Semiclassical Cauchy estimates and applications

Jin, Long

doi:10.1090/tran/6715

Cited by 9 publications

(13 citation statements)

References 19 publications

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“…By Theorem 1, provided ε, δ, n are sufficiently small, for almost every (19) together with the Hausdorff measure estimates in (16) imply Theorem 4.…”

Section: Proof Of Theorems 3 Andmentioning

confidence: 74%

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Level Spacings and Nodal Sets at Infinity for Radial Perturbations of the Harmonic Oscillator

Beck

Hanin

2018

International Mathematics Research Notices

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We study properties of the nodal sets of high frequency eigenfunctions and quasimodes for radial perturbations of the Harmonic Oscillator. In particular, we consider nodal sets on spheres of large radius (in the classically forbidden region) for quasimodes with energies lying in intervals around a fixed energy E. For well chosen intervals we show that these nodal sets exhibit quantitatively different behavior compared to those of the unperturbed Harmonic Oscillator. These energy intervals are defined via a careful analysis of the eigenvalue spacings for the perturbed operator, based on analytic perturbation theory and linearization formulas for Laguerre polynomials.

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“…By Theorem 1, provided ε, δ, n are sufficiently small, for almost every (19) together with the Hausdorff measure estimates in (16) imply Theorem 4.…”

Section: Proof Of Theorems 3 Andmentioning

confidence: 74%

“…In A E , the eigenfunction ψ ,E behaves much like an eigenfunction of the Laplacian. For instance, if V is real analytic, then Jin [19] (2) Figure 1. Nodal sets of energy E eigenfunctions of HO have qualitatively different behavior in A E and F E .…”

Section: Introductionmentioning

confidence: 99%

Level Spacings and Nodal Sets at Infinity for Radial Perturbations of the Harmonic Oscillator

Beck

Hanin

2018

International Mathematics Research Notices

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“…The first results in this direction are due to Brüning and Gromes [6,7] who show that the length of the nodal set N (u) is bounded from below by a constant times √ λ. For further results in this direction (Yau's conjecture), we refer to [15,29,28,34,35]. In Section 7, we investigate this question for the harmonic oscillator.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…As a matter of fact, we prove a lower bound for more general Schrödinger operators in R 2 (Propositions 7.2 and 7.8), shedding some light on the exponent 3 2 in the above estimate. In Section 7.3, we investigate upper and lower bounds for the length of the nodal sets, using the method of Long Jin [29].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Some nodal properties of the quantum harmonic oscillator and other Schrödinger operators in ℝ²

Bérard¹,

Helffer²

2017

Geometric and Computational Spectral Theory

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For the spherical Laplacian on the sphere and for the Dirichlet Laplacian in the square, Antonie Stern claimed in her PhD thesis (1924) the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with exactly two nodal domains. These results were given complete proofs respectively by Hans Lewy in 1977, and the authors in 2014 (see also Gauthier-Shalom-Przybytkowski, 2006). In this paper, we obtain similar results for the two dimensional isotropic quantum harmonic oscillator. In the opposite direction, we construct an infinite sequence of regular eigenfunctions with as many nodal domains as allowed by Courant's theorem, up to a factor 1 4 . A classical question for a 2-dimensional bounded domain is to estimate the length of the nodal set of a Dirichlet eigenfunction in terms of the square root of the energy. In the last section, we consider some Schrödinger operators −∆ + V in R 2 and we provide bounds for the length of the nodal set of an eigenfunction with energy λ in the classically permitted region {V (x) < λ}.Arrange the eigenvalues in increasing order, λ 1 (q) < λ 2 (q) < · · · . A classical theorem of C. Sturm [22] states that an eigenfunction u of (1.1) associated with λ k (q) has exactly (k − 1) zeros in ]a, b[ or, equivalently, that the zeros of u divide ]a, b[ into k sub-intervals.In higher dimensions, one can consider the eigenvalue problem for the Laplace-Beltrami operator −∆ g on a compact connected Riemannian manifold (M, g), with Dirichlet condition in case M has a boundary ∂M , − ∆u = λu in M, u| ∂M = 0.( 1.2) Arrange the eigenvalues in non-decreasing order, with multiplicities,Denote by M 0 the interior of M , M 0 := M \ ∂M . Given an eigenfunction u of −∆ g , denote by N (u) := {x ∈ M 0 | u(x) = 0} (1.3) the nodal set of u, and by µ(u) := # { connected components of M 0 \ N (u)} (1.4) the number of nodal domains of u i.e., the number of connected components of the complement of N (u).Courant's theorem [12] states that if −∆ g u = λ k (M, g)u, then µ(u) ≤ k.In this paper, we investigate three natural questions about Courant's theorem in the framework of the 2D isotropic quantum harmonic oscillator.Question 1. In view of Sturm's theorem, it is natural to ask whether Courant's upper bound is sharp, and to look for lower bounds for the number of nodal domains, depending on the geometry of (M, g) and the eigenvalue. Note that for orthogonality reasons, for any k ≥ 2 and any eigenfunction associated with λ k (M, g), we have µ(u) ≥ 2.We shall say that λ k (M, g) is Courant-sharp if there exists an eigenfunction u, such that −∆ g u = λ k (M, g)u and µ(u) = k. Clearly, λ 1 (M, g) and λ 2 (M, g) are always Courantsharp eigenvalues. Note that if λ 3 (M, g) = λ 2 (M, g), then λ 3 (M, g) is not Courant-sharp.The first results concerning Question 1 were stated by Antonie Stern in her 1924 PhD thesis [36] written under the supervision of R. Courant.

