Abstract
We consider the limit measures induced by the rescaled eigenfunctions of Schrödinger operators with even confining potentials. 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.
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1 Introduction
Let A be a Schrödinger operator acting in \(L^2({\mathbb {R}})\)
where Q is a real-valued, even potential which tends to \(+\infty \) as \(|x| \rightarrow \infty \). More precisely, we suppose that \(Q=V+W\), where V is sufficiently regular (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
where
As explained in [17, Sec. 1.3], the existence of the limiting function in (1.2) already implies that \(\omega _\beta \) is a power of |x| or zero; functions V satisfying (1.2) with \(\beta <\infty \) 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 \(\{\lambda _k\}\) of A are simple, thus they can be ordered increasingly and the corresponding eigenspaces are one-dimensional. Since the potential Q is real, eigenfunctions \(\{\psi _k\}\) related to \(\{\lambda _k\}\) can be selected as real functions satisfying
These conditions do not determine \(\psi _k\) uniquely, since \(-\psi _k\) satisfies the same conditions; nonetheless, the squares \(\{\psi _k^2\}\) are already uniquely determined.
Let \(x_{\lambda _k}\) be positive turning points of V corresponding to eigenvalues \(\{\lambda _k\}\), i.e.
here \(k_0 \in \mathbb {N}\) is sufficiently large so that \(x_{\lambda _k}\) are well-defined by (1.5), see also Assumption I. We define non-negative normalized measures on \({\mathbb {R}}\) induced by the eigenfunctions \(\{\psi _k\}\) by
This rescaling transforms the classically forbidden region \(\{x \, : \, V(x) > \lambda _k\}\) with (super)-exponential decay of \(\psi _k\) to \({\mathbb {R}}\setminus [-1,1]\) while the rescaled functions \(\psi _k(x_{\lambda _k}\cdot )\) oscillate in \([-1,1]\). Notice that W enters the definition of \(\{x_{\lambda _k}\}\), and thus the rescaling of eigenfunctions, since \(\{\lambda _k\}\) are eigenvalues of the operator with the potential \(Q = V+W\); the assumptions on the size of W comparing to V, see Assumption II and Proposition 2.2, allow for treating W perturbatively.
In this paper, we prove that measures (1.6) converges (as \(k \rightarrow \infty \)) to a limiting concentration measure supported on \([-1,1]\)
see Theorem 2.3. This generalizes the classical result for the harmonic oscillator, i.e. \(Q(x)=x^2\), namely the arcsine law for the concentration measure
of the Hermite functions. Limiting measures of the type (1.7) were found for rescaled eigenfunctions with a different normalization for polynomial, possibly complex, potentials in [3, Thm. 2]. The concentration of eigenfunctions is in particular used in estimates of norms of the spectral projections of non-self-adjoint Schrödinger operators obtained through conjugation, see [14], in particular, Sect. 3.
Notice that the condition (1.2) does not require V to be a polynomial. For instance, the potentials below satisfy both technical Assumption I and the condition (1.2):
lead to the limit
while for the fast-growing potentials
the limit reads
the latter is not a special case, see Proposition 2.1.i). Moreover, one can include further, possibly irregular and unbounded perturbations W, see Proposition 2.2 for examples of admissible W.
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 Sect. 5.2 below.
This paper is organized as follows. Our results with precise assumptions are formulated in Sect. 2 and they are proved in Sect. 3 relying on asymptotic formulas for the eigenfunctions \(\{\psi _k\}\) summarized in Sect. 3.1. In Sect. 4 we prove the asymptotic formulas following and slightly extending the ideas and results in the book [18, §22.27] and in [7]. Finally, in Sect. 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 Sect. 3.1. In particular, to avoid many appearing constants, for \(a,b \ge 0\), we write \(a\lesssim b\) if there exists a constant \(C>0\), independent of any relevant variable or parameter, such that \(a\le Cb\); the relation \(a\gtrsim b\) is introduced analogously. By \(a\approx b\) it is meant that \(a\lesssim b\) and \(a\gtrsim b\). The natural numbers are denoted by \(\mathbb {N}= \{1,2,\dots \}\) and \(\mathbb {N}_0=\mathbb {N}\cup \{0\}\).
2 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 [7, 18], guarantee that V is an even confining potential with sufficient regularity to obtain convenient asymptotic formulas for eigenfunctions (associated with large eigenvalues) of the corresponding Schrödinger operator, see Sects. 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
Let \(V: {\mathbb {R}}\rightarrow {\mathbb {R}}\) satisfy the following conditions.
