Spectral asymptotics of harmonic oscillator perturbed by bounded potential
نویسندگان
چکیده
Consider the operator T = − d dx2 + x2 + q(x) in L2(R), where real functions q, q′ and ∫ x 0 q(s) ds are bounded. In particular, q is periodic or almost periodic. The spectrum of T is purely discrete and consists of the simple eigenvalues {μn}n=0, μn < μn+1. We determine their asymptotics μn = (2n+1)+(2π) −1 ∫ π −π q( √ 2n + 1 sin θ) dθ+O(n−1/3).
منابع مشابه
High energy asymptotics and trace formulas for the perturbed harmonic oscillator
A one-dimensional quantum harmonic oscillator perturbed by a smooth compactly supported potential is considered. For the corresponding eigenvalues λn, a complete asymptotic expansion for large n is obtained, and the coefficients of this expansion are expressed in terms of the heat invariants. A sequence of trace formulas is obtained, expressing regularised sums of integer powers of eigenvalues ...
متن کاملHigh energy asymptotics and trace formulae for the perturbed harmonic oscillator
A one-dimensional quantum harmonic oscillator perturbed by a smooth compactly supported potential is considered. For the corresponding eigenvalues λn, a complete asymptotic expansion for large n is obtained, and the coefficients of this expansion are expressed in terms of the heat invariants. A sequence of trace formulas is obtained, expressing regularised sums of integer powers of eigenvalues ...
متن کامل56 v 1 2 9 A ug 2 00 5 The inverse problem for perturbed harmonic oscillator on the half - line
We consider the perturbed harmonic oscillator TDψ = −ψ′′ + x2ψ + q(x)ψ, ψ(0) = 0 in L(R+), where q ∈ H+ = {q′, xq ∈ L(R+)} is a real-valued potential. We prove that the mapping q 7→ spectral data = {eigenvalues of TD} ⊕ {norming constants} is one-to-one and onto. The complete characterization of the set of spectral data which corresponds to q ∈ H+ is given. Moreover, we solve the similar invers...
متن کاملA semiclassical heat trace expansion for the perturbed harmonic oscillator
In this paper we study the heat trace expansion of the perturbed harmonic oscillator by adapting to the semiclassical setting techniques developed by Hitrick-Polterovich in [HP]. We use the expansion to obtain certain inverse spectral results.
متن کاملA remarkable spectral feature of the Schrödinger Hamiltonian of the harmonic oscillator perturbed by an attractive δ′-interaction centred at the origin: double degeneracy and level crossing
We rigorously define the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ′-interaction, of strength β, centred at 0 (the bottom of the confining parabolic potential), by explicitly providing its resolvent. Our approach is based on a ‘coupling constant renormalization’, related to a technique originated in quantum field theory and implemented in the rigorous mathe...
متن کامل