Concentration of eigenfunctions of Schroedinger operators

We consider the limit measures induced by the rescaled eigenfunctions of single-well Schr\"odinger operators. We show that the limit measure is supported on $[-1,1]$ and with the density proportional to $(1-|x|^\beta)^{-1/2}$ when the non-perturbed potential resembles $|x|^\beta$, $\beta>0$, for large $x$, and with the uniform density for super-polynomially growing potentials. We compare these results to analogous results in orthogonal polynomials and semiclassical defect measures.


Introduction
Let A be a Schrödinger operator acting in L 2 (R) where Q is a real-valued, even, unbounded single-well potential. More precisely, we suppose that Q = V + W , where V is a sufficiently regular single-well (see Assumptions I) and W is its possibly irregular perturbation (satisfying Assumption II that guarantees that W is small in a suitable sense). Our main condition on the potential is that V satisfies ∃β ∈ (0, ∞], ∀x ∈ (−1, 1), lim where ω β (x) := |x| β , β ∈ (0, ∞), 0, β = ∞. (1.3) As explained in [17,Sec. 1.3], the existence of the limiting function in (1.2) already implies that ω β is a power of |x| or zero; functions V satisfying (1.2) with β < ∞ are called regularly varying. It is well-known (also under much weaker assumptions on Q) that the operator A, defined via its quadratic form, is self-adjoint with compact resolvent, hence its spectrum is real and discrete. In fact, all eigenvalues {λ k } of A are simple, thus they can be ordered increasingly and the corresponding eigenspaces are one-dimensional.
Since the potential Q is real, eigenfunctions {ψ k } related to {λ k } can be selected as real functions satisfying Aψ k = λ k ψ k , ψ k = ψ k L 2 (R) = 1, k ∈ N. (1.4) These conditions do not determine ψ k uniquely, since −ψ k satisfies the same conditions; nonetheless, the squares {ψ 2 k } are already uniquely determined. Let x λ k be positive turning points of V corresponding to eigenvalues {λ k }, i.e.
(1.6) This rescaling transforms the classically forbidden region {x : V (x) > λ k } with (super)-exponential decay of ψ k to R \ [−1, 1] while the rescaled functions ψ k (x λ k ·) oscillate in [− 1,1]. Notice that we do not include W in the definition of x λ k and thus in the rescaling of eigenfunctions; the assumptions on the size of W comparing to V , see Assumption II and Proposition 2.2, allow for treating W perturbatively.
We emphasize that while the limiting function, if exists, is always homogeneous, this not required for V ; see examples (1.9) and (1.11) above. Thus rescaling leads to a semi-classical operator only in very special cases; a relation of our result and so called semi-classical defect measures in these special cases can be found in Section 5.2 below. This paper is organized as follows.
Our results with precise assumptions are formulated in Section 2 and they are proved in Section 3 relying on asymptotic formulas for the eigenfunctions {ψ k } summarized in Section 3.1. In Section 4 we prove the asymptotic formulas following and slightly extending the ideas and results in the book [18, §22.27] and in [8]. Finally, in Section 5 our results are compared to the existing literature in more detail.
1.1. Notation. Throughout the paper, we employ notations and results summarized in Section 3.1. In particular, to avoid many appearing constants, for a, b ≥ 0, we write a b if there exists a constant C > 0, independent of any relevant variable or parameter, such that a ≤ Cb; the relation a b is introduced analogously. By a ≈ b it is meant that a b and a b. The natural numbers are denoted by N = {1, 2, . . . } and N 0 = N ∪ {0}.

