Schrödinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues

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dc.creatorShin, Kwang C.
dc.date2004-11-07
dc.date.accessioned2026-07-07T05:14:03Z
dc.date.available2026-07-07T05:14:03Z
dc.descriptionFor integers $m\geq 3$ and $1\leq\ell\leq m-1$, we study the eigenvalue problem $-u^{\prime\prime}(z)+[(-1)^{\ell}(iz)^m-P(iz)]u(z)=λu(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\fracπ{2}\pm \frac{(\ell+1)π}{m+2}$ in the complex plane, where $P(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues $λ_{n}$. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues. Also, we show for arbitrary $\mathcal{PT}$-symmetric polynomial potentials of degree $m\geq 3$ and all symmetric decaying boundary conditions that the eigenvalues are all real and positive, with only finitely many exceptions.
dc.description31 pages, 1 figure
dc.identifierhttps://arxiv.org/abs/math/0411143
dc.identifierhttp://arxiv.org/abs/math/0411143
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/73131
dc.subjectSpectral Theory
dc.subjectHigh Energy Physics - Theory
dc.subjectMathematical Physics
dc.subjectQuantum Physics
dc.subject34L40; 34L20; 34M40
dc.titleSchrödinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues
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