Counting eigenvalues of Schr¨odinger operators using the landscape function

We prove an upper and a lower bound on the rank of the spectral projections of the Schr¨odinger operator ´ ∆ ` V in terms of the volume of the sublevel sets of an eﬀective potential 1 u . Here, u is the ‘landscape function’ of [10], namely a solution of p´ ∆ ` V q u “ 1 in R d . We prove the result for non-negative potentials satisfying a Kato-type and a doubling condition, in all spatial dimensions, in inﬁnite volume, and show that no coarse graining is required. Our result yields in particular a necessary and suﬃcient condition for discreteness of the spectrum.


Introduction
In a celebrated body of work, Fefferman and Phong [12] carried out an extensive analysis of the spectrum of self-adjoint differential operators based on the uncertainty principle, namely the fact that there is lower bound on the localization of the Fourier transform of a function that is well localized in space.Among the far reaching consequences of this old observation, they show that the number of eigenvalues E j of a Schrödinger operator with positive polynomial potential V , below an energy µ, is equivalent to a coarse-grained notion of the volume of the sublevel sets of the potential at µ. Precisely, they count the number of boxes of side length of order µ ´1{2 inside which V is less than or equal to µ.This coarse graining is shown to be a necessary feature that arises from the uncertainty principle: for a test function to fit into a very narrow box, its kinetic energy must be large.
The Fefferman-Phong result is a wide generalization of the classical Weyl law, which is an asymptotic result for the number of eigenvalues of ´∆ ´λV as λ Ñ 8, see [50,51,52,53], and [20] for many extensions and variations.It is also closely related to the Lieb-Thirring inequalities [26] (see also [15] for a recent overview) V pxq γ`d{2 dx.
The case γ " 0 of interest to us in this work was obtained in dimensions d ě 3 independently and by very different techniques by Cwickl [8], Lieb [24], and Rozenblum [37,38], and we shall refer to it as the CLR inequality.It is well-known that the CLR inequality cannot hold in complete generality for dimensions d " 1, 2. A general result in one dimension is the classical Calogero inequality [5].It can further be used to obtain CLR-type bounds in some two-dimensional cases, see e.g.[22,21,43,16,23].For more general kinetic energies and simplified proofs of the CLR inequality see [14,18,19] and the references therein.Shen generalized the results of Fefferman-Phong to potentials in some reverse Hölder class and also in the presence of a magnetic field [40,41].For this he used previously established L p estimates for such Schrödinger operators [39] (in the case of nonnegative polynomials, see also Smith [46] and Zhong [54]).These results rely on estimates on the Green's function, see Davey-Hill-Mayboroda [9], as well as Mayboroda-Poggi [34] for optimal kernel estimates for more general elliptic operators and in the presence of a magnetic field, in dimensions d ě 3. Poggi [33] considered potentials of Kato-type defined in more general domains.In dimension d " 2 Christ [7] obtained kernel estimates under stronger assumptions.We point out that Otelbaev obtained similar two-sided estimates earlier [30,31], see also [28,29] for related bounds.
The Fefferman-Phong approach was also extended by David-Filoche-Mayboroda [10] who introduced a new technique which is central to the present work.While the bounds of Fefferman-Phong only depend on the dimension and the degree of polynomial, the dependence on the potential in DFM is more subtle and relies on the so-called landscape function.One considers ´∆ `V on finite boxes Λ L of side length L and define the landscape function as the solution of p´∆ `V qu L " 1 (1.1) with suitable boundary conditions.A formal computation shows that the operator ´∆ `V on L 2 pΛ L , dxq is unitarily equivalent to the elliptic operator L pxqdxq.We shall henceforth call 1{u L the effective potential.The landscape function encodes (via the effective potential) a substantial part of the spectral information of the original Schrödinger operator.Among the many rigorous results using the effective potential [2,48,4,13,10,47,3,49,6], we are interested in the following, see [10]: If N V L pµq is the number of eigenvalues of ´∆ `V , counted with multiplicity, on the box with sidelength L (with periodic boundary conditions) that are smaller than µ, then there are constants c, C ą 0 such that where N pµ, Lq is the coarsed-grained volume corresponding to the effective potential, namely it is the number of boxes of sidelength µ ´1{2 in which 1{u L ď µ.While C only depends on the dimension, the constant c depends on the oscillation of u L .
