <
0. Recalling (72) we obtain the complex energies E = E n ,Re E n ≍ n N , Im E n ≍ | n | N log | n | N , and those energies with c N log N ≤ | n | ≤ C N log N lie in the rectangle Σ (see Theo-rem 4). This completes the proof of the lower bound (10).8. Technical tools
Lower bounds on moduli of holomorphic functions.
We collect somewell known results about the modulus of holomorphic functions away from zeros,based on Cartan’s bound for polynomials (see e.g. [50]).Let U ⋐ U ⋐ C , with U simply connected. Assume that f is holomorphicin a neighborhood of U and ζ ∈ U . Let z , z . . . , z n , be the zeros of f in U .Define Z f,δ,U := n [ j =1 D ( z j , δ ) . The following version can be found in [24, Appendix D].
Lemma 32.
There exists a constant C = C ( U , U , z ) such that for any suffi-ciently small δ > , log | f ( z ) | ≥ − C log 1 δ (cid:0) max z ∈ U log | f ( z ) | − log | f ( ζ ) | (cid:1) for all z ∈ U \ Z f,δ,U . We need also use a more precise version, where f is holomorphic in a neighbor-hood of U , U j = D (0 , r j ) , j = 1 , , , (85)with r < r < r and z = 0. The proof is a straightforward adaptation of [50,Chapter 1, Theorem 11], but we include it for completeness. Lemma 33.
Assume (85) . Then there exists an absolute constant C such that forany sufficiently small δ > , log | f ( z ) | ≥ − C log 1 δ (cid:0) ( r − r ) − + log − (cid:0) r r (cid:1)(cid:1) max | z | = r log | f ( z ) | for all z ∈ U \ Z f,δr ,U .Proof. Consider the function ϕ ( z ) := ( − r ) n z z . . . z n n Y k =1 r ( z − z k ) r − z k z . We recall that z , z , . . . , z n are the zeros of f in D (0 , r ). Observe that ϕ (0) = 1and ϕ ( r e i θ ) = r n | z z . . . z n | for θ ∈ R . The function Ψ( z ) := f ( z ) ϕ ( z )has no zeros in D (0 , r ), and therefore by Carath´eodory’s theorem [50, Theorem9], for | z | ≤ r ,log | ψ ( z ) | ≥ − r r − r (cid:0) log max | z | = r log | f ( z ) | − log 1 | z z . . . z n | (cid:1) ≥ − r r − r log max | z | = r log | f ( z ) | . To estimate ϕ from below for | z | ≤ r we use n Y k =1 | r − z k z | < (2 r ) n , n Y k =1 | r ( z − z k ) | > (cid:18) δr e (cid:19) n r n , z / ∈ Z f, δr ,U . The second inequality follows from Cartan’s estimate [50, Theorem 10]. We thusobtain the lower bound | ϕ ( z ) | > (2 r ) − n (cid:18) δr e (cid:19) n r n | z z . . . z n | > (cid:18) δ (cid:19) n , z / ∈ Z f, δr ,U . By Jensen’s formula [50, Lemma 4], since f (0) = 1, n ≤ log − (cid:0) r r (cid:1) max | z | = r log | f ( z ) | , and consequentlylog | ϕ ( z ) | > log − (cid:0) r r (cid:1) max | z | = r log | f ( z ) | log (cid:18) δ (cid:19) , z / ∈ Z f, δr ,U . Together with the lower bound for log | ψ | this leads to the claimed estimate, uponredefining δ and absorbing an error into the constant C . (cid:3) Next we state a version of Lemma 32 for “wedges” of the form W ( ϕ, θ ; r, R ) := { z ∈ C \ [0 , ∞ ) : arg( z ) ∈ (2 ϕ, θ ) , | z | ∈ ( r , R ) } and W ( ϕ, θ ; R ) := W ( ϕ, θ ; R, ∞ ), where 0 ≤ ϕ < θ ≤ π . In the following we fix0 ≤ ϕ < ϕ < ϕ < θ < θ < θ ≤ π and 0 < r < r < r < R < R , anddefine U := W ( ϕ , θ ; r , R ) , U := W ( ϕ , θ ; r , R ) , U := W ( ϕ , θ ; r ) . Lemma 34.