-
(i)
\(V \in C({\mathbb {R}}) \cap C^2({\mathbb {R}}\setminus \{0\})\) is even,
$$\begin{aligned} \lim _{|x| \rightarrow +\infty } V(x) = +\infty , \end{aligned}$$(2.1) -
(ii)
there exists \(\xi _0>0\) such that \(V \in C^3({\mathbb {R}}\setminus [-\xi _0,\xi _0])\),
$$\begin{aligned} V(x)>0, \ V'(x)>0, \qquad x \ge \xi _0, \end{aligned}$$(2.2)and
$$\begin{aligned} \frac{V'^2}{V^\frac{5}{2}} \in L^1((\xi _0,\infty )), \qquad \frac{V''}{V^\frac{3}{2}} \in L^1((\xi _0,\infty )), \end{aligned}$$(2.3) -
(iii)
there exists \(\nu \ge -1\) such that for all \(x \ge \xi _0\)
$$\begin{aligned} \begin{aligned} V'(x)&\approx V(x) x^\nu , \\ |V''(x)|&\lesssim V'(x) x^\nu , \quad |V'''(x)| \lesssim V'(x) x^{2\nu }. \end{aligned} \end{aligned}$$(2.4)\(\square \)
Assumption I is an extension of conditions in [18, §22.27] where the case \(\nu =-1\), i.e. polynomial-like potentials, is analyzed; conditions analogous to Assumption I are used also in [1, 9] where the resolvent estimates of non-self-adjoint Schrödinger operators are given. The assumptions of [7] allow for fast growing potentials and are based on suitable restrictions of \(V'''\), see [7, Condition 2].
The first assumption (2.4) implies there are two constants \(0< c_1 \le c_2 < \infty \) such that for all \(x \ge \xi _0\)
This can be seen from (with \(\xi _0\le x_1 \le x_2\))
The crucial technical observation used frequently in the proofs is that (2.4) imply that, for any \(\varepsilon \in (0,1)\) 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 \(x^{-\nu }\), 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 \(B=-{\mathrm{d}}^2/{\mathrm{d}}x^2 +V(x)\), namely,
see [4, 5, 8]; note that the separation property might be lost for A due to the possibly irregular W.
The following proposition relates the parameter \(\nu \) and the condition (1.2).
Proposition 2.1
Let V satisfy Assumption I.
-
(i)
If \(\nu >-1\), then V satisfies the condition (1.2) with \(\beta = \infty \).
-
(ii)
If \(\nu =-1\) and V satisfies the condition (1.2), then \(\beta \in (0,\infty )\).
Proof
Let \(x \in (0,1)\) be fixed. From (2.6), we have that for all \(t \ge \xi _0/x\)
Thus, if \(\nu >-1\), we get that for every \(x \in (0,1)\)
If \(\nu =-1\) and the condition (1.2) holds, then for every \(x \in (0,1)\)
where \(\beta _1, \beta _2 \in (0,\infty )\) are independent of x. \(\square \)
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 \(w_1^{-2}\) 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 \(x_\lambda \) of V, the quantity \(a_\lambda \) (the value of \(V'\) at the turning point) and a “natural small region” around the turning point (characterized by \(\delta \) and \(\delta _1\)), see Sect. 3.1 for details. Examples of perturbations satisfying Assumption II are given in Proposition 2.2 below.
Assumption II
Let \(w_1\) be as in (3.24) below. Let \(W: {\mathbb {R}}\rightarrow {\mathbb {R}}\) be even, locally integrable and satisfy
\(\square \)
Proposition 2.2
Let \(V(x) = |x|^\beta \), \(\beta >0\), and let \(W = W_1+W_2\) where \({\text {supp}}W_1\) is compact, \(W_1 \in L^1({\mathbb {R}})\), \(W_2 \in L^\infty _\mathrm{\mathrm {loc}}({\mathbb {R}})\) and let \(|W_2(x)| \lesssim |x|^\gamma \), \(x \in {\mathbb {R}}\), for some \(\gamma \in {\mathbb {R}}\). Then (2.13) is satisfied if \(\beta >2\gamma +2\). Moreover, if \(\beta >1\), already \(W_1 \in L^1({\mathbb {R}})\) suffices (one can omit the condition on the compactness of support of \(W_1\)).
Proof
For all large \(\lambda >0\), we get (let \({\text {supp}}W_1 \subset [-x_1,x_1]\))
For \(\beta >1\) and \(W_1\in L^1({\mathbb {R}})\) without the condition on \({\text {supp}}W_1\), one can use (3.20) and (3.19) to obtain
Next, changing the integration variable \(s = x_\lambda t\) and using (3.19), we get (with the assumption \(\beta > 2\gamma +2\))
\(\square \)
Conditions on W in Proposition 2.2, in particular \(\beta >2\gamma +2\) or \(W \in L^1({\mathbb {R}})\) when \(\beta >1\), arise also in [9, 13], where the Riesz basis property of eigenfunctions, eigenvalue asymptotics and resolvent estimates are analyzed for complex W.
