Half-Line non-self-adjoint Schrödinger operators with polynomial potentials: Asymptotics of eigenvalues

2005/02/24 by Kwang C. Shin, Shin, Kwang C.
Mathematics · Physics and Astronomy · #34E05 #34E10 #34L20 #34L40 #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Spectral Theory (math.SP) #hep-th #math-ph #math.MP #math.SP #msc:34E05 #msc:34E10 #msc:34L20 #msc:34L40 #quant-ph

paper · pdf · doi:10.48550/arxiv.math/0502522

15 pages, 1 figure

arxiv created 2005/02/24 · arxiv updated 2009/12/01

Abstract

For integers m≥ 3, we study the non-self-adjoint eigenvalue problems -u^′′(x)+(x^m+P(x))u(x)=E u(x), 0≤ x<+∞, with the boundary conditions u(+∞)=0 and αu(0)+βu^′(0)=0 for some α, β∈\C with |α|+|β|\not=0, where P(x)=a₁ x^m-1+a₂ x^m-2+...+a_m-1 x is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues E_n. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues.

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