Asymptotic inverse spectral problem for anharmonic oscillators |
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Authors: | David Gurarie |
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Institution: | 1. Department of Mathematics and Statistics, Case Western Reserve University, 44106, Cleveland, OH, USA
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Abstract: | We study perturbationsL=A+B of the harmonic oscillatorA=1/2(??2+x 2?1) on ?, when potentialB(x) has a prescribed asymptotics at ∞,B(x)~|x|?α V(x) with a trigonometric even functionV(x)=Σa mcosω m x. The eigenvalues ofL are shown to be λ k =k+μ k with small μ k =O(k ?γ), γ=1/2+1/4. The main result of the paper is an asymptotic formula for spectral fluctuations {μ k }, $$\mu _k \sim k^{ - \gamma } \tilde V(\sqrt {2k} ) + c/\sqrt {2k} ask \to \infty ,$$ whose leading term \(\tilde V\) represents the so-called “Radon transform” ofV, $$\tilde V(x) = const\sum {\frac{{a_m }}{{\sqrt {\omega _m } }}\cos (\omega _m x - \pi /4)} .$$ as a consequence we are able to solve explicitly the inverse spectral problem, i.e., recover asymptotic part |x ?α|V(x) ofB from asymptotics of {µ k }. 1 ∞ |
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