Our main result reads as follows.
Theorem 2.3
Let \(Q =V + W\) where V and W satisfy Assumptions I and II, respectively. Let V satisfy in addition the condition (1.2) and let \(\{\mu _k\}\), \(\mu _*\) be as in (1.6), (1.7), respectively. Let
Then, for every \(f \in \mathcal F_V\), we have
Hence, in particular, the measures \(\{\mu _k\}\) converge weakly to the limit measure \(\mu _*\) as \(k \rightarrow \infty \).
2.1 Distribution of Zeros
We remark that the related result on the number of zeros of the eigenfunction \(\psi _k\) in \([-\varepsilon x_{\lambda _k}, \varepsilon x_{\lambda _k}]\), \(\varepsilon \in (0,1]\), denoted by \(N_k(\varepsilon x_{\lambda _k})\), 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. [6, 11, 16]. A generalization of (2.19) for polynomial, possibly complex, potentials has been given in [3].
The distribution of zeros of eigenfunctions \(\psi _k\), see (2.19), is closely related to the distribution of eigenvalues of A and it is essentially proved in [19, Sect. 7]. Indeed, without the perturbation W, i.e. \(W=0\), the eigenvalues of A satisfy
see [19, Sect. 7], [7, Theorem. 2], so (2.19) follows from [19, Lemma. 7.3, Theorem. 7.4]. To include W, one could check that (2.21) remains valid for \(V+W\), e.g. like in [13, Theorem. 6.6], and adjust the arguments in [19, Sect. 7]. Alternatively, one can use the asymptotic formulas for \(\{\psi _k\}\) and \(\{\psi _k'\}\) in Sect. 3.1; the latter can be derived by differentiating (4.43). The zeros of \(\psi _k\) for \(|x|<x_{\lambda _k}\) are in a neighborhood of the zeros of
and, for large \(\zeta \), using asymptotic formulas for Bessel functions, see [15, §10.17], these are in a neighborhood of zeros of
3 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 \(g \in L^\infty ((-1,1))\),
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.
First let \(x \in [0,1/2]\) and let \(\xi _0>0\) be as in Assumption I. Since \(V \in C({\mathbb {R}})\) and V(y) is positive and increasing for \(y \ge \xi _0\), see (2.2), we get that
Thus (2.1) and (1.2) imply that there exists \(\varepsilon _0>0\) such that for all \(x \in [0,1/2]\) and all \(t>t_0\) with \(t_0 \ge 2 \xi _0\) (independent of x) we have
Combining (3.3) and the assumption that V is eventually increasing on \({\mathbb {R}}_+\), see (2.2), we have that \(V(xt) \le V(t)\) for all \(x \in [0,1]\) 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 \(\varsigma >0\) (depending only on \(\nu \)) such that for all \(x \in [1/2,1)\) and all \(t \ge 2\xi _0\)
For \(\nu =-1\), combining (3.3) and (3.4) for \(x \in [\frac{1}{2},1)\), we arrive at the integrable bound
For \(\nu >-1\), we show that for all \(x \in [\frac{1}{2},1]\) and all sufficiently large \(t \ge 2\xi _0\) (independently of x)
To see this, we introduce \(y=1-x^{\nu +1} \in [0,y_0]\) with \(y_0=1-(1/2)^{\nu +1}<1\) and \(s=\varsigma t^{\nu +1}\). Then (3.6) holds if
for all \(y \in [0,y_0]\) and all large \(s>0\) (independently of y). Since \(e^{sy} \ge 1+sy\), we get
thus (3.7) holds if
Hence the sought integrable bound reads
\(\square \)
3.1 Summary of Properties of Eigenfunctions of Schrödinger Operators
We summarize properties eigenfunctions of Schrödinger operators with even confining potentials \(Q=V+W\) satisfying Assumptions I and II. The details and proofs are given in Sect. 4; this slightly extends the reasoning in [18, §22.27] and [7].
Since Q is an even function by assumption, we can restrict ourselves to \((0,+\infty )\). Following the notations of [7], we introduce (for large enough \(\lambda >0\))
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
where
moreover, the Wronskian of u and v satisfies
The \(L^2\)-solution of Schrödinger equation \(-y''+Qy=\lambda y\) is then found by solving the integral equation (obtained by variation of constants)
where \(G(x,s) = u(x) v(s) - v(x) u(s)\), see Theorem 3.3 and its proof in Sect. 4.