In this paper, we consider the setting of [42] where the potentials are Kato-class and satisfy a doubling condition (the precise assumptions are (2.1,2.2)).We first study the existence of the landscape function in the whole space R d , namely the existence and positivity of solutions of (1.1).This purely PDE question is set in an apriori inconvenient space since, unlike in the case of finite boxes considered in [10], the right hand side belongs to no L p space but for p " 8.Not surprisingly, this can be addressed by considering a sequence of compactly supported functions converging pointwise to 1.A similar approach was used by Poggi [33,Theorem 1.18] in dimension d ě 3.With the landscape function and therefore the effective potential 1  u in hand, we turn to the problem of the counting of eigenvalues.We extend the DFM result to the infinite volume setting and without coarse graining.We show that the effective potential is confining if and only if the spectrum of the Schrödinger operator is discrete, in which case the measure of its sublevel sets is finite.We then prove that this measure controls the eigenvalue counting function, namely where Vpµq " Crucially, this CLR-type bound is valid in all spatial dimensions, and for all µ P R. The latter is one of the advantages of working immediately in infinite volume, since otherwise the size of the domain Λ L yields a lower bound on the energy levels µ that can be considered.As already seen in other applications of the DFM landscape function, this bound where the volume is not coarse-grained reflects the fact that the transformation 'transfers' some of the kinetic energy to the effective potential.
One may wonder about the relationship of the effective potential with the semiclassical limit.Not surprisingly, one partial answer is provided by microlocal analysis.Indeed, the effective potential is given by the resolvent acting on the constant function.The inverse of ´∆ `V is a pseudo-differential operator whose symbol is given to highest order by 1 |ξ| 2 `V pxq .In this approximation, we conclude that, formally, In other words, the effective potential 1 u is equal to the physical potential V up to lower order corrections in the sense of microlocal analysis.It is precisely these corrections however that remove the need for coarse-graining.
We conclude this introduction by commenting on the specific case V px, yq " x 2 y 2 in two dimensions.In [44], Simon provided five proofs that this operator has discrete spectrum with strictly positive first eigenvalue, that the number of eigenvalues below any fixed energy µ follows the Fefferman-Phong estimate (in particular, the coarse-grained volume is finite), and established the precise asymptotics in [45].These results are particularly remarkable given that the Lebesgue measure of tpx, yq P R 2 : x 2 y 2 ď µu is infinite for every µ ą 0. This shows in particular that the measure of the sublevel sets of the potential do not capture the spectral information, unlike the coarsed-grained volume associated with the potential x 2 y 2 .Using pseudo-differential calculus [36] obtains similar results for more general degenerate polynomials.Our result removes the need for coarse graining, provided V is replaced with the DFM effective potential 1 u ; Since it is simple to see that the effective potential is confining in this case (by Corollary 2.3), we obtain yet another proof of discreteness of the spectrum of ´∆ `x2 y 2 in R 2 .

Main results
We denote by u a particular weak solution of p´∆ `V qu " 1 in R d , which can be realized as the pointwise limit of specific Lax-Milgram solutions.It will be constructed in details in Section 3. Our main result is that the volume of * is comparable to the rank of the spectral projection of (the Friedrich's extension of) ´∆ `V at energies less than or equal to µ.
Theorem 2.1.Assume that V ě 0, V ı 0 satisfies the following conditions: 1. (Kato-type condition) There exists C K , δ ą 0, such that for all x P R d and all r, R with 0 ă r ă R.

(Doubling condition) There exists
for all x P R d and all r ą 0.
We denote by H the Friedrichs extension of the positive symmetric operator ´∆ `V defined on C 8 c pR d q.Let N V pµq be the rank of the spectral projection ½ p8,µs pHq.Then there exist constants c, C ą 0 such that for all µ P R, where Vpµq " The constants c, C depend only on C K , C D , δ and the spatial dimension d.
The assumption (2.1) is a scale-invariant variant of the standard Kato condition.For d ě 3 one obtains via Fubini's theorem that the condition (2.1) is equivalent to ż V pyqdy for all 0 ă R, all x P R d and some C independent of x, R. Conditions (2.1),(2.2) are satisfied by potentials in the reverse Hölder class pRHq d{2 (see [39]).In particular, this include nonnegative polynomials and fractional power functions |x| α for α ą ´2 for d ě 3. On the other hand, potentials with compact support or exponential growth violate (2.1), respectively (2.2).