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“…Then, since −h 2 ∆ g + 1 is h-elliptic, in view of (26), by the semiclassical Cauchy estimates [Jin,Theorem 2.6], for each x 0 ∈ M \B y ( ) there is a coordinate neighborhood U ⊂ (M \B y ( )) with x 0 ∈ U and a positive constant C 0 , such that for all x ∈ U and α ∈ N n ,…”

Section: The Green's Operator G(h) and Its Complexificationmentioning

confidence: 99%

Nodal Sets of Schrödinger Eigenfunctions in Forbidden Regions

Canzani

Toth

2016

Ann. Henri Poincaré

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Abstract. This note concerns the nodal sets of eigenfunctions of semiclassical Schrödinger operators acting on compact, smooth, Riemannian manifolds, with no boundary. We prove that if H is a separating hypersurface that lies inside the classically forbidden region, then H cannot persist as a component of the zero set of infinitely many eigenfunctions. In addition, on real analytic surfaces, we obtain sharp upper bounds for the number of intersections of the zero sets of the Schrödinger eigenfunctions with a fixed curve that lies inside the classically forbidden region.Let (M, g) be a smooth, compact, Riemannian manifold with no boundary. Write ∆ g for the Laplace operator, and given any smooth potential V ∈ C ∞ (M ; R), consider the Schrödinger operator acting on L 2 (M ) defined aswhere h ∈ (0, 1]. Let E ∈ R be a regular value for the total energy function p(x, ξ) = |ξ| 2 gx + V (x) defined on T * M , and write Ω E for the the classically forbidden region Ω E := {x ∈ M : V (x) > E}.In this paper we study the nodal sets of Schrödinger eigenfunctions (with energy close to E) inside the classically forbidden region, in the semiclassical limit h → 0 + . Consider L 2 -normalized Schrödinger eigenfunctions {φ h } withThere is a large literature devoted to the study of the zero sets of Laplace eigenfunctions, Z φ h = {x ∈ M : φ h (x) = 0}, on compact manifolds. We refer the reader to [Z2] for a detailed list of references. The Hausdorff measure of the zero sets, their distribution properties, the number of nodal domains and their inner radius, have been extensively studied (although many open problems remain, even for surfaces). More generally, it is natural to study the properties of zero sets of Schrödinger eigenfunctions inside the classically allowed region where V < E. Many of the known results in the homogeneous case where V = 0 extend to Schrödinger eigenfunctions in the allowable region (see [Jin]). In contrast, very little is known about the zero sets of Schrödinger eigenfunctions inside the classically forbidden region where V > E. In dimension one, it is known that the eigenfunctions of the Harmonic Oscillator have no zeros in the forbidden region and in recent work, Hanin-Zelditch-Zhou [HZZ] have proved that in any higher dimension Y

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Semiclassical Cauchy estimates and applications

Cited by 9 publications

References 19 publications

Level Spacings and Nodal Sets at Infinity for Radial Perturbations of the Harmonic Oscillator

Level Spacings and Nodal Sets at Infinity for Radial Perturbations of the Harmonic Oscillator

Some nodal properties of the quantum harmonic oscillator and other Schrödinger operators in ℝ²

Nodal Sets of Schrödinger Eigenfunctions in Forbidden Regions

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