Next, for \(0\le x < x_\lambda \), one gets
The positive numbers \(\delta \) and \(\delta _1\) are defined by
and they satisfy
see Lemma 4.1 and its proof for details. As \(\lambda \rightarrow +\infty \), we have
see Lemma 4.1 below.
If \(|x| < x_\lambda \) stays away from turning points, \(\zeta \) is large and so it is useful to employ asymptotic formulas for Bessel functions with large argument, see [15, §10.17]. In particular, one obtains
where (see also [7, Sec. 7])
For the absolute values of u and v, we have that, for all large enough \(\lambda >0\),
with the weights
see Lemma 4.2 below. Notice that \(\arg \zeta (x) = \pi /2\) for \(x > x_\lambda \) thus |u(x)| is exponentially decreasing while |v(x)| is allowed to be exponentially increasing as \(x \rightarrow +\infty \).
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 \(x_\lambda \) and \(a_\lambda \) be as in (3.11). Then, as \(\lambda \rightarrow + \infty \),
Proof
The claims follow from \(V'(x) \approx V(x) x^\nu \) for x sufficiently large, see (2.4), and
together with (2.3). \(\square \)
Finally, we have that
see Lemma 4.3 below.
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 Sect. 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 \(\lambda >0\) 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 \(x_\lambda \) and u be as in (3.11), let \(w_1\), \(w_2\) be as in (3.24), let \(\kappa _\lambda \) as in (3.12) and let \(\mathcal J_W\) be as in (2.13). Then, for every sufficiently large \(\lambda >0\), there is a solution of
on \((0,+\infty )\) such that
where
and
Moreover
3.2 Proof of Theorem 2.3
Since the eigenfunctions \(\{\psi _k\}\) are even or odd, we consider only \(x \in (0,\infty )\). We select the eigenfunctions \(\{\psi _k\}\) such that
where \(y_k=y(\cdot ,\lambda _k)\), \(u_k=u(\cdot ,\lambda _k)\) and \(r_k=y_k-u_k\), see Sect. 3.1 and in particular Theorem 3.3. Hence, the densities \(\{\phi _k\}\) of the measures \(\{\mu _k\}\), see (1.6), satisfy
In the sequel, notations and results summarized in Sect. 3.1 are used, moreover, we introduce the constant (for \(\beta \in (0,\infty ]\))
We also drop the subscript k and work with quantities like \(y=y(\cdot ,\lambda )\) as \(\lambda \rightarrow +\infty \).
First, Lemma 3.1, (3.32) and the change of integration variables \(x=x_{\lambda } t\) imply
Thus with \(f \in \mathcal F_V\), see (2.17), and the change of integration variables, we get
the integral indeed converges for \(f \in \mathcal F_V\) as can be seen from (3.42), (3.43) below and the behavior of y at infinity, see (3.29), (3.30), (3.23) and (3.24).
First we show that the contribution from the region around the turning point is negligible. It follows from (3.19) and (3.25) that
hence, since \(f \in L^\infty _\mathrm{loc}({\mathbb {R}})\),
Employing estimates (3.23), (3.30), (3.39) and (3.19) in the last step, we obtain
Similarly, since \(x_\lambda ^{-\nu } \le x_\lambda \) and \(\delta _1 = o(x_\lambda ^{-\nu })\) as \(\lambda \rightarrow +\infty \), see (3.19), we get (using (3.23), (3.20), changing the integration variables \(-\mathrm{i}\zeta (x) = |\zeta (x)|=t\) and observing that \(|\zeta (x)|'=(V(x)-\lambda )^{1/2}\))
We investigate the region \((x_{\lambda }+x_{\lambda }^{-\nu }/2,\infty )\) and also explain the convergence of the integral in (3.37). To this end, we recall that by assumption \(f \in \mathcal F_V\), see (2.17), thus with some \(M>0\)
and we show below that
To prove (3.43), notice that for \(x > x_\lambda \) and assuming that \(\lambda \) is sufficiently large that \(x_\lambda > \xi _0\)
Thus, for \(x>x_{\lambda }+x_{\lambda }^{-\nu }/2\),
Hence for \(\nu <0\) we immediately arrive at
For \(\nu \ge 0\), we use (2.6) to get (with \(\xi _0>0\) 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
which we use in the estimate of integral
In detail, employing (3.49), (3.23), (3.30), changing the integration variables \(-\mathrm{i}\zeta (x) = |\zeta (x)|=t\) and using (2.7) and (2.4) in the last steps, we get
Thus in summary, using (2.4), (3.25) and \(\nu \ge -1\), we get
We continue with the integral over \((0,x_\lambda -\delta )\), see (3.37), where we use the representation of \(u^2\) from (3.21), i.e.
The main contribution in (3.37) reads (employing Lemma 3.1)
Thus, to prove (2.18), we need to show that the remaining terms are negligible.