Combining the above theorem with the property that u varies slowly, we further derive an analogous result to [40,Corollary 0.11].
Corollary 2.2.Let V be as in Theorem 2.1.Then the spectrum of H is discrete if and only if lim RÑ8 }u} L 8 pR d zBp0,Rqq " 0.
We will present the proofs in Section 4, while in Section 5 we concentrate on the case where V is a polynomial.For polynomial potentials one can further analyze the landscape function.In particular, one has the following: Corollary 2.3.Let V be a polynomial that is bounded from below.Then the spectrum of H is discrete if and only if none of the directional derivatives of V vanishes identically.
If the condition of the last corollary is violated, then the corresponding operator has no eigenvalues.Indeed, after conjugating with a suitable rotation, we can assume without loss of generality that the polynomial does not depend on the last variable.We define r V px 1 , . . ., x d´1 q " V px 1 , . . ., x d q.By taking a Fourier transform in the last variable, we get that the Friedrichs extension where r H " ´∆R d´1 `r V .It follows from [35,Theorem XIII.85] that σpHq " rmin σp r Hq, 8q and H admits no eigenvalue.
Finally, we point out that Theorem 2.1 together with (3.7) below recover the result of Shen [40, Theorem 0.9] in our class of potentials and in the absence of a magnetic potential.

Existence of the landscape function in infinite volume
In this section we show the existence of the landscape function in infinite volume and establish some estimates of the landscape function in terms of the Fefferman-Phong-Shen maximal function.In [33, Theorem 1.18, Theorem 1.31] Poggi proves this for d ě 3. We briefly recall the construction and explain how to extend this to the case d " 1, 2. For this we will rely on extensions of results for d ě 3 due to Shen (see [42,Proposition 1.8]).One of the key objects is the Fefferman-Phong-Shen maximal function mp¨, V q, which is defined as where C D is the constant in (2.2).This maximal function satisfies the following properties.
Lemma 3.1.Let V satisfy the conditions of Theorem 2.1.Then we have 1.0 ă mpx, V q ă 8 for every x P R d .
2. For every C 1 , there exists C, depending only on C K , C D , δ and C 1 , such that for all x, y P R with |x ´y| ď C 1 mpx,V q .3. There exists k 0 , C ą 0, depending only on C K , C D , δ and d, such that for all x, y P R d we have Then for all px, tq P R d`1 and all 0 ă r ă R, for all px, tq P R d`1 and all r ą 0.
Finally, there exists C ą 0 depending on C K , C D , δ and d such that for all px, tq P R d ˆR.
Note that in (3.4), the exponent in the maximal function involving V is d, while it is d `1 in the one involving Ṽ .
The validity of 2. and 3. for d ě 3 is proved in [42,Proposition 1.8].Hence, it suffices to prove 4. for 2. and 3. to hold for all d ě 1.
The fact that r V satisfies both the Kato-type and the doubling condition are simple computations.Since V satisfies the doubling condition in dimension d, which yields the claim upon noting that the last integral is bounded above by π.For the Kato-type condition, we first consider the case 0 ă ?2r ă R. Then we have The bound is immediate if, on the other hand R ?
To show (3.4) we introduce the following maximal function which is defined over cubes Qpx, rq centered at x and of sidelength r, rather than over balls.Clearly (3.4) holds true with C " 1 for m replaced by m Q .Thus, we only need to show that m and m Q are equivalent.For d ď 2, the inclusion Qpx, rq Ă Bpx, rq and the positivity of V yield immediately and hence m Q px, V q ď mpx, V q.Reciprocally, let x P R d and let r " 2 mpx,V q .Then for any R ą V pyqdy where we used (2.1) in the second inequality, and the fact that Bpx, R{2q Ď Qpx, Rq in the third.It follows that f pyqvpyqdy for all v P H.Note that H " For every x P R d there exists a function Γ V px, ¨q P L p loc pR d q, for 1 ă p ă d{pd ´2q, such that for all f P L 8 pR d q with compact support and f ě 0, the unique Lax-Milgram solution u f of p´∆ `V qu " f can be written as for almost every x P R d .Furthermore, one has the kernel estimate In what follows, we will also consider weak solutions of p´∆ `V qu " f for f P L 1 loc pR d q, namely a function u f such that ż f pyqϕpyqdy for all ϕ P C 8 c pR d q.We shall now construct the landscape function in infinite volume.This is an alternative and simpler approach, valid in the present setting, than that of [33,Theorem 1.18].Proposition 3.3.Let V be as in Theorem 2.1.Then there exists constants c, C ą 0 depending only on C K , C D , δ and d, and a weak solution for almost every x P R d .