Assume f is a bounded holomorphic function on U . Assume that r ≪ r , d( ∂U , ∂U )d( ∂U , ∂U ) ≪ (cid:18) r R (cid:19) πθ − ϕ +2 , d( ∂U , ∂U )( θ − ϕ ) R (cid:18) r R (cid:19) πθ − ϕ ≪ . (86) CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 35
Then there exists an absolute constant C such that for any ζ ∈ U and anysufficiently small δ > , log | f ( z ) | ≥ − C R d( ∂U , ∂U ) (cid:18) R r (cid:19) πθ − ϕ log 1 δ (cid:0) max z ∈ U log | f ( z ) | − log | f ( ζ ) | (cid:1) for all z ∈ U \ Z f,δ,U .Remark . The constant C only depends on the implicit constants in (86). Hence,one can optimize the inequality with respect to ζ , subject to the conditions above. Proof.
We map U conformally onto the unit disk, using a composition of thefollowing conformal maps (where, by abuse of notation, we denote the variableand the map by the same letter):i) U → κ ( U ) ⊂ H , κ ( z ) := √ z ;ii) κ ( U ) → S := { σ ∈ C : 0 < Im σ < π, Re σ > } , σ ( κ ) := log(e − i πϕ θ − ϕ ( κ/r ) πθ − ϕ ) , where we select the principal branch of the logarithm on C \ [0 , − i ∞ );iii) The Schwarz-Christoffel transformation S → H , τ ( σ ) := cosh( σ ).iv) H → D (0 , w ( τ ) := τ − τ τ + τ , where τ := τ ( σ ( √ ζ )).The choice of τ has been made in such a way that w ( z ) = 0 if z = ζ . Here weagain abuse notation and write w ( z ) = w ( τ ( σ ( √ z ))). Note that τ ( σ ( κ )) = 12 ( α ( κ ) + α ( κ ) − ) , α ( κ ) := e − i πϕ θ − ϕ ( κ/r ) πθ − ϕ . By distortion bounds [60, Cor. 1.4], (cid:12)(cid:12)(cid:12)(cid:12) d w ( z )d z (cid:12)(cid:12)(cid:12)(cid:12) d( z, U ) ≤ − | w ( z ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) d w ( z )d z (cid:12)(cid:12)(cid:12)(cid:12) d( z, U ) . (87)We compute the differential of w at z ∈ U by the chain rule, (cid:12)(cid:12)(cid:12)(cid:12) d w d z (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) d w d τ d τ d σ d σ d κ d κ d z (cid:12)(cid:12)(cid:12)(cid:12) = π | τ || sinh( σ ) | ( θ − ϕ ) | z | ( τ + τ ) = π ( θ − ϕ ) | z | | α ( κ ) + α ( κ ) − || α ( κ ) + α ( κ ) − || α ( κ ) + α ( κ ) − + α ( κ ) + α ( κ ) − | . Since | κ | ≥ r ≫ r , we have | α ( κ ) | ≫
1, whence | α ( κ ) + α ( κ ) − || α ( κ ) + α ( κ ) − || α ( κ ) + α ( κ ) − + α ( κ ) + α ( κ ) − | ≍ | α ( κ ) || α ( κ ) | ( | α ( κ ) | + | α ( κ ) | ) , which, in view of | α ( κ ) | = ( | κ | /r ) πθ − ϕ , leads to1( θ − ϕ ) R (cid:18) r R (cid:19) πθ − ϕ . (cid:12)(cid:12)(cid:12)(cid:12) d w d z (cid:12)(cid:12)(cid:12)(cid:12) . θ − ϕ ) r
226 J.-C. CUENIN for z ∈ U . Denoting the numbers r j in Lemma 33 by ρ j instead (with ρ = 1),we then find, using (87),1 − ρ & d( ∂U , ∂U )( θ − ϕ ) R (cid:18) r R (cid:19) πθ − ϕ ,ρ − ρ & d( ∂U , ∂U )( θ − ϕ ) R (cid:18) r R (cid:19) πθ − ϕ − d( ∂U , ∂U )( θ − ϕ ) r & d( ∂U , ∂U )( θ − ϕ ) R (cid:18) r R (cid:19) πθ − ϕ , where in the second line we used the triangle inequality and the second inequalityin (86). By the third inequality in (86) we can Taylor expandlog( 1 ρ ) = − log(1 − (1 − ρ )) ≍ − ρ . Lemma 33 applied to the function z f ( z ) f ( ζ ) now yields the claim. (cid:3) Distribution function.
