High Energy Asymptotics and Trace Formulas for the Perturbed Harmonic Oscillator

Abstract

A one-dimensional quantum harmonic oscillator perturbed by a smooth compactly supported potential is considered. For the corresponding eigenvalues λ n , 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 λ n in terms of the heat invariants.

Summation formula inequalities for eigenvalues of Schrödinger operators

“…we have the convergence of the series on the left-hand side of (1.2). This and other examples of this type of identity may be found in the books by Levitan and Sargsjan [13,14] (see also [7]) and have more recently been shown to have extensions to the case of the perturbed harmonic oscillator on the whole real line -see [17] and the references therein.…”

“…is the spectral zeta function associated with (1.3), the second equality being valid for Re s > 1, and ζ( . ) is the Riemann zeta function; see [ABP,Theorem 2] or [PS,Eq. (1.12)].…”

Inverse spectral theory as influenced by Barry Simon

“…A particularly interesting open problem in inverse spectral theory concerns the characterization of the isospectral class of potentials V with purely discrete spectra (e.g., the harmonic oscillator V (x) = x 2 , cf. [32]- [36], [99], [160], [180], [193]).…”

Trace formulas for the modified Mathieu equation

“…as → −∞, where potential ( ) grows as | | → ∞, and function ( ) is rapidly decaying. The interesting example ( ) = 2 was considered by A. Pushnitski and I. Sorrell in [19].…”