For later purposes, we immediately note that the proof of the proposition yields the following 'finite volume' result.If d ě 3 we denote by u L the Lax-Milgram solutions of p´∆ `V qu L " ½ Bp0,Lq .
for almost every x P R d .We remark that all the results work equally well if we replace the indicator function over balls by indicator function over other compact sets tΩ L : Proof.First we consider the case d ě 3. Denote by u L the Lax-Milgram solution (3.8) given by Proposition 3.2.As ½ Bp0,L 2 q ´½Bp0,L 1 q ě 0 for L 2 ě L 1 , we get from (3.5) that pu L q Lě1 is monotone increasing almost everywhere.On the other hand, it is essentially bounded since for almost every x P R d by (3.5,3.6).Thus, we can define upxq " lim LÑ8 u L pxq.
As u L are Lax-Milgram solutions of p´∆ `V qu L " ½ Bp0,Lq , one easily checks that u is a weak solution of p´∆ `V qu " 1.The lower bound for u follows from the lower bound in (3.6).
We show now that u P H 1 loc pR d q for d ě 3. Fix any ball B Ď R d and a smooth cut-off function χ B P C 8 c pR d q such that χ B " 1 on B. As χ B u L P dompH 1{2 q and u L is a Lax-Milgram solution of (3.8), we get ż The product rule for Sobolev functions yields ż Using integration by parts for the second term on the RHS yields ż Thus, we get ż Hence, there exists a constant C ą 0 depending only on the dimension such that ż L for all balls B and all L ą 0. By (3.9) and (3.3) we get that p∇u L q Lě1 is uniformly bounded in L 2 pBq for fixed B. Therefore, by Banach-Alaoglu, there exists a subsequence u L k converging weakly to some g B P L 2 pR d q.One readily checks that g B is the weak derivative of u and hence u P H 1 loc pR d q.Next we consider the case d " 2. For this we use Hadamard's method of descent.Recall that r V px, tq " V pxq for px, tq P R 2 ˆR.By Lemma 3.1 the function r V satisfies (2.1,2.2), and therefore the first part yields a weak solution r u of p´∆ `r V qr u " ,αq,Lq .Thus, by (3.5), we have Γ r V ppx, tq, yq½ Bpp0,0,αq,Lq pyqdy.
Hence, for almost every x P R 2 there exists C x such that for almost every t P R we have r upx, tq " C x and we define u on R 2 by upxq " C x .Let ϕ P C 8 c pR 2 q and ψ P C 8 c pRq with ş R ψptqdt " 1.Then, as ş R ψ 2 ptqdt " 0 and r u is a weak solution of p´∆ `r V qr u " 1 on R 3 , we get ż Therefore, p´∆ `V qu " 1 is a weak solution on R 2 .The inequality (3.7) follows from (3.4).As shown before, we have r u P H 1 loc pR 3 q and |∇upxq| " |∇r upx, tq| as upx, tq is independent of t.Hence u P H 1 loc pR 2 q.The case d " 1 follows similarly as the case d " 2. Finally, continuity follows from [27, Corollary 1.5].
We point out that the weak solution u constructed above does in general not belong to the form domain of H, and we will therefore often have to work with the Lax-Milgram solution u L instead of u.If V is a polynomial, the maximal function mp¨, V q is equivalent to the function introduced in [46,54] M px, V q " ÿ αPN n 0 |B α V pxq| 1{p|α|`2q , (3.10) see (5.5) below.The sum is of course finite for a polynomial.We now consider V pxq " |x| 2 on R d .Then M px, V q is comparable to 1 `|x| and hence, by (3.7) and (5.5), upxq is comparable to p1 `|x|q ´2 which is not square integrable for d ą 2. In Lemma 5.3 we show that the landscape function, for polynomial potentials, belongs to the form domain if and only if the landscape function is integrable.The equivalence of the landscape function and the Fefferman-Phong-Shen maximal function exhibited in Proposition 3.3 allows one to prove a Harnack inequality for the landscape function, see also [33,Corollary 1.38] for the case d ě 3. Proof.This follows immediately from (3.2) and 3.7.