Employing the estimates on |u|, |r|, see (3.23), (3.30), we get by changing the integration variables \(x=x_{\lambda } t\) and applying Lemma 3.1 that (recall that \(f \in L^\infty _\mathrm{loc}({\mathbb {R}})\))
Thus the contribution from the integrals with \(2 u r+r^2\) is indeed negligible.
Using (3.22), (4.7), (4.4), (2.4) and (3.25), we obtain (recall that \(f \in L^\infty _\mathrm{loc}({\mathbb {R}})\), \(-\zeta ' = (\lambda -V)^\frac{1}{2}\) and see also (4.20))
Finally, we analyze the term with \(\sin 2 \zeta \), see (3.53). For every \(\varepsilon >0\) there is \(g \in C_0^\infty ((0,1))\) such that \(\Vert f-g\Vert _{L^1((0,1))}<\varepsilon \). With this \(\varepsilon >0\), we define \(\delta _\varepsilon {:}{=}\varepsilon x_\lambda ^{-\nu }\); notice that \(\delta = o(\delta _\varepsilon )\) as \(\lambda \rightarrow +\infty \), see (3.19). Then
Using that \(f \in L^\infty _\mathrm{loc}({\mathbb {R}})\), (2.7), (2.4) and the mean value theorem (with \(\eta _\lambda \in (x_\lambda -\delta _\varepsilon ,x_\lambda )\)),
From \(\Vert f-g\Vert _{L^1((0,1))}<\varepsilon \), (2.7) and (2.4), we get
By integration by parts and (3.20),
Putting the estimates from above together, we finally obtain
thus the claim (2.18) follows since \(\varepsilon >0\) was arbitrary. \(\square \)
4 Eigenfunctions of Schrödinger Operators with Even Confining Potentials
In this section, we collect technical lemmas and proofs of results summarized in Sect. 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 [7].
Lemma 4.1
Let V satisfy Assumption I, let \(\xi _0\) be as in (2.2), let \(x_\lambda \), \(a_\lambda \), \(\zeta \) be as in (3.11) and \(\delta \), \(\delta _1\) as in (3.18). Let \(\varepsilon \in (0,1)\). Then, for all sufficiently large \(\lambda >0\) and all sufficiently large x, the following hold.
Proof
Using Assumption I, for \(\nu >-1\), we have
for \(\nu =-1\),
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 \(x_\lambda + \varepsilon x_\lambda ^{-\nu }\) is analogous.
The number \(\delta \) must satisfy
for otherwise \(\zeta (x_\lambda -\delta ) \rightarrow + \infty \) by (4.2) and (3.25). Then, using the definition of \(\delta \), see (3.18), we get similarly as in (4.7),
and thus (4.3) follows. The reasoning for \(\delta _1\) is analogous.
Relations (4.4) follow by the mean value theorem, (4.8), (4.1) and (4.3). \(\square \)
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 \(\lambda >0\), we have
Proof
For \(x \in (0,x_\lambda -\delta ) \cup (x_\lambda +\delta _1, \infty )\), where \(|\zeta |>1\), the inequalities (4.10) follow from the definitions of u and v and asymptotic expansions of the corresponding Bessel functions for a large argument, see e.g. [15, Chap. 10]; we omit details.
In the region around the turning point \(x_\lambda \), one has \(|\zeta | \le 1\) and so expansions of Bessel functions for a small argument are used, see e.g. [15, Chap. 10]. More precisely, for u and \(x_\lambda -\delta \le x \le x_\lambda \), one has, see (3.17),
Similarly as in (4.7), we obtain
thus \(|u(x)| \approx a_\lambda ^{-\frac{1}{6}}\). The case \(x_\lambda< x < x_\lambda + \delta _1\) is similar.
The estimates for v are obtained analogously. \(\square \)
Lemma 4.3
Let V satisfy Assumption I and u, \(x_\lambda \) and \(a_\lambda \) be as in (3.11). Then
Proof
Using (3.21), we obtain
First we notice that
Using (4.10) and (4.3), we get
Since \(\delta \approx a_\lambda ^{-\frac{1}{3}} = o(x_\lambda ^{-\nu })\) as \(\lambda \rightarrow +\infty \), see (4.3) and (4.8), using (4.1), we get
Using (4.10), the definition (3.18) of \(\delta _1\) and (4.4), we have
The second mean value theorem for integrals (from which the point \(\xi _1=\xi _1(\lambda )\) arises below), the fact that V is increasing for \(x>\xi _0\) (see (2.2)) and (4.4) yield (recall that by (3.11) \(-\zeta ' = (\lambda -V)^\frac{1}{2}\))
Using (3.22), (4.7) and (4.4), we have
From (2.4) we have
thus the claim (4.13) follows by putting together all estimates from above (and (3.25)). \(\square \)
Lemma 4.4
Let V satisfy Assumption I, let K be as in (3.14), let \(w_1\) be as in (3.24) and let \(\kappa _\lambda \) be as in (3.12). Then
Proof
We follow and extend the strategy in [18, §22.27]. We split the integral into several regions; we define \(\delta _\lambda '{:}{=}\varepsilon _1 x_\lambda ^{-\nu }\) and \(\delta _\lambda ''{:}{=}\varepsilon _2 x_\lambda ^{-\nu }\), where \(\varepsilon _1, \varepsilon _2 \in (0,1)\) will be determined below.