Assumptions and results
Our results are obtained under the following assumptions on the potential Q = V +W . The conditions on V , similar to those used in [18,8], guarantee that V is an even single-well potential with sufficient regularity to obtain convenient asymptotic formulas for eigenfunctions (associated with large eigenvalues) of the corresponding Schrödinger operator, see Section 3.1 and 4 for details. The conditions on W ensure that it is indeed a small perturbation which does not essentially affect the shape of the eigenfunctions.
Assumption I is an extension of conditions in [18, §22.27] where the case ν = −1, i.e. polynomial-like potentials, is analyzed; conditions analogous to Assumption I are used also in [10,1] where the resolvent estimates of non-self-adjoint Schrödinger operators are given. The assumptions of [8] allow for fast growing potentials and are based on suitable restrictions of V ′′′ , see [8,Condition 2].
The first assumption (2.4) implies there are two constants 0 < c 1 ≤ c 2 < ∞ such that for all x ≥ ξ 0 This can be seen from (with (2.6) The crucial technical observation used frequently in the proofs is that (2.4) imply that, for any ε ∈ (0, 1) and all sufficiently large x > 0, we have i.e. we have a control of how much V and V ′ varies over the intervals of size x −ν , see Lemma 4.1. Assumptions (2.3) and (2.4) also imply that see Lemma 3.2, which is almost optimal condition for the separation property of the domain of the self-adjoint Schrödinger operator see [6,5,9]; note that the separation property might be lost for A due to the possibly irregular W .
In the next step, we formulate a condition on the perturbation W that guarantees that it is small in a suitable sense (arising in the proof of Theorem 3.3). The appearing weight w −2 1 is naturally related with the main part of the potential V , although, the precise formula (3.24) might seem more complicated to grasp. It includes the turning point x λ of V , the quantity a λ (the value of V ′ at the turning point) and a "natural small region" around the turning point (characterized by δ and δ 1 ), see Section 3.1 for details. Examples of perturbations satisfying Assumption II are given in Proposition 2.2 below.
Our main result reads as follows.  Then, for every f ∈ F V , we have Hence, in particular, the measures {µ k } converge weakly to the limit measure µ * as k → ∞.
2.1. Distribution of zeros. We remark that the related result on the number of zeros of the eigenfunction ψ k in [−εx λ k , εx λ k ], ε ∈ (0, 1], denoted by N k (εx λ k ), is This generalizes the classical results for the harmonic oscillator, i.e. Q(x) = x 2 , namely the semi-circle law for the limiting distribution of the number of zeros of Hermite functions, see e.g. [16,7,12]. A generalization of (2.19) for polynomial, possibly complex, potentials has been given in [4]. The distribution of zeros of eigenfunctions ψ k , see (2.19), is closely related to the distribution of eigenvalues of A and it is essentially proved in [19,Sec. 7]. Indeed, without the perturbation W , i.e. W = 0, the eigenvalues of A satisfy  [14,Thm. 6.6], and adjust the arguments in [19,Sec. 7]. Alternatively, one can use the asymptotic formulas for {ψ k } and {ψ ′ k } in Section 3.1; the latter can be derived by differentiating (4.42). The zeros of ψ k for |x| < x λ k are in a neighborhood of the zeros of and, for large ζ, using asymptotic formulas for Bessel functions, see [3, §10.17], these are in a neighborhood of zeros of

The proofs
We start with an implication of the condition (1.2) for integrals frequently appearing in our analysis and proceed with the proof of Theorem 2.3.
Lemma 3.1. Let V satisfy Assumption I and the condition (1.2). Then, for every g ∈ L ∞ ((−1, 1)), Proof. Both statements follow by (1.2) and the dominated convergence theorem. Since V is even, it suffices to consider the integrals on (0, 1) only.

(3.2)
Thus (2.1) and (1.2) imply that there exists ε 0 > 0 such that for all x ∈ [0, 1/2] and all t > t 0 with t 0 ≥ 2ξ 0 (independent of x) we have Combining (3.3) and the assumption that V is eventually increasing on R + , see and all t > t 0 . Thus the existence of an integrable bound in the first limit follows. For the second limit, we use inequalities (2.10). These imply in particular that there is a constant c > 0 (depending only on ν) such that for all x ∈ [1/2, 1) and To see this, we introduce for all y ∈ [0, y 0 ] and all large s > 0 (independently of y). Since e sy ≥ 1 + sy, we get Hence the sought integrable bound reads (3.10)