For s >
0, we define h L ( s ) = |{ k ∈ [ N ] : η L k ≤ /s }| ∈ Z + , where η − is an arbitrary length scale. Note that h L is decreasing and tends toinfinity as s →
0. In fact, h L is the distribution function of the sequence ( η L k ) − ).Since we assume that L k is increasing, we also have h L ( s ) = min { k ∈ Z + : η L k +1 > /s } . We will show that, under the assumption ∃ λ ∈ (0 ,
1) such that lim sup s → h L ( λs )e h L ( s ) < , (88)the potential V ( L ) is strongly separating in the sense of Definition 2. Proposition 35.
Assume (88) . Then sep(
L, η ) . exp( − ηL ) h h L ( η/η ) i . (89)In particular, this implies that the examples in Subsection 2.1 are stronglyseparating:a) If η L k & k α for α >
0, then h L ( s ) . s − /α .b) If η L k & exp( k ), then h L ( s ) . log(1 /s ).c) If η L k & exp(exp( k )), then h L ( s ) . log log(1 /s ). Lemma 36.
Assume (88) . Then for any δ > and for all s > , h h L ( δs ) i . δ h h L ( s ) i . Proof.
We may restrict our attention to the case δ < δ ≥ λ ∈ (0 ,
1) and s > h L ( λs ) < e h L ( s ) (90)holds for all s ∈ (0 , s ]. Now let n be the smallest integer such that λ n ≤ δ .Iterating (90) n times, we get h L ( δs ) < e n h L ( s )for all s ∈ (0 , s ]. For s > s , the inequality holds trivially. (cid:3) CHR ¨ODINGER OPERATORS WITH COMPLEX SPARSE POTENTIALS 37
Proof of Proposition 35.
Without loss of generality we may assume that η = η .We first consider the case ηL ≤
1, and hence h L (1) ≥
1. Then ∞ X k =1 exp( − ηL k ) ≤ h L (1) exp( − ηL ) + ∞ X k = h L (1) exp( − ηL k ) . It remains to show that the second term is bounded by the right hand side of (89).To this end, we decompose the sum into dyadic intervals I j = [ h L (2 − j ) , h L (2 − j − )], j ∈ Z + . Then X k ∈ I j exp( − ηL k ) ≤ exp( − j ) h L (2 − j − ) . Summing over j and using Cauchy’s condensation test yields ∞ X k = h L (1) exp( − ηL k ) . ∞ X n =1 exp( − n ) h L ( n ) . (91)By the quotient test, the series converges provided thatlim sup n →∞ h L ( λ n n )e h L ( n ) < , where λ n = nn +1 . This follows from assumption (88). Indeed, since lim n →∞ λ n =1, we have λ < λ n for large n and hence h L ( λ n n ) ≤ h L ( λ n ). The series (91)is thus bounded by h h L (1) i , where we have used Lemma 36 with δ = 1 /
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