Asymptotic inverse spectral problem for anharmonic oscillators with odd potentials
- Creators
- Gurarie, David
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.
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.Additional details
- Eprint ID
- 31731
- Resolver ID
- CaltechAUTHORS:20120531-073158269
- DMS-8620231
- NSF
- Case Institute of Technology
- Created
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2012-05-31Created from EPrint's datestamp field
- Updated
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2022-07-12Created from EPrint's last_modified field