Summary of properties of eigenfunctions of Schrödinger operators.
We summarize properties eigenfunctions of Schrödinger operators with even singlewell potentials Q = V + W satisfying Assumptions I and II. The details and proofs are given in Section 4; this slightly extends the reasoning in [18, §22.27] and [8].
Since Q is an even function by assumption, we can restrict ourselves to (0, +∞). Following the notations of [8], we introduce (for enough large λ > 0) (3.11) here K 1/3 , I 1/3 are modified Bessel functions of order 1/3. Furthermore, we define The functions u and v are known to be two linearly independent solutions of the differential equation moreover, the Wronskian of u and v satisfies The L 2 -solution of Schrödinger equation −y ′′ + Qy = λy is then found by solving the integral equation (obtained by variation of constants) 3 and its proof in Section 4. Next, for 0 ≤ x < x λ , one gets The positive numbers δ and δ 1 are defined by and they satisfy see Lemma 4.1 and its proof for details. As λ → +∞, we have If |x| < x λ stays away from turning points, ζ is large and so it is useful to employ asymptotic formulas for Bessel functions with large argument, see [3, §10.17]. In particular, one obtains where (see also [8,Sec. 7]) For the absolute values of u and v, we have that, for all large enough λ > 0, with the weights see Lemma 4.2 below. Notice that arg ζ(x) = π/2 for x > x λ thus |u(x)| is exponentially decreasing while |v(x)| is allowed to be exponentially increasing as x → +∞. Next, from Assumption I we obtain the following estimates, frequently occurring in our statements and proofs.
Lemma 3.2. Let V satisfy Assumption I and let x λ and a λ be as in (3.11). Then, as λ → +∞, Finally, we have that The following theorem shows that the function u is the main term in the asymptotic formula for eigenfunctions of the operator A from (1.1). The proof is given at the end of Section 4. One can check that the eigenvalues of A are simple and eigenfunctions are even or odd functions (since Q is assumed to be even). Thus the eigenvalues and eigenfunctions of A can be found by determining λ > 0 for which solutions y in (3.29) of the differential equation (3.28) satisfy a Dirichlet (y(0) = 0) or a Neumann (y ′ (0) = 0) boundary condition at 0. Theorem 3.3. Let Q = V + W where V and W satisfy Assumptions I and II, respectively. Let x λ and u be as in (3.11), let w 1 , w 2 be as in (3.24), let κ λ as in (3.12) and let J W be as in (2.13). Then, for every sufficiently large and Proof of Theorem 2.3. Since the eigenfunctions {ψ k } are even or odd, we consider only x ∈ (0, ∞). We select the eigenfunctions {ψ k } such that where y k = y(·, λ k ), u k = u(·, λ k ) and r k = y k −u k , see Section 3.1 and in particular Theorem 3.3. Hence, the densities {φ k } of the measures {µ k }, see (1.6), satisfy In the sequel, notations and results summarized in Section 3.1 are used, moreover, we introduce the constant (for β ∈ (0, ∞]) .

(3.35)
We also drop the subscript k and work with quantities like y = y(·, λ) as λ → +∞. First, Lemma 3.1, (3.32) and the change of integration variables x = x λ t imply Thus with f ∈ F V , see (2.17), and the change of integration variables, we get (1) Employing estimates (3.23), (3.30), (3.39) and (3.19) in the last step, we obtain We investigate the region (x λ + x −ν λ /2, ∞) and also explain the convergence of the integral in (3.37). To this end, we recall that by assumption f ∈ F V , see (2.17), thus with some M > 0 and we show below that To prove (3.43), notice that for x > x λ and assuming that λ is sufficiently large that and, using (2.4) and (2.7), (3.46) Hence for ν < 0 we immediately arrive at For ν ≥ 0, we use (2.6) to get (with ξ 0 > 0 from Assumption I and some c > 0) thus (3.43) follows also in this case (recall (3.25)). As a consequence of (3.42) and (3.43) we obtain in particular that ess sup which we use in the estimate of integral In detail, employing (3.49), (3.23), (3.30), changing the integration variables −iζ(x) = |ζ(x)| = t and using (2.7) and (2.4) in the last steps, we get (3.51) Thus in summary, using (2.4), (3.25) and ν ≥ −1, we get We continue with the integral over (0, x λ − δ), see (3.37), where we use the representation of u 2 from (3.21), i.e.