\(\bullet \) \(0 \le s \le \xi _0\): Notice that \(\zeta (s) \gtrsim \lambda ^\frac{1}{2}\), hence (recall that \(-\zeta ' = (\lambda -V)^\frac{1}{2}\))
\(\bullet \) \(\xi _0 \le s \le x_\lambda -\delta _\lambda '\): We give the estimate for any value of \(\varepsilon _1 \in (0,1)\); \(\varepsilon _1\) will be specified below, see (4.39),
The first integral on the r.h.s. is estimated using (4.7)
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 \(\delta _\lambda '\), (4.1) and (4.27) lead to (with \(\xi _0 >0\) as in (2.2))
Putting together the estimates above, we arrive at
\(\bullet \) \(x_\lambda +\delta _\lambda '' \le s\): The estimates are again obtained for any value of \(\varepsilon _2 \in (0,1)\) which will be specified later. The important observations are (based on the choice of \(\delta _\lambda ''\) and (2.4))
Moreover, since \(V'(x)>0\) for all sufficiently large \(x>0\),
and (see (2.4))
we obtain (recall (3.25))
\(\bullet \) \(x_\lambda -\delta _\lambda ' \le s \le x_\lambda \): We integrate by parts twice in the formula for \(\zeta \) and obtain
where
Using (2.4) and (4.1), we obtain
To estimate T, we first notice that by (2.4), (4.1) and (3.25)
Thus, inserting \(V'(t)/V'(t)\) and using (4.1),
Hence it is possible to select \(\varepsilon _1 \in (0,1)\) so small that
Using Taylor’s theorem for \(\zeta ^{-2}\) and cancellations in K, one arrives at (employing (4.38), (2.4) and (3.24))
in the first step we use in addition (see (3.24) and (3.20))
Hence,
\(\bullet \) \(x_\lambda \le s \le x_\lambda + \delta _\lambda ''\): The estimate and the choice of \(\varepsilon _2\) 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). \(\square \)
Proof of Theorem 3.3
We follow the steps in [7]; the main differences are the additional perturbation W and new estimate of \(\mathcal J_K(\lambda )\) from Lemma 4.4.
Using (3.15) and variation of constants, we can find a solution (distributional, since \(W \in L_\mathrm{loc}^1({\mathbb {R}})\) only) of (3.28) by solving the integral equation
where \(G(x,s) = u(x) v(s) - v(x) u(s)\). Using the notation \(\hat{f}\) for a function f multiplied by \(w_1 w_2\), we rewrite the integral equation (4.43) as
here
and \(|H(x,s)| \lesssim 1\) in \(0 \le x \le s\), see (3.23).
Let
If \(\mathcal J_{K+W}(\lambda ) = o(1)\) as \(\lambda \rightarrow +\infty \), then the norm of the integral operator in (4.44) in \(L^\infty ({\mathbb {R}}_+)\) decays as \(\lambda \rightarrow +\infty \). Thus we can solve the equation (4.44), moreover, the solution can be expressed as
Returning back to y, we obtain (3.29) and (3.30).
The estimate on \(\mathcal J_K\) is the main technical step of the proof, see Lemma 4.4 above, the decay of \(\mathcal J_W\) is guaranteed by Assumption II.
Finally, the formula (3.32) for the \(L^2\)-norm of y follows from (3.27) as in [7, Thm. 1]. Namely,
and
see the proof of Lemma 4.3 for more details on the estimates. The claim (3.32) then follows from (3.27), (4.49) and \(\Vert \hat{r}(2 \hat{u} +\hat{r})\Vert _{L^\infty } \lesssim C(\lambda )\), see (4.47) and (3.23).
