Localization of Quantum States and Landscape Functions
LLOCALIZATION OF QUANTUM STATESAND LANDSCAPE FUNCTIONS
STEFAN STEINERBERGER
Abstract.
Eigenfunctions in inhomogeneous media can have strong localization properties.Filoche & Mayboroda showed that the function u solving ( − ∆ + V ) u = 1 controls the behaviorof eigenfunctions ( − ∆ + V ) φ = λφ via the inequality | φ ( x ) | ≤ λu ( x ) (cid:107) φ (cid:107) L ∞ . This inequality has proven to be remarkably effective in predicting localization and recentlyArnold, David, Jerison, Mayboroda & Filoche connected 1 /u to decay properties of eigenfunc-tions. We aim to clarify properties of the landscape: the main ingredient is a localized variationestimate obtained from writing φ ( x ) as an average over Brownian motion ω ( · ) in started in xφ ( x ) = E x (cid:16) φ ( ω ( t )) e λt − (cid:82) t V ( ω ( z )) dz (cid:17) . This variation estimate will guarantee that φ has to change at least by a factor of 2 in a smallball, which implicitly creates a landscape whose relationship with 1 /u we discuss. Introduction
The Landscape function.
It is well known that physical systems comprised of inhomoge-neous materials can exhibit peculiar vibration properties: let Ω ∈ R n be open, bounded and( − ∆ + V ) φ = λφ in Ω with Dirichlet boundary conditions,where V : Ω → R ≥ is a real-valued, nonnegative potential. Anderson [2] noticed that for somepotentials the low-lying eigenfunctions tend to strongly localize in a subregion of space in a verycomplicated manner. It seemed difficult to get any information about the localization behavior ofthese first few eigenfunctions without explicitely computing them.In a truly remarkable contribution, Filoche & Mayboroda [11] have given a simple but astonishinglyeffective method to predict the behavior of low-energy eigenfunctions. Their approach is based onthe following inequality (originally due to Moler & Payne [14]): if we associate to the problem a landscape function u : Ω : R → R given as the solution of( − ∆ + V ) u = 1 in Ω ⊂ R n with Dirichlet boundary conditions,then there is the inequality | φ ( x ) | ≤ λu ( x ) (cid:107) φ (cid:107) L ∞ (Ω) . The regions where u is small will be of particular interest because an eigenfunction φ can onlylocalize in { x : λu ( x ) ≥ } ⊂ Ω. The landscape function turns out to be more effective than that:it is instructive to regard the graph of u ( x ) as a landscape comprised of ’peaks’ and ’valleys’; thevalleys may then be understood as inducing a partition of the domain. Numerical experiments[11] suggest that low-lying eigenfunctions respect that partition and favor localization in one orat most a few elements in that partition. Moreover, these localized eigenfunctions are ’almost’compactly supported in the sense that in crossing from one element of the partition to anothereigenfunctions seem to experience exponential decay when crossing the valley (see [11]).1.2. The effective potential.
Concerning this exponential drop in size of an eigenfunction whencrossing valley, this was recently made precise by Arnold, David, Jerison, Mayboroda & Filoche[3] who point out that the inverse of the landscape function 1 /u ( x ) acts as an effective potentialresponsible for the exponential decay of the localized states (the connection being that u ( x ) is small a r X i v : . [ m a t h . SP ] O c t in valleys, which makes 1 /u ( x ) large and large potentials induce large decay). Their approach isbased on writing an eigenfunction as φ = uψ for some unknown function ψ . The equation( − ∆ + V ) φ = λφ then transforms into (cid:20) u div( u ∇ ψ ) (cid:21) + 1 u ψ = λψ. The new dominating potential W ≡ /u is now responsible for the underlying dynamics. Thenext step is to build an Agmon distance ρ ( r , r ) = min γ (cid:18)(cid:90) γ (cid:112) ( W ( r ) − λ ) + ds (cid:19) , where γ ranges over all paths from r to r and use Agmon’s inequality [1] to deduce that foreigenfunctions φ localized in r ∈ Ω | φ ( r ) | (cid:46) e − ρ ( r ,r ) . This indicates that W ≡ /u is playing a distinguished role. The paper [3] also gives convincingnumerical evidence that W − λ seems to predict decay more accurately than the classical quantity V − λ . This might seem surprising because clearly V determines the behavior of the eigenfunctions.1.3. Organization.