Proof of Theorem 2.1
In this section we show that we can estimate the rank N V pµq of the spectral projection of H in terms of the measure of the sublevel set Vpµq of the effective potential 1 u , both defined in Theorem 2.1.
For this we introduce two types of coarse-grained volumes.A box of sidelength ℓ is a set of the form ˆd i"1 ra i , b i s where b i ´ai " ℓ.For any ℓ ą 0, we consider a collection Q ℓ of boxes of sidelength ℓ such that We define for any µ ą 0 where ess inf, ess sup denote the essential infimum, respectively the essential supremum.
For the class of potentials considered here, namely those satisfying the Kato-type and doubling conditions, both coarse-grained volumes are directly related to the measure Vpµq of the sublevel set.Proof.The first two inequalities are immediate as, up to null sets, npµq{µ d{2 is the measure of all boxes that are strictly contained in the sublevel set t1{u ď µu and N pµq{µ d{2 is the measure of all the boxes that intersect the sublevel set.
Let Q be a box such that inf Q 1{u ď µ, then by (3.11) we have sup Q 1{u ď C H µ. Hence, N pµq ď npC H µq.
We now turn to the proof of the main theorem, namely the bounds (2.3).Our arguments are variational and adapted from the proofs of [10], which are themselves inspired by Fefferman-Phong [12].We start with the upper bound.Proof.In order to have that N V pµq ď N it suffices, by the Min-Max Principle (see [35, Theorem XIII.2]), to exhibit a subspace H N Ď dompH 1{2 q with codimension at most N such that ż for all v P H N .Let F be the collection of boxes such that F " where C ą 0 will be chosen later, and let Since the cubes are disjoint, the codimension of H N is equal to |F| " N pCµq.
First we want to show that for all ϕ P C 8 c pR d q.We start by considering d ě 3. Denote by u L the Lax-Milgram solution of (3.8).By (3.3), (3.9) we know that 1{u L P L 8 loc pR d q X H 1 loc pR d q, using the chain rule for Sobolev functions [25,Theorem 6.16].This readily implies that |ϕ| 2 {u L is in the form domain of H for all ϕ P C 8 c pR d q.As u L is a Lax-Milgram solution of (3.8), we get ż Furthermore, using the product rule [25,Lemma 7.4] yields ∇u L ¨∇p|ϕ| 2 {u L q " |∇ϕ| 2 ´u2 L |∇pϕ{u L q| 2 .Combining the last two equalities and taking L Ñ 8 implies (4.1) for d ě 3.
For d ď 2 we set r V px, tq " V pxq for all px, tq P R d ˆR3´d and denote by r u the landscape function of r V .Let ϕ P C 8 c pR d q, ψ P C 8 c pR 3´d q with ş R 3´d ψptqdt " 1 and pϕ b ψqpx, tq " ϕpxqψptq for all px, tq P R d ˆR3´d .Then we have by the previous computations for d " 3 The bound (4.1) extends, for all d ě 1, by density of C 8 c pR d q in the form domain of H (see [11,Theorem 8.2.1.])to all v P dompH 1{2 q.This implies that for all v P dompH 1{2 q.With this, the statement of the lemma follows from the claim that if v P H N zt0u, then ż We check this inequality using the partition into boxes.In any box Q R F, we simply use the bound min Q 1{u ą Cµ.If Q P F, we recall that the integral of v vanishes and use the Poincaré inequality with optimal constant π 2 d pCµq since the boxes have sidelength pCµq ´1{2 , see [32].Hence, the claimed lower bound holds for all C ą maxt2, 2d π 2 u indeed.
Next we turn to the lower bound in (2.3).
Proof.For a lower bound N ď N V pCµq it suffices, again by the Min-Max Principle, to find a subspace We define Furthermore, for a box Q we pick χ Q P H 1 pR d q with 0 ď χ Q ď 1, }∇χ Q } L 8 pR d q ď 4µ 1{2 , χ Q " 1 on Q{2 and χ Q " 0 on R d zQ (a possible choice for χ Q is to interpolate linearly from BpQ{2q to BQ).Since the functions χ Q u are non-zero and orthogonal to each other, the space By Proposition 3.3 we have u P H 1 loc pR d q X L 8 loc pR d q and thus χ Q u is in the form domain of H. Using the product rule for Sobolev function [25,Lemma 7.4] and the fact that u solves the landscape equation we get for all ϕ, ψ P C 8 c pR d q x∇pψuq, ∇ϕy L 2 pR d q `xψu, V ϕy L 2 pR d q " x∇u, ∇pψϕqy L 2 pR d q `xu, V ψϕy L 2 pR d q ´x∇u, p∇ψqϕy L 2 pR d q `xp∇ψqu, ∇ϕy L 2 pR d q " xψ, ϕy L 2 pR d q ´x∇u, p∇ψqϕy L 2 pR d q `xp∇ψqu, ∇ϕy L 2 pR d q .
