High energy asymptotics and trace formulae for the perturbed harmonic oscillator

Summation formula inequalities for eigenvalues of the perturbed harmonic oscillator

We derive explicit inequalities for sums of eigenvalues of one-dimensional Schrödinger operators on the whole line. In the case of the perturbed harmonic oscillator, these bounds converge to the corresponding trace formula in the limit as the number of eigenvalues covers the whole spectrum.

A semiclassical heat trace expansion for the perturbed harmonic oscillator

In this paper we study a semiclassical heat trace expansion for perturbations of the harmonic oscillator, by adapting to the semiclassical setting techniques developed by Hitrik and Polterovich in [HP]. We use the expansion to obtain certain inverse spectral results.

The spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions

We consider the operator $ L = - (d/dx)^2 + x^2 y + w(x) y , y \in L^2(\mathbb{R}) $, where $ w(x) = s [ δ(x - b) - δ(x + b)], b \neq 0,$ real, $s \in \mathbb{C}$. This operator has a discrete spectrum: eventually the eigenvalues are simple and $λ_n = (2n + 1) + s^2 (κ(n) / n) + ρ(n)$, where $ κ(n) = \frac{1}{2π} [(-1)^{n + 1} \sin ( 2 b \sqrt{2n} ) - \frac{1}{2} \sin ( 4 b \sqrt{2n} ) ]$ and $ |ρ(n) | \leq C (\log n) / (n^{3/2})$ If $s = i γ$, $γ$ real, the number $T(γ)$ of non-real eigenvalues is finite, and $T(γ) \leq [ C (1 + | γ|) \log (e + | γ|)]^2.$ The analogue of the above equations i…

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

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 analy…

谐振子型算子奇异摄动的根系统

Root system of singular perturbations of the harmonic oscillator type operators

本研究针对希尔伯特空间中谐振子型自伴算子的二次型摄动，在局部从属假设下证明摄动算子特征值最终单根化且根系统构成Riesz基。

Spectral Properties of the Massless Relativistic Harmonic Oscillator

The spectral properties of the pseudo-differential operator $(-d^2/dx^2)^{1/2}+x^2$ are analyzed by a combination of functional integration methods and direct analysis. We obtain a representation of its eigenvalues and eigenfunctions, prove precise asymptotic formulae, and establish various analytic properties. We also derive trace asymptotics and heat kernel estimates.

Asymptotic Analysis of Orthogonal Polynomials via the Transfer Matrix Approach

In this paper, we present a new method via the transfer matrix approach to obtain asymptotic formulae of orthogonal polynomials with asymptotically identical coefficients of bounded variation. We make use of the hyperbolicity of the recurrence matrices and employ Kooman's Theorem to diagonalize them simultaneously. The method introduced in this paper allows one to consider products of matrices such that entries of consecutive matrices are of bounded variation. Finally, we apply the asymptotic formulae obtained to solve the point mass problem on the real line when the measure is essentially sup…

Refined Weyl law for homogeneous perturbations of the harmonic oscillator

Let $H$ denote the harmonic oscillator Hamiltonian on $\mathbb{R}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schrödinger propagator $U(t)=e^{-itH},$ and find that while $\operatorname{singsupp} \operatorname{Tr} U(t) \subset 2 π\mathbb{Z}$ as in the unperturbed case, there exists a large class of perturbations in dimension $d \geq 2$ for which the singularities of $\operatorname{Tr} U(t)$ at nonzero multiples of $2 π$ are weaker than the singularity at $t=0$. The remainder term in the Weyl law is of order $o(λ^{d-1})$, improving in these cases the $…

Spectral Properties of the Massless Relativistic Quartic Oscillator

An explicit solution of the spectral problem of the non-local Schrödinger operator obtained as the sum of the square root of the Laplacian and a quartic potential in one dimension is presented. The eigenvalues are obtained as zeroes of special functions related to the fourth order Airy function, and closed formulae for the Fourier transform of the eigenfunctions are derived. These representations allow to derive further spectral propertiessuch as estimates of spectral gaps, heat trace and the asymptotic distribution of eigenvalues, as well as a detailed analysis of the eigenfunctions. A subtle…

Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem

We extend the perturbation theory of Vishik, Ljusternik and Lidskii for eigenvalues of matrices, using methods of min-plus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturbation formulae, extending the classical ones and solving cases which where singular in previous approaches. Our results include general weak majorisation inequalities, relating leading exponents of eigenvalues of perturbed matrices and min-plus analogues of eigenvalues.

Grothendieck-Lidskii trace formula for mixed-norm and variable Lebesgue spaces

In this note we present the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ and variable exponent Lebesgue type spaces. As a consequence, this also implies the same property for modulation and Wiener-Amalgam spaces. We then characterise nuclear operators on such spaces and state the corresponding Grothendieck-Lidskii trace formulae. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb R^n$ and functions of the harmonic oscillator in terms of the global symbol.