The purpose of our paper is to further clarify these observations and theinterplay between an eigenfunction doubling its size in a small ball and the landscape function;the main tool is an identity following from the Feynman-Kac formula. More precisely, we • derive and discuss the relevant identity, • use it to prove a variation estimate localized in a small ball, • show how w.r.t. decay V is not as important as a suitable mollification of V , • compare how 1 /u ( x ) fits into that framework • and discuss some refinements of the landscape function u ( x ).We always assume that Ω ⊂ R n is bounded with a smooth boundary and V ∈ C (Ω) to becontinuous. Technically, this excludes ’block potentials’ (which are only L ∞ ) but it is clear thatthe first few eigenfunctions hardly change if a potential is replaced by a suitably mollification andtherefore the assumption is without loss of generality.2. Local analysis of the heat flow
The torision function.
The landscape function arising from V = 0, i.e. the solution of − ∆ v = 1 in Ω with Dirichlet boundary conditions,is a classical object in shape optimization called the torsion function . It appears in elasticitytheory [5], heat conduction [17] and geometry [13]. A version of a landscape function with potentialalready appeared in the context of homogenization in work of Coifman & Meyer (unpublished,but see the application to parabolic operators by S. Wu [18]). The most prominent role of thetorsion function in the field of shape optimization (see e.g. [6]) is that v ( x ) gives the expectedlifetime of Brownian motion started in x until it hits the boundary. This suggests to interpret thelandscape function in that language; the idea of using Brownian motion to analyze decay propertiesof eigenfunctions is classical and was very successfully used in seminal papers by Carmona [7],Carmona & Simon [8], Carmona, Masters & Simon [9], Simon [15] and others; recently, a similartechnique was used by the author [16] to obtain bounds on the size of nodal sets of Laplacianeigenfunctions { x : φ ( x ) = 0 } on compact manifolds. The idea.
The crucial ingredient is a simple equation representing an eigenfunction φ ( x ) asa localized average over local Brownian motion paths running for a short time. This equation isnot new and has been used earlier for very similar purposes, see e.g. Carmona & Simon [8]. Thisequation is obtained by looking at the effect of the semigroup e t (∆ − V ) on the eigenfunction. Sinceeigenfunctions diagonalize the semigroup, we have that if( − ∆ + V ) φ = λφ then e t (∆ − V ) φ = e − λt φ. At the same time, there is another interpretation of the action of the semigroup in terms ofBrownian motion. The
Feynman-Kac formula states that for an arbitrary function fe t (∆ − V ) f ( x ) = E x (cid:16) f ( ω ( t )) e − (cid:82) t V ( ω ( z )) dz (cid:17) , where the expectation E x is taken with respect to Brownian motion ω ( · ) started in x , running fortime t and destroyed upon impact on the boundary. Combining these two equations, we get ∀ t ≥ φ ( x ) = E x (cid:16) φ ( ω ( t )) e λt − (cid:82) t V ( ω ( z )) dz (cid:17) . This equation describes a complicated relationship between φ ( x ), λ and V , however, it is perfectlysuited for establishing a variation estimate in a small ball: assuming the eigenfunction φ to beessentially constant on a small scale allows us to move the eigenfunction φ out of the expectationat the cost of a very small error. We sketch a non-rigorous version of the argument. x B Figure 1.