Now we pick a sequence pϕ n q nPN Ď C 8 c pR d q such that supppϕ n q Ď 2Q, sup n }ϕ n } L 8 pR d q ă 8 and ϕ n Ñ χ Q u in H 1 pR d q and a similar approximation ψ n Ñ χ Q u and we get ż where the last inequality follows from the properties of χ Q .This yields the claim we had set to prove.
Proof of Corollary 2.2.If u vanishes at infinity, i.e. lim sup RÑ8 sup R d zBp0,Rq u " 0, then each sublevel set of 1{u is bounded up to a null set and thus H has discrete spectrum by (2.3).Assume on the other hand that u does not vanish at infinity.There is µ ą 0 and a sequence of points px n q ně1 such that lim nÑ8 |x n | " 8 and lim inf εÑ0 `inf Bpxn,εq u ě C H µ for all n.Then by (3.11) we have and hence, by (2.3), the spectrum of H is not discrete.

The case of polynomial potentials
When the potential V is a polynomial, as in the original setting of Fefferman-Phong, one can obtain more precise information of the landscape function.We start by giving the proof for Corollary 2.3.
Proof of Corollary 2.3.Since the addition of a constant does not change the structure of the spectrum, we assume that the polynomial satisfies V ě 1.We check first that these polynomials satisfy (2.1) and (2.2).Condition (2.1) holds with δ " 2 due to the inequality where c can be chosen to depend only on d and the total degree of V , but neither x nor r.The upper bound is immediate.It is enough to show the lower bound for r " 1 and x " 0 by scaling and translation.In that case, the claim follows from the fact that the space of all polynomials in d variables and total degree at most D is a finite dimensional vector space and thus all norms are equivalent.
For the same reason and since polynomials are analytic functions, there exists a constant C ą 0 depending only on d and D such that ż Bp0,2q V pyqdy, which implies doubling after rescaling and translation.In particular, C D can be chosen to only depend on d and D. Now, Corollary 2.2 and (3.7) imply that the spectrum of H is discrete if and only if lim |x|Ñ8 mpx, V q " 8.For polynomials the Fefferman-Phong-Shen maximal function mpx, V q is in fact equivalent to M px, V q introduced in (3.10), in the sense that cM px, V q ď mpx, V q ď CM px, V q. (5.1) The equivalence was already noted in [40] and we provide a proof below for completeness, see Lemma 5.4.
With these preliminaries, we can now turn to the central claim of the corollary.If one of the directional derivative vanishes, then ´∆ `V is unitarily equivalent (via a suitable rotation) to ´∆`W where B 1 W " 0. In this case M ppt, 0, . . ., 0q, W q " M p0, W q, which implies by the remarks above that the spectrum of ´∆ `W is not discrete and hence also the spectrum of ´∆ `V is not discrete.
Next we are going to show that if ´∆ `V does not have discrete spectrum, then some directional derivative of V vanishes identically.As ´∆ `V does not have discrete spectrum we must have that lim inf |x|Ñ8 M px, V q ": M 0 ă 8. (5.2) We consider the semi-algebraic set ) and the polynomial function 2) implies that A is an unbounded set and we can therefore pick a sequence px pnq q nPN Ď A such that |x pnq | Ñ 8 and lim nÑ8 F px pnq q ": y " py α q αPr0,Ds .Next we would like to pass from a mere sequence to an analytic curve.This is done by the following curve selection lemma at infinity.Lemma 5.1.[17, Lemma 2.17] Let A Ă R d be a semi-algebraic set, and let F : R d Ñ R N be a semi-algebraic map.Assume that there exists a sequence px pnq q nPN Ă A such that lim nÑ8 |x pnq | " 8 and lim nÑ8 F px pnq q " y P pR Y t˘8uq N .Then there exists an analytic curve γ : p0, δq Ñ A of the form γptq " 8 ÿ j"´m a pjq t j (5.3) such that a p´mq P R N zt0u, m P Z ą0 and lim tÑ0 `F pγptqq " y.