Eigenfunctions of Schrödinger operators with even single-well potentials
In this section, we collect technical lemmas and proofs of results summarized in Section 3.1; these are used in the proof of the main Theorem 2.3. Notice that in this section we do not assume that (1.2) holds. The proofs follow mostly the reasoning in [18, §22.27] and [8].  .2), let x λ , a λ , ζ be as in (3.11) and δ, δ 1 as in (3.18). Let ε ∈ (0, 1). Then, for all sufficiently large λ > 0 and all sufficiently large x, the following hold.
Proof. Using Assumption I, for ν > −1, we have (4.6) the case with j = 1 is similar.
Using (4.1) for V ′ and the mean value theorem in the last step, we get the case with x λ + εx −ν λ is analogous. The number δ must satisfy for otherwise ζ(x λ − δ) → +∞ by (4.2) and (3.25). Then, using the definition of δ, see (3.18), we get similarly as in (4.7), Lemma 4.2. Let V satisfy Assumption I, let u, v be as in (3.11) and let w 1 , w 2 be as in (3.24). Then, for all sufficiently large λ > 0, we have In the region around the turning point x λ , one has |ζ| ≤ 1 and so expansions of Bessel functions for a small argument are used, see e.g. [3,Chap. 10]. More precisely, for u and x λ − δ ≤ x ≤ x λ , one has, see (3.17), (4.11) Similarly as in (4.7), we obtain The case x λ < x < x λ + δ 1 is similar. The estimates for v are obtained analogously. Lemma 4.3. Let V satisfy Assumption I and u, x λ and a λ be as in (3.11).
• 0 ≤ s ≤ ξ 0 : Notice that ζ(s) λ (4.23) We give the estimate for any value of ε 1 ∈ (0, 1); ε 1 will be specified below, see (4.39), (4.24) The first integral on the r.h.s. is estimated using (4.7) (4.25) Since by (2.4) we have for the third integral on the r.h.s. in (4.24) that (we use (2.4) and (3.25)) Integration by parts in the second integral on the r.h.s. in (4.24), the choice of δ ′ λ and (4.1) lead to (4.28) Putting together the estimates above, we arrive at (4.29) • x λ + δ ′′ λ ≤ s: The estimates are again obtained for any value of ε 2 ∈ (0, 1) which will be specified later. The important observations are (based on the choice of δ ′′ λ and (2.4)) V Moreover, since V ′ (x) > 0 for all sufficiently large x > 0, and (see (2.4)) (4.33) We integrate by parts twice in the formula for ζ and obtain where Using (2.4), we obtain To estimate T , we first notice that by (2.4), (4.1) and (3.25) (4.38) Hence it is possible to select ε 1 ∈ (0, 1) so small that and so, using Taylor's theorem for ζ −2 and cancellations in K, one arrives at (using (4.38), (2.4) and (3.24)) Hence, The estimate and the choice of ε 2 in this region is analogous to the previous case. We omit the details.
In summary, putting all estimates together and using (3.25), we obtain the claim (4.22).
Proof of Theorem 3.3. We follow the steps in [8]; the main differences are the additional perturbation W and new estimate of J (λ) from Lemma 4.4.
Using (3.15) and variation of constants, we can find a solution (distributional, since W ∈ L 1 loc (R) only) of (3.28) by solving the integral equation G(x, s)(K(s) + W (s))y(s) ds, (4.42) where G(x, s) = u(x)v(s) − v(x)u(s). Using the notationf for a function f multiplied by w 1 w 2 , we rewrite the integral equation (4.42) aŝ Returning back to y, we obtain (3.29) and (3.30).
The estimate on J K is the main technical step of the proof, see Lemma 4.4 above, the decay of J W is guaranteed by Assumption II.