Brownian motion started in x .Assume w.l.o.g. that φ ( x ) >
0. Let B = B ( x , r ) be the ball centered at x with maximal radius r > ∀ x ∈ B : 12 | u ( x ) | ≤ | u ( x ) | ≤ | u ( x ) | . Assume furthermore that V ( x ) ≥ λ and that V is essentially constant on B . We now consider theequation above for t = c n r with c n a small universal constant depending only on the dimensionof Ω. By making c n sufficiently small, we can ensure that 99% of all Brownian paths spend arefully contained in B up to time t . Furthermore, φ ( x ) = E x (cid:16) φ ( ω ( t )) e λt − (cid:82) t V ( ω ( z )) dz (cid:17) ∼ φ ( x ) E x (cid:16) e λt − (cid:82) t V ( ω ( z )) dz (cid:17) . In order for this to hold, we clearly require E x e λt − (cid:82) t V ( ω ( z )) dz ∼ − (cid:28) λt − (cid:90) t V ( ω ( z )) dz ≤ . However, this last quantity can be approximated by0 ≥ λt − (cid:90) t V ( ω ( z )) dz ∼ t ( λ − V ( x )) ∼ r ( λ − V ( x )) (cid:29) − r depending on λ − V ( x ). We will now make thisheuristic sketch precise and phrase it in classical terms. However, we emphasize that the mostuseful way of thinking about the setup to be in terms of path integrals: the condition V ( x ) − λ ≥ c could, for example, be replaced by V ( x ) − λ ≥ V ( x ) − λ ≥ c true ’on average’. Theorem 1 (Variation estimate) . Suppose ( − ∆ + V ) φ = λφ for V ≥ . There exists a universalconstant c n (depending only on the dimension) such that the following holds: if, for some c > , V ( x ) − λ ≥ c or V ( x ) − λ ≤ − c uniformly on the ball B = B (cid:32) x , c n √ c (cid:115) λc + log (cid:18) c n (cid:107) φ (cid:107) L ∞ | φ ( x ) | (cid:19)(cid:33) ⊂ Ω , then we have sup x ∈ B | φ ( x ) | inf x ∈ B | φ ( x ) | ≥ . Opposite statements (especially for the case V = 0) are usually called ’doubling estimates’ andguarantee that an eigenfunction can at most double its size in a certain region of space whichthen bounds the order with which it can vanish around a root (see e.g. Bakri [4] or the survey ofZelditch [19]). Let us first discuss the case V ( x ) − λ ≥ c . Then the result has the expected scalingand translates into the classical (cid:112) ( V − λ ) + factor in the Agmon metric. The proof of Theorem 1yields a stronger result: the uniform estimate V − λ ≥ c is not necessary (it is only required ’w.r.t.to path integrals’) and we can rephrase the condition. Local. φ locally varies by a constant factor on scale ∼ / (cid:112) ( V ( x ) − λ ) + . Nonlocal. φ varies locally by a constant factor on the scale ∼ √ t x where t x = inf (cid:26) t > E x e λt − (cid:82) t V ( ω ( z )) dz ≤ (cid:27) . The non-locality of the second formulation incorporates the diffusive action of the partial differ-ential equation. Moreover, the nonlocal formulation allows for t to be large. At the crudest level,the two estimates coincide because E x e λt − (cid:82) t V ( ω ( z )) dz ∼ − ( λ − V ( x )) t. This estimate is only correct up to first order; one might suspect that the estimate should holdup to second order because Brownian motion is isotropic and therefore by symmetry E x (cid:104)∇ V ( x ) , ω ( t ) − x (cid:105) = 0 . However, there is a curious contribution: Brownian motion moves to distance ∼ √ t within time t and this has the curious effect of turning the local geometry of V at second order into a first ordercontribution in time E x (cid:10) ω ( t ) − x, ( D V )( x ) ω ( t ) − x (cid:11) = t ∆ V ( x ) . This gives E x e − (cid:82) t V ( ω ( z )) dz = 1 − V ( x ) t + t (cid:0) V ( x ) − ∆ V ( x ) (cid:1) + o ( t )A geometric interpretation would be as follows: local convexity (∆ V ( x ) >
0) yields to a strongerdecay of the expectation and thus enforces a stronger decay of the eigenfunction; conversely, localconcavity enforces slightly less decay (compared to flat potentials at the same numerical scale). V ( x ) Figure 2.
Two potentials having comparable numerical value. Local convexityenforces stronger decay of the eigenfunction, local concavity flatter decay.