5 Comparison with Existing Results
5.1 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 \(\exp (-|x|^\alpha )\), \(\alpha >0\), or even more general non-even weights \(w(x)= \exp (-\tilde{w}(x))\) with properly chosen \(\tilde{w}\). Following [10, 12], let
the corresponding system of orthogonal polynomials \(\{p_n(x)\}\)
has the property that, for sufficiently small \(\delta > 0\) and for every \(x \in [\delta , 1-\delta ]\), as \(n \rightarrow \infty \),
where
and where the implicit constant in \(\mathcal {O}(n^{-1})\) is allowed to depend on \(\delta \). Formula (5.3) and elementary trigonometry imply that, as \(n \rightarrow \infty \),
Thus, for any \(f \in C([-1,1])\), Riemann-Lebesgue lemma gives
The analogous limit holds on the interval \([-1+\delta , -\delta ]\) because the polynomials \(p_n\) are either even or odd. Moreover, by [10, Thm.1.16],
so
On the whole real line, one can use the following inequalities, see [12, 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 \(n \ge 1\); the constants \(C_1\), \(C_2\) depend on a, but not on n or P. These inequalities imply
for any bounded continuous function on \({\mathbb {R}}\).
A striking difference between (5.10) and (2.18) is that in the case of orthogonal polynomials the concentration measure does not depend on \(\alpha \), or \(\tilde{w}\) in a more general case of weights \(\exp (-\tilde{w}(x))\).
5.2 Semi-classical Defect Measures
In classical mechanics, cf. [2], a particle with position x(t) subject to the differential equation
remains for all times on the energy surface
and travels along the trajectory \((\dot{x}(t), \dot{\xi }(t))\) obeying
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 \([x_*, x_* + {\mathrm{d}}x]\) in physical space with velocity near \(\eta (x_*)\) or \(-\eta (x_*)\), where
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 \(V:{\mathbb {R}}\rightarrow {\mathbb {R}}\) be even, smooth and suppose that there exists some \(\beta > 0\) such that
Suppose also that
and that there exists \(x_0 > 0\) such that
the latter implies that, for |x| sufficiently large,
We consider the semiclassical Schrödinger operator
in the limit \(\hbar \rightarrow 0^+\).
For example, if \(V(x) = |x|^\beta \) for \(\beta \in 2\mathbb {N}\) (so that V(x) is infinitely differentiable), 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 \(\lambda _0 > \inf V(x)\), there exists a sequence \(\{\hbar _k\}_{k \in \mathbb {N}}\) of positive numbers tending to zero and eigenfunctions \(\{u_{k}\}_{k \in \mathbb {N}}\) obeying \(\Vert u_k\Vert = 1\) and
For each \(u_{k}\), one can define the functional
Here, \(D_x=-\mathrm{i}\frac{{\mathrm{d}}}{{\mathrm{d}}x}\) and \(b_\hbar ^w(x,\hbar D_x)\) is the Weyl quantization (see e.g. [20, Ch. 4]); when \(b\in C_c^\infty ({\mathbb {R}}^2)\), the Weyl quantization of b is a compact operator on \(L^2({\mathbb {R}})\) which takes \(\mathscr {S}'({\mathbb {R}})\) to \(\mathscr {S}({\mathbb {R}})\).
Following [20, Thm. 5.2] there is a subsequence \(\{u_{k_j}\}_{j \in \mathbb {N}}\) with \(\hbar _{k_j} \rightarrow 0^+\) for which the functionals \(\varphi _{k}\) converge to a non-negative Radon measure \(\mu \) in the sense that, for each \(b\in C_c^\infty ({\mathbb {R}}^2)\),
We will show that this \(\mu \) is unique and that therefore \(\varphi _{k} \rightarrow \mu \) in the same sense since every subsequence admits a further subsequence tending to \(\mu \).
By [20, Thm. 5.3 or Thm. 6.4],
so let us define, in analogy with (5.12),
for those x such that \(V(x) < \lambda _0\). There exists a measure \(\nu _+\) such that, when \({\text {supp}}b \subset \{\xi > 0\}\), then
By [20, Thm. 5.4], for any \(b \in C_c^\infty ({\mathbb {R}}^2)\),
where the Poisson bracket \(\{a, b\}\) of the symbol \(a(x,\xi ) = \xi ^2 + V(x)\) of \(A_\hbar \) with b is
This corresponds to invariance of \(\mu \) under the classical Hamilton flow associated to \(a(x,\xi )\), which in the case of a Schrödinger operator corresponds to (5.11).
Finally, since in our situation the support of \(\mu \) is compact, we show that
as follows. For any \(b(x,\xi ) \in C_c^\infty ({\mathbb {R}}^2)\) such that \(b \equiv 1\) on \(\{\xi ^2 + V(x) = \lambda _0\}\), we use that the Weyl quantization of the constant 1 function is the identity operator to write
By [20, Thm. 6.4],
meaning that its \(L^2({\mathbb {R}})\) norm is smaller than any power of \(\hbar _{k_j}\) as \(\hbar _{k_j} \rightarrow 0^+\), and by the definition (5.17) of \(\mu (x, \xi )\) and the fact that \(b \equiv 1\) on \({\text {supp}} \mu \),
Taking (5.23), (5.24), and (5.25) together proves (5.22).