Let γ be a curve as given by the previous lemma.We would like to say that V remains constant along γ and thus get a direction in which the gradient of V vanishes identically.However, analytic functions can remain bounded on an unbounded set without being constant.Thus, we truncate the series (5.3) at j " 0, thereby obtaining a polynomial approximation of the curve γ, and F will still remain bounded along the truncation.Lemma 5.2.For every ε ą 0 there exists C ą 0 such that for all v P R d with |v| ă ε we have for all t P p0, δ{2q 0 ď V pγptq `vq ď C. (5.4) Proof.By Taylor's theorem, V pγptq`vq " ř α! pB α V qpγptqq.The claim follows from the fact that |pB α V qpγptqq| are all uniformly bounded for t P p0, δ{2q.

With this, we define polynomial function
Gpsq " m ÿ j"0 a p´jq s j .
For s ą 1 δε and |x| ă ε 2 we get 0 ď P ps, x 1 , . . ., x d q ď C again by Lemma 5.2.Now, for any |x| ă ε 2 , the function s Þ Ñ P ps, xq is a polynomial that is bounded on an unbounded interval and thus constant.Therefore, B s P ps, xq " 0 on R ˆBε{2 p0q.By the identity theorem we get that B s P ps, xq " 0 on R d`1 .But 0 " B s P ps, xq " p∇V qpGpsq `xq ¨G1 psq.
As G is not constant, there is s 0 P R such that G 1 ps 0 q ‰ 0 and so the derivative of V in direction G 1 ps 0 q vanishes identically.
As mentioned before, the landscape function will not belong to the form domain of H.For polynomial potentials, there is an easy criterion to check whether u P dompH 1{2 q.
Lemma 5.3.Let V ě 0 be a non-zero polynomial.Then u P L 1 pR d q if and only if u P dompH 1{2 q.
Proof.As V is smooth, we get by standard elliptic regularity theory that the landscape function is a classical solution of the landscape equation.Multiplying the landscape equation by ϕ P C 8 c pR d q and integrating yields after integration by parts ż We saw in the proof of Corollary 2.3 that either lim |x|Ñ8 upxq " 0 or that there is one spatial coordinate along which u is constant.Hence, if u P L 1 pR d q, then u vanishes at infinity and automatically u P L 2 pR d q.However, then we can choose a sequence of ϕ n P C 8 c pR d q converging to 1 and obtain by monotone convergence ż and therefore u P dompH 1{2 q.
On the other hand, if u P dompH 1{2 q, then u P H 1 pR d q and in particular u P L 2 pR d q.Thus, we can take again a suitable sequence of test functions to obtain by dominated convergence ż Let us now return to the concrete example of Simon's potential V px, yq " x 2 y 2 .First of all, we can now prove that the corresponding landscape function is in the form domain of ´∆ `V .Combining Proposition 3.3 and Lemma 5.4, it is enough to check that M p¨, x 2 y 2 q ´2 P L 1 pR 2 q.An explicit calculation yields M ppx, yq, x 2 y 2 q ě |xy| à|x| `a|y| `1, and thus its inverse is indeed square integrable in R 2 .Similarly, we obtain the following two-sided estimate on the effective potential: This yields for µ sufficiently large Vpµq.
On the other hand, we have The combination of those two estimates with Theorem 2.1 recovers Simon's asymptotics [45,Theorem 1.4] up to multiplicative constants.We conclude this section with a proof of the equivalence of the functions mp¨, V q and M p¨, V q in the case of polynomials.We point out that the arguments in this section show that in the case of polynomials, the constants appearing in Theorem 2.1, and a fortiori the Harnack constant, depend only on the spatial dimension and the degree of the polynomial.Lemma 5.4.Let V ě 0 a polynomial on R d of total degree D ě 0. Then there exist constants C, c ą 0 depending only on d, D such that cM px, V q ď mpx, V q ď CM px, V q, (5.5) where mp¨, V q, M p¨, V q were defined in (3.1) and (3.10).