The case V ≤ λ . The description above only discusses the cases, where locally V ≥ λ (inorder to stress the similarity to Agmon’s inequality), however, it is clear that a similar argumentapplies whenever V ≤ λ . In that case the expectatation over the path integrals will grow and thiswill imply variation at scale √ t x where t x = inf (cid:110) t > E x e λt − (cid:82) t V ( ω ( z )) dz ≥ (cid:111) . Put differently, if V ≤ λ in such a way that there is local growth in t of E x e λt − (cid:82) t V ( ω ( z )) dz , thenthe only way for φ ( x ) = E x (cid:16) φ ( ω ( t )) e λt − (cid:82) t V ( ω ( z )) dz (cid:17) to be valid is for φ ( ω ( t )) to be, on average, smaller than φ ( x ) which indicates doubling of sizeon the length scale √ t x . In particular, the nonlocal formulation also applies to the setting wherethe classical (cid:112) ( V − λ ) + yields no more information. Using this formulation, we can recover theclassical intuition for Laplacian eigenfunctions: if − ∆ ψ = κψ , then one expects ψ to oscillate onthe wavelength ∼ κ − / . In our case, if V (cid:28) λ , then − ∆ ψ = ( λ − V ) ψ ∼ λψ and the variation estimate guarantees oscillation on scale √ λ/c , where c ∼ λ − V ∼ λ and thus √ λ/c ∼ λ − / . Another way of seeing this is that we expect a doubling on the scale ∼ √ t x andwhenever V (cid:28) λ , then t x = inf (cid:110) t > E x e λt − (cid:82) t V ( ω ( z )) dz ≥ (cid:111) ∼ inf (cid:8) t > E x e λt ≥ (cid:9) ∼ λ . Landscape from variation.
So far, we have only discussed the variation estimate; therelationship with landscapes is easily visualized: pick a sequence of points x , x , . . . , x k · · · ∈ Ω,compute t x j and draw circles with radius (cid:112) t x j around the points. Smaller circles correspond tovariation by a factor on a smaller spatial scale (i.e. faster growth/decay). The ’valleys’ of thelandscape correspond to regions with smaller circles (crossing them causes a variation by a factor2 and thus frequent crossing is equivalent to either large growth or large decay) whereas ’hills’correspond to regions larger circles. This essentially reproduces the landscape generated by u sinceboth √ t x and u may be regarded as mollifications of 1 /V . Figure 3.
An implicit description of the landscape by (cid:112) t x j balls around points x j .The computation of t x j is clearly nontrivial, however, up to first order it is easy, since E x e − (cid:82) t V ( B ( ω ( z ))) dz ∼ − E x (cid:90) t V ( B ( ω ( z ))) dz and, by linearity, we can exchange expectation and integration E x (cid:90) t V ( B ( ω ( z ))) dz = (cid:90) t E x V ( B ( ω ( z ))) dz ∼ (cid:90) t ( e z ∆ V )( x ) dz = (cid:20)(cid:18)(cid:90) t e z ∆ dz (cid:19) ∗ V (cid:21) ( x ) . This is essentially accurate as along as d ( x, ∂ Ω) (cid:29) √ t while it overestimates the quantity as soonas d ( x, ∂ Ω) (cid:46) √ t . In contrast, the landscape function λu ( x ) has a simpler and more convenientlinear scaling in the eigenvalue λ . Indeed, λu ( x ) arises naturally from the following heuristic, which nicely clarifies how u ( x ) interacts locally with the semigroup induced by ( − ∆ + V ). Heuristic.
We compute an expansion of e t (∆ − V ) u ( x ) for t small in two different ways. Note that( − ∆ + V ) u = 1 and thus, for t small, e t (∆ − V ) u ( x ) = u ( x ) − t + o ( t ) . However, the semigroup may also be expanded using Feynman-Kac and for t small( e t (∆ − V ) u )( x ) = E x (cid:16) u ( ω ( t )) e − (cid:82) t V ( ω ( z )) dz (cid:17) = E x (cid:18) u ( ω ( t )) (cid:18) − (cid:90) t V ( ω ( z )) dz (cid:19)(cid:19) ∼ u ( x ) − u ( x ) E x (cid:90) t V ( ω ( z )) dz. By matching the coefficient in the linear term, we get that u ( x ) E x (cid:90) t V ( ω ( z )) dz ∼ t. Therefore, for t sufficiently small E x (cid:90) t V ( ω ( z )) dz ∼ tu ( x )and thus E x e λt − (cid:82) t V ( ω ( z )) dz ∼ (cid:18) λ − u ( x ) (cid:19) t. This heuristic naturally recovers decay in the region { x : 1 /u ( x ) > λ } . The only inaccuracy is thatwe actually have E x u ( ω ( t )) ∼ u ( x ) + t ∆ u ( x ) . Indeed, unless | ∆ u ( x ) | is very big, we get from( − ∆ + V ) u ( x ) = 1 that u ( x ) ∼ V ( x ) . In that case it is not very surprising that 1 /u ( x ) is effective at predicting decay. However, therelationship runs deeper than that: using again Feynman-Kac we get e t (∆ − V ) E x e − (cid:82) t V ( ω ( z )) dz e t (∆ − V ) − − tu ( x ) + o ( t )Furthermore, information up to second order appears with the correct sign. We can rewrite( − ∆ + V ) u = 1 as u = 1 V + ∆ uV . Assuming ∆ u ∼ u ∼ /V . The next natural step is to iteratethis u ∼ V + ∆ (cid:0) V (cid:1) V = 1 V + 2 |∇ V | V − ∆ VV . Restricting to local extrema, we see that (up to lower order terms) 1 /u is bigger than V in localminima and smaller than V in local maxima (thus recreating the behavior from above).3. New landscape functions
Nonlocal refinement.