We now prove that a measure \(\mu \) satisfying the properties of a semiclassical defect measure must have the form matching the classical heuristic (5.13) generalized in Theorem 2.3.
Proposition 5.1
Let \(V(x) \in C^\infty ({\mathbb {R}}; {\mathbb {R}})\) satisfy (5.14), (5.16), and (5.15). Let \(\lambda _0 > V(0) = \inf V(x)\), and let \(\mu \) be a measure satisfying (5.18), (5.21), and (5.22) and let \(\eta \) be as in (5.19). Then the measure \(\mu \) obeys for all \(b \in C_c^\infty ({\mathbb {R}}^2)\)
where the normalization constant \(c_0\) is such that \(\int {\mathrm{d}}\mu = 1\).
Proof
We observe that
Letting \(b\in C_c^\infty ({\mathbb {R}}^2)\) be such that \({\text {supp}}b \subset \{\xi > \delta \}\) for some \(\delta > 0\), we obtain from (5.20), (5.21), and (5.26) that
vanishes. Taking \(b(x,\xi ) = f(x)\chi _{[\delta , \delta ^{-1}]}(\xi )\) for \(f\in C_c^\infty ({\mathbb {R}})\) arbitrary and for \(\chi \) a cutoff function, letting \(\delta \rightarrow 0^+\) allows us to conclude that
for all \(f\in C_c^\infty ({\mathbb {R}})\). Therefore along \(\{\xi ^2 + V(x) = \lambda _0\}\),
for some \(c_+\) which is positive because \(\mu \) is a positive measure.
When \({\text {supp}}b(x, \xi ) \subset \{\xi < 0\}\), the same argument shows that there is some \(c_- > 0\) such that
One can show then that \(c_+ = c_-\) by projecting onto the \(\xi \) variable instead of the x variable: let
where the inverse image is chosen nonnegative. Note that because V(x) is strictly increasing and unbounded on \([0, \infty )\), one may define \(\tilde{x}(\xi )\) if and only if \(\lambda _0 - \xi ^2 \ge V(0)\). Let \({\mathrm{d}}\rho _+(\xi )\) be such that when \({\text {supp}}b \subset \{x > 0\}\),
Then \(\tilde{x}'(\xi ) = -\frac{2\xi }{V'(\tilde{x}(\xi ))}\),
The earlier argument (along with the fact that \(V'(x) > 0\) for \(x > 0\)) shows that there is some \(d_+ > 0\) such that
On \(\{\xi ^2 + V(x) = \lambda _0\}\), note that
Since the pull-backs of \({\mathrm{d}}\nu _\pm (x) = \frac{c_\pm }{|\xi |}{\mathrm{d}}x\) and \({\mathrm{d}}\rho _+\) agree on \(a^{-1}(\{\lambda _0\}) \cap \{x>0, \xi >0\}\) and since \({\mathrm{d}}\rho _+\) and \({\mathrm{d}}\nu _-\) agree on \(\{x > 0, \xi < 0\}\) we can conclude that \(c_+ = d_+/2 = c_-\). We remark that this argument is not available in the case \(\omega _\beta = 0\) corresponding to a very rapidly-growing potential.
Finally, we conclude that \(c_0 = c_+\) is such that \(\int {\mathrm{d}}\mu = 1\) by the hypothesis (5.22). \(\square \)
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Communicated by Fabio Nicola.
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P.S. acknowledges the support of the Swiss National Science Foundation, SNSF Ambizione Grant No. PZ00P2_154786, till December 2017 and of the OSU for his stays there in October 2017 and May 2018. J.V. acknowledges the support of the Région Pays de la Loire through the project EONE (Évolution des Opérateurs Non-Elliptiques). The authors acknowledge the Research in Paris stay at the Institute of Henri Poincaré, Paris, September 3–23, 2018, during which an essential part of the work has been done. We are grateful to our colleagues, Doron Lubinsky, Georgia Tech., Atlanta, Georgia; Andrei Martinez-Finkelshtein, Baylor University, Waco, Texas; Paul Nevai, the Ohio State University, Columbus, Ohio; Gabriel Rivière, Université de Nantes, for helpful discussions on literature and related results.
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Mityagin, B., Siegl, P. & Viola, J. Concentration of Eigenfunctions of Schrödinger Operators. J Fourier Anal Appl 28, 67 (2022). https://doi.org/10.1007/s00041-022-09961-3
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DOI: https://doi.org/10.1007/s00041-022-09961-3