Gurarie, David
(1989)
*Asymptotic inverse spectral problem for anharmonic
oscillators with odd potentials.*
Inverse Problems, 5
(3).
pp. 293-306.
ISSN 0266-5611.
http://resolver.caltech.edu/CaltechAUTHORS:20120531-073158269

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## Abstract

The present paper continues our earlier work on asymptotic spectral problems for perturbations of the harmonic oscillator L =A + B on R, where A = ½(-∂^2 + x^2). The class of functions B is defined by their asymptotics at {∞} B(x)~ │x│^(-ɑ) x trigonometric V(x). The even-potential case of the author's earlier paper, i.e. V=Σ ɑ_m cos ω_mx, is now extended to a more difficult odd case, V = Σ b_m sin ω_mx. Precisely, the eigenvalues of L, λ_k = k + µ_k are shown to admit an asymptotic expansion µ_k ~ constant xk^(-y)V(√2k+... as k → ∞ where the exponent y = ½ ɑ + ¾, and V denotes the 'odd version' of the so-called 'Radon transform' of V. As a consequence we get a unique and explicit solution of the asymptotic inverse spectral problem for odd perturbations of A, similar to the even case previously consider by the author. The methods employed here combine some ideas of the n-sphere Schrödinger theory as well as the techniques developed by the author.

Item Type: | Article | |||||||||
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Additional Information: | © 1989 Institute of Physics. Received 18 February 1988. This research was partly supported by US NSF grant DMS-8620231 and a Case Institute of Technology research initiation grant. We acknowledge the hospitality extended to us by B Simon and D Wales during our stay at CALTECH. | |||||||||

Funders: |
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Subject Keywords: | Quantum information and quantum mechanics | |||||||||

Classification Code: | PACS: 03.65.Ge; 03.65.Db; 03.65.Fd. MSC: 81Q10; 41A60; 15A18; 81Q05; 44A12 | |||||||||

Record Number: | CaltechAUTHORS:20120531-073158269 | |||||||||

Persistent URL: | http://resolver.caltech.edu/CaltechAUTHORS:20120531-073158269 | |||||||||

Official Citation: | Asymptotic inverse spectral problem for anharmonic oscillators with odd potentials D Gurarie 1989 Inverse Problems 5 293 | |||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||

ID Code: | 31731 | |||||||||

Collection: | CaltechAUTHORS | |||||||||

Deposited By: | Ruth Sustaita | |||||||||

Deposited On: | 31 May 2012 15:50 | |||||||||

Last Modified: | 31 May 2012 15:50 |

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