From now on, a ’landscape function’ refers to any function h ( x ) withthe property that for a fixed eigenvalue λ the eigenfunction ( − ∆ + V ) φ = λφ satisfies | φ ( x ) | ≤ h ( x ) (cid:107) φ (cid:107) L ∞ .h ( x ) = 1 is trivially admissible. We will continue to use u ( x ) to denote the classical landscapefunction given as the solution of ( − ∆ + V ) u = 1. It satisfies the inequality with h ( x ) = λu ( x ). Theorem 2 (Landscape bootstrapping) . Suppose ( − ∆ + V ) φ = λφ and h ( x ) satisfies | φ ( x ) | ≤ h ( x ) (cid:107) φ (cid:107) L ∞ , then the same inequality holds for h ( x ) replaced by h ( x ) = inf t ≥ E x (cid:16) h (( ω t ) e λt − (cid:82) t V ( ω ( s )) ds (cid:17) = inf t ≥ e λt e t (∆ − V ) h ( x ) . By letting t →
0, we get h ( x ) ≤ h ( x ). There is a delicate balance between two terms: e t (∆ − V ) h ( x )is a diffusion semigroup inducing exponential decay which is counteracted by the exponentialgrowth e λt . It is interesting to note that the function λu ( x ) plays a distinguished role and ischaracterized by the fact that no purely local (i.e. t →
0) considerations have any effects.
Proposition.
Suppose f ∈ C (Ω) satisfies ddt e λt e t (∆ − V ) f ( x ) (cid:12)(cid:12) t =0 = 0 then f ( x ) = λu ( x ) . Proof.
The argument is immediate. Note that for t small e λt e t (∆ − V ) f ( x ) = (1 + λt + O ( t )))(1 + t (∆ − V ) f ( x ) + O ( t ))= 1 + t ( λ + (∆ − V ) f ( x )) + O ( t ))and therefore f ( x ) = ( − ∆ + V ) − λ = λu ( x ) . (cid:3) Computational tricks.
The main results from the previous section allow for the generationof new landscape functions out of λu ( x ), however, one should consider that solving e t (∆ − V ) u ( x )may be as hard or harder than directly computing eigenfunctions. The purpose of this section isto suggest a cheap way of creating computationally feasible improvements. The original proofs[11, 14] demonstrating | φ ( x ) | ≤ λu ( x ) (cid:107) φ (cid:107) L ∞ use Green’s functions. A very simple argument that we could not find in the literature is as follows. Proof. φ is an eigenfunction and − ∆ + V is elliptic operator. The maximum principle yields φ ( x ) = ( − ∆ + V ) − λφ ( x ) ≤ ( − ∆ + V ) − λ (cid:107) φ (cid:107) L ∞ = λ (cid:107) φ (cid:107) L ∞ ( − ∆ + V ) − , where the last term is precisely u ( x ). (cid:3) The very simple proof immediately suggests two improvements. The first improvement would beto iterate the inequality: let k ∈ N be arbitrary. Then we have the inequality φ ( x ) = ( − ∆ + V ) − k λ k φ ( x ) ≤ ( − ∆ + V ) − k λ k (cid:107) φ (cid:107) L ∞ = λ k (cid:107) φ (cid:107) L ∞ ( − ∆ + V ) − k . This variant was already known to Filoche & Mayboroda; its downside is that the increased poweron the eigenvalue tends to make the bounds worse for higher eigenfunctions. There exists anelementary improvement that preserves linear scaling in the eigenvalue.