Concentration measures for orthogonal polynomials.
It is interesting to compare the concentration phenomenon (2.18) of measures (1.6) with its analogue in the case of orthogonal polynomials {p n (x)} for the weights exp(−|x| α ), α > 0, or even more general non-even weights w(x) = exp(−w(x)) with properly choseñ w. Following [11,13], let has the property, as n → ∞, where 0 < δ ≤ x ≤ 1 − δ with δ arbitrarily small and Formula (5.3) and elementary trigonometry imply that, as n → ∞, Thus, for any f ∈ C([−1, 1]), Riemann-Lebesgue lemma gives On the whole real line, one can use the following inequalities, see [13,Thm.19, p.16, Eq.(1.66)]. Let a > 1 and P be a polynomial of degree smaller than or equal to n. Then for all n ≥ 1; the constants C 1 , C 2 depend on a, but not on n or P . These inequalities imply for any bounded continuous function on R.
A striking difference between (5.10) and (2.18) is that in the case of orthogonal polynomials the concentration measure does not depend on α, orw in a more general case of weights exp(−w(x)).
The classical-quantum correspondence suggests that, in the high-energy limit, the L 2 -mass of an eigenfunction should be distributed in the same way as the average position of a classical particle: since a classical particle passes through an interval [x * , x * + dx] in physical space with velocity near η(x * ) or −η(x * ), where η(x * ) = λ − V (x * ), (5.12) we obtain the heuristic (for a normalization constant c 0 ) which agrees with Theorem 2.3 after the corresponding scaling.
To make this correspondence precise, one can use the notion of semiclassical defect measures (see, for instance, [20,Ch. 5]). The following discussion will be under weaker hypotheses than Theorem 2.3, because our goal is only to show that the precise asymptotics obtained agree with the semiclassical prediction.
Let V : R → R be even, smooth and suppose that there exists some β > 0 such that (5.14) Suppose also that V ′ (x) > 0, x > 0. (5.15) and that there exists x 0 > 0 such that the latter implies that, for |x| sufficiently large, We consider the semiclassical Schrödinger operator For instance, if V (x) = |x| β for β ∈ 2N, scaling gives a unitary equivalence Other potentials can be treated by rescaling and controlling the error, but this analysis is outside the aim of this work. We emphasize that the assumptions on Q in Theorem 2.3 are significantly weaker than the hypotheses on V here, cf. (1.2), Assumption I and II and comments in Introduction. Suppose that for λ 0 > inf V (x), there exists a sequence { k } k∈N of positive numbers tending to zero and eigenfunctions {u k } k∈N obeying u k = 1 and For each u k , one can define the functional Here, D x = −i d dx and b w (x, D x ) is the Weyl quantization (see e.g. [20,Ch. 4]); when b ∈ C ∞ c (R), the Weyl quantization of b is a compact operator on L 2 (R) which takes S ′ (R) to S (R).
Following [20,Thm. 5.2] there is a subsequence {u kj } j∈N with kj → 0 + for which the functionals ϕ k converge to a non-negative Radon measure µ in the sense that, for each b ∈ C ∞ c (R), We will show that this µ is unique and that therefore ϕ k → µ in the same sense since every subsequence admits a further subsequence tending to µ. for those x such that V (x) < λ 0 . There exists a measure ν + such that, when supp b ⊂ {ξ > 0}, then This corresponds to invariance of µ under the classical Hamilton flow associated to a(x, ξ), which in the case of a Schrödinger operator corresponds to (5.11).
Finally, since in our situation the support of µ is compact, we show that R 2 dµ(x, ξ) = 1 (5.22) as follows. For any b(x, ξ) ∈ C ∞ c (R) such that b ≡ 1 on {ξ 2 + V (x) = λ 0 }, we use that the Weyl quantization of the constant 1 function is the identity operator to write (5.23) By [20,Thm. 6.4], meaning that its L 2 (R) norm is smaller than any power of kj as kj → 0 + , and by the definition (5.17) of µ(x, ξ) and the fact that b ≡ 1 on supp µ, dµ(x, ξ). We now prove that a measure µ satisfying the properties of a semiclassical defect measure must have the form matching the classical heuristic (5.13) generalized in Theorem 2.3.
Finally, we conclude that c 0 = c + is such that dµ = 1 by the hypothesis (5.22).