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Asymptotic inverse spectral problem for anharmonic oscillators with odd potentials

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
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1088/0266-5611/5/3/007DOIUNSPECIFIED
http://iopscience.iop.org/0266-5611/5/3/007/PublisherUNSPECIFIED
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.
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Funding AgencyGrant Number
NSFDMS-8620231
Case Institute of TechnologyUNSPECIFIED
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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