Proposition. If ( − ∆ + V ) φ = λφ , then | φ ( x ) | ≤ (cid:0) λ ( − ∆ + V ) − min( λu ( x ) , (cid:1) (cid:107) φ (cid:107) L ∞ Proof.
We bootstrap the original inequality and have φ = ( − ∆ + V ) − λφ ≤ λ ( − ∆ + V ) − | φ | ≤ λ ( − ∆ + V ) − min( λu ( x ) , (cid:107) φ (cid:107) L ∞ (cid:3)
500 1000 15000.20.40.60.81.01.2
Figure 4.
The profile φ / (cid:107) φ (cid:107) L ∞ (blue), the landscape function u ( x ) (purple)and the first two iterations (yellow, green) of Proposition 3.The reason why such a simple argument could indeed be effective is that the landscape functionis sometimes bigger than 1; in that case we can perform a simple cut-off at 1 and iterate toget additional information; alternatively, one could use the bound from the previous section ina neighborhood to propagate the gain from the cutoff to nearby regions. Clearly, that simpleargument could also be iterated and, by the same token, | φ ( x ) | ≤ (cid:0) λ ( − ∆ + V ) − min (cid:0) , λ ( − ∆ + V ) − min( λu ( x ) , (cid:1)(cid:1) (cid:107) φ (cid:107) L ∞ . Like the original landscape function, these improvements can only be effective for λ small: for λ large, we will have λu ( x ) ≥ λu ( x ) , ∼ u .4. Proofs
Proof of Theorem 1.
The main idea has already been outlined above. We use φ ( x ) = E x (cid:16) φ ( ω ( t )) e λt − (cid:82) t V ( ω ( z )) dz (cid:17) for a suitable time t . Almost all paths are contained within a ball of radius ∼ √ t (suggesting toset t = 1 /c ). However, some Brownian motion paths will actually leave that ball and may givea large additional contribution but the likelihood of that happening is small. Using the standardreflection principle implies for one-dimensional Brownian motion B ( t ) that P (cid:18) max t > P ( (cid:107) ω ( t ) − ω (0) (cid:107) ≥ δt / ) ≤ c e − c δ . This holds at a much greater level of generality [10]. The proof consists of showing that all thequantities work together as described by doing the algebra.
Proof.
We assume that ∀ x ∈ B : 12 | φ ( x ) | ≤ | φ ( x ) | ≤ | φ ( x ) | with B = B (cid:32) x , c √ c (cid:115) λc + log (cid:18) c (cid:107) φ (cid:107) L ∞ | φ ( x ) | (cid:19)(cid:33) ⊂ Ω , where c is some constant we are allowed to choose depending on c , c . We start with assumingthat V − λ ≥ c uniformly on B. The case V ( x ) − λ ≤ − c is similar and will be described afterwards. We can assume without lossof generality that φ ( x ) >
0. The time scale of the argument will be t = 1 /c . For a Brownianmotion ω ( s ) started in x we distinguish two cases: • Case 1 (generic). { ω ( s ) : 0 ≤ s ≤ t } ⊂ B • Case 2 (rare). { ω ( s ) : 0 ≤ s ≤ t } (cid:54)⊂ B. Case 1 is very easy to deal with: in that case, we can easily bound any single path fully via φ ( ω ( t )) e λt − (cid:82) t V ( ω ( z )) dz ≤ φ ( x ) e (cid:82) t ( λ − V ( ω ( z ))) dz ≤ e − tc φ ( x )and the same argument holds in expectation for all paths conditioned