Proof.By translating the potential, we can pick x " 0. Furthermore, for all λ ą 0 we have mpx, V λ q " λmpλx, V q and M px, V λ q " λM pλx, V q, where V λ pxq " λ 2 V pλxq.Hence, we can assume that M p0, V q " 1 and then need to show that there exists c ą 0 depending only on d, D such that mp0, V q ě c.Since V Þ Ñ B α V is a linear map on a finite dimensional space |V pxq|, and so V pxq ¯1{pD`2q ¯.
Hence, we have sup xPBp0,1q V pxq ě c ą 0. For r ě 1 we get Recall that C D can be chosen to only depend on d, D, therefore the right hand side is greater than C D for r large enough, which yields an upper bound on 1 mp0,V q .Hence, mp0, V q ě c " cM p0, V q.
We turn to the lower bound.First of all, a simple Taylor expansion yields (see [46,Lemma 2.5]) for all x, y P R d .Thus, if mp0, V q " 1 then M py, V q 2 ď C 2 M p0, V q 2`D , by (5.6), which again yields the desired estimate by translating and rescaling.

The potential well
In this section we explicitly compute the landscape function for potential wells ε½ Bp0,δq c , where Bp0, δq c " R d zBp0, δq and ε, δ ą 0. We shall observe first of all that the minimum of the effective potential properly reflects the value of the bottom of the spectrum in the sense that both are of order ε as ε Ñ 0. Secondly, we will see that the estimates of the main theorem are not tight enough to distinguish the difference between d " 1, 2, where an eigenvalue is present for all ε ą 0, and d ě 3 where this is not the case.We start by observing that the landscape function corresponding to the spherical well are radially symmetric.Indeed, all the Lax-Milgram solutions (3.8) are invariant under rotation of the first d variables and thus the landscape function, given as a pointwise limit of those solutions, shares the same symmetry.Passing to spherical coordinates we see that the radial part f p|x|q " upxq solves the ODE ´f 2 prq ´d ´1 r f 1 prq `ε½ rδ,8q prqf prq " 1 on p0, 8q.The general solution of this ODE on p0, δq is given, for d ‰ 2, by f prq " ´r2 2d `a1 `a2 r d´2 , respectively by the same expression with r ´pd´2q replaced by logprq for d " 2. As lim rÑ0 `f prq " lim rÑ0 `upre 1 q " up0q, we conclude in the case d ě 2 that a 2 " 0. The same follows for d " 1 as u is even and C 1 pRq.
Finally, the coefficients can be determined by the fact that f P C 1 pR ą0 q.In dimensions d " 1, 3 the Bessel functions can be expressed in elementary functions and the solutions are given by upxq " In all dimensions we have that u is radially symmetric and its radial part is monotone decreasing (even exponentially).Furthermore, we have lim |x|Ñ8 upxq " 1 ε .This implies that the sublevel set Vpµq of the effective potential 1{u is monotone increasing, remains finite for µ ă ε and lim µÑε ´V pµq " 8.This is consistent with the fact that the bottom of the essential spectrum is ǫ and c ă 1, C ą 1 in Theorem 2.1.
Thus, for δ ą 0 fixed, we obtain µ 0 " εp1 ´Op ?εqq as ε Ñ 0 `.As discussed at the beginning of the section, this is the same asymptotic behaviour as that of the minimum of 1{u, see (6.1).The same holds for d " 3 where however the bottom of the spectrum is the bottom of the essential spectrum, namely N pµq " 0 for µ ă ε and N pµq " 8 for µ ě ε.Here, Vpµq is arbitrarily large for µ Ñ ε ´, showing that c ă 1 in (2.3).

Corollary 3 . 4 .
Let V be as in Theorem 2.1.Then there exists a constant C H ě 1, depending only on C K , C D , δ and d, such that for almost every x P R d and almost every y P Qpx, 2 a upxqq we have C ´1 H upxq ď upyq ď C H upxq. (3.11)

1 ε
`δ ?ε e ´?εp|x|´δq , |x| ą δ for d " 1 and by upxq " .6) Proof.Existence and uniqueness of Lax-Milgram solution follows directly from the Lax-Milgram theorem on the form domain H equipped with its standard inner product xv, wy H " x∇v, ∇wy L 2 pR d q `x? V v, ?V wy L 2 pR d q .The representation of the Lax-Milgram solution in terms of the integral kernel Γ V was shown in [42, Theorem 2.16] and the estimate in terms of the Fefferman-Phong-Shen maximal function follow from [42, Theorem 3.11, Remark 3.21, Theorem 4.15].