on Case 1. It remains toconsider Case 2. If it occurs, then we only have the trivial bound (using V ≥ φ ( ω ( t )) e λt − (cid:82) t V ( ω ( z )) dz ≤ (cid:107) φ (cid:107) L ∞ e λt , which is large but the likelihood of that event is small. Combining this with the upper bound onthe likelihood of Case 2 gives φ ( x ) = E x (cid:16) φ ( ω ( t )) e λt − (cid:82) t V ( ω ( z )) dz (cid:17) ≤ e − tc φ ( x ) + c e − c δ e λt (cid:107) φ (cid:107) L ∞ . We derive a contradiction by setting t = 1 /c , which gives δ = c (cid:118)(cid:117)(cid:117)(cid:116) log (cid:32) c e λc (cid:107) φ (cid:107) L ∞ | φ ( x ) | (cid:33) . Simple algebra allows us to reformulate the inequality as φ ( x ) ≤ e − φ ( x ) + ε c ,c ,c φ ( x ) , where ε c ,c ,c = c c c c e − λc ( c c − (cid:18) | φ ( x ) |(cid:107) φ (cid:107) L ∞ (cid:19) c c − can be made arbitrarily small by making c sufficiently large (depending only on c , c ). This givesa contradiction since 2 e − < V − λ ≤ − c uniformly on B. Here, essentially all signs are reversed and we show that there is too much local growth. We canagain distinguish Case 1 and Case 2 and get for Case 1 that φ ( ω ( t )) e λt − (cid:82) t V ( ω ( z )) dz ≥ φ ( x ) e (cid:82) t ( λ − V ( ω ( z ))) dz ≥ e tc φ ( x ) . The second case is again completely without control and we can only use the trivial estimate φ ( ω ( t )) e λt − (cid:82) t V ( ω ( z )) dz ≥ − e λt (cid:107) φ (cid:107) L ∞ . Altogether, this yields φ ( x ) = E x (cid:16) φ ( ω ( t )) e λt − (cid:82) t V ( ω ( z )) dz (cid:17) ≥ P (Case 1) 12 e tc φ ( x ) − c e − c δ e λt (cid:107) φ (cid:107) L ∞ . For c sufficiently large, we can ensure that P (Case 1) ≥ /
2. Thus, for c sufficiently large φ ( x ) = E x (cid:16) φ ( ω ( t )) e λt − (cid:82) t V ( ω ( z )) dz (cid:17) ≥ e tc φ ( x ) − c e − c δ e λt (cid:107) φ (cid:107) L ∞ . Plugging things in as before yields (using again t = 1 /c and therefore same value of δ as before) φ ( x ) ≥ e δ φ ( x ) − ε c ,c ,c φ ( x ) , where by the same computation as above e δ / c sufficientlylarge and ε c ,c ,c = c c c c e − λc ( c c − (cid:18) | φ ( x ) |(cid:107) φ (cid:107) L ∞ (cid:19) c c − can be made arbitrarily small by making c sufficiently large. (cid:3) Proof of the Theorem 2.
Proof.
The proof used uses the identity for all t ≥ φ ( x ) = ( e λt e t (∆ − V ) φ )( x ) = E x (cid:16) φ ( B t ( ω )) e λt − (cid:82) t V ( B ( ω ( s ))) ds (cid:17) = inf t ≥ E x (cid:16) φ ( B t ( ω )) e λt − (cid:82) t V ( B ( ω ( s ))) ds (cid:17) . By assumption φ is dominated by a landscape function h ( x ) and thusinf t ≥ E x (cid:16) φ ( B t ( ω )) e λt − (cid:82) t V ( B ( ω ( s ))) ds (cid:17) ≤ (cid:107) φ (cid:107) L ∞ inf t ≥ E x (cid:16) h ( B t ( ω )) e λt − (cid:82) t V ( B ( ω ( s ))) ds (cid:17) = (cid:107) φ (cid:107) L ∞ inf t ≥ e λt e t (∆ − V ) h ( x ) . This concludes the argument. (cid:3)
Refined variation estimate.
Since Brownian motion is isotropic, the proof of the variationestimate can cover a stronger result: simply put, the variation estimate is true because of curvatureof the graph. More generally, we can show that if h ( x ) : Ω → R is another function satisfying h ( x ) = 0. ∀ x ∈ B (cid:12)(cid:12)(cid:12) e λt e t (∆ − V ) h ( x ) (cid:12)(cid:12)(cid:12) ≤ φ ( x )for the value of t for which we wish to apply the variation estimate, then the function φ ( x ) − h ( x ) doubles its sizes on B. The best example is perhaps given by V = 0 (eigenfunctions of the Laplace operator − ∆ φ = λφ )and h ( x ) being the best linear approximation of φ in x h ( x ) = (cid:104)∇ φ ( x ) , x − x (cid:105) . This function is essentially invariant under heat flow for d ( x , ∂ Ω) (cid:29) √ t (with a negligible contri-bution from |∇ φ ( x ) | that can be made precise). The variation estimate implies that φ ( x ) − h ( x )doubles its size and since we have removed the tangent plane this implies that φ ( x ) − h ( x ) can’t betoo small because there is curvature in the graph. This also explains why the variation estimatedoes not apply when V ∼ λ : then the functions have no guaranteed curvature − ∆ φ = ( λ − V ) φ ∼ Acknowledgement.
I am grateful to Ronald R. Coifman and Peter W. Jones for extensivediscussions. The author was partially supported by an AMS-Simons travel grant and INET grant References [1] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunc-tions of N-body Schr¨odinger operators. Mathematical Notes, 29. Princeton University Press, Princeton, NJ;University of Tokyo Press, Tokyo, 1982. 118 pp.[2] P. W. Anderson, Absence of Diffusion in Certain Random Lattices, Physical Review 109 (1958), 1492–1505.[3] D. Arnold, G. David, D. Jerison, S. Mayboroda, M. Filoche, The effective confining potential of quantum statesin disordered media, arXiv :1505.02684[4] L. Bakri, Quantitative uniqueness for Schr¨odinger operator. Indiana Univ. Math. J. 61 (2012), no. 4, 1565-1580.[5] C. Bandle, Isoperimetric inequalities and applications. Monographs and Studies in Mathematics, 7. Pitman(Advanced Publishing Program), Boston, Mass.-London, 1980.[6] R. Banuelos and T. Carroll, Brownian motion and the fundamental frequency of a drum. Duke Math. J. 75(1994), no. 3, 575-602.[7] R. Carmona, Pointwise bounds for Schr¨odinger eigenstates. Comm. Math. Phys. 62 (1978), no. 2, 97-106.[8] R. Carmona and B. Simon, Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems.V. Lower bounds and path integrals. Comm. Math. Phys. 80 (1981), no. 1, 59-98.[9] R. Carmona, W. C. Masters and B. Simon, Relativistic Schr¨odinger operators: asymptotic behavior of theeigenfunctions. J. Funct. Anal. 91 (1990), no. 1, 117-142.[10] X. Fernique, Intgrabilit des vecteurs gaussiens. C. R. Acad. Sci. Paris Sr. A-B 270 1970 A1698-A1699.[11] M. Filoche and S. Mayboroda, Universal mechanism for Anderson and weak localization. Proc. Natl. Acad.Sci. USA 109 (2012), no. 37, 14761-14766.[12] M. Filoche and S. Mayboroda, The landscape of Anderson localization in a disordered medium, ContemporaryMathematics, to appear.[13] S. Markvorsen, and V. Palmer, Torsional rigidity of minimal submanifolds. (English summary) Proc. LondonMath. Soc. (3) 93 (2006), no. 1, 253-272.[14] C. Moler and L. Payne, Bounds for eigenvalues and eigenvectors of symmetric operators. SIAM J. Numer.Anal. 5 1968 64-70.[15] B. Simon, Brownian motion, L p properties of Schr¨odinger operators and the localization of binding, Journalof Functional Analysis 35 (1980), 215–229.[16] S. Steinerberger, Lower bounds on nodal sets of eigenfunctions via the heat flow. Comm. Partial DifferentialEquations 39 (2014), no. 12, 2240-2261.[17] M. van den Berg, Large time asymptotics of the heat flow. Quart. J. Math. Oxford Ser. (2) 41 (1990), no. 162,245-253.[18] S. Wu, Homogenization of differential operators. Acta Math. Appl. Sin. Engl. Ser. 18 (2002), no. 1, 9-14.[19] S. Zelditch, Local and global analysis of eigenfunctions on Riemannian manifolds. (English summary) Hand-book of geometric analysis. No. 1, 545-658, Adv. Lect. Math., Int. Press, Somerville, MA, 2008. Stefan Steinerberger, Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven,CT 06511, USA
E-mail address ::