Schrödinger operator with periodic plus compactly supported potentials on the half-line

Simple item page
dc.creatorKorotyaev, Evgeny
dc.date2007-10-15
dc.date2009-05-07
dc.date.accessioned2026-07-07T13:11:57Z
dc.date.available2026-07-07T13:11:57Z
dc.descriptionWe consider the Schrödinger operator $H$ with a periodic potential $p$ plus a compactly supported potential $q$ on the half-line. We prove the following results: 1) a forbidden domain for the resonances is specified, 2) asymptotics of the resonance-counting function is determined, 3) in each nondegenerate gap $\g_n$ for $n$ large enough there is exactly an eigenvalue or an antibound state, 4) the asymptotics of eigenvalues and antibound states are determined at high energy, 5) the number of eigenvalues plus antibound states is odd $\ge 1$ in each gap, 6) between any two eigenvalues there is an odd number $\ge 1$ of antibound states, 7) for any potential $q$ and for any sequences $(\s_n)_{1}^\iy, \s_n\in \{0,1\}$ and $(\vk_n)_1^\iy\in \ell^2, \vk_n\ge 0$, there exists a potential $p$ such that each gap length $|\g_n|=\vk_n, n\ge 1$ and $H$ has exactly $\s_n$ eigenvalues and $1-\s_n$ antibound state in each gap $\g_n\ne \es$ for $n$ large enough, 8) if unperturbed operator (at $q=0$) has infinitely many virtual states, then for any sequence $(\s)_1^\iy, \s_n\in \{0,1\}$, there exists a potential $q$ such that $H$ has $\s_n$ bound states and $1-\s_n$ antibound states in each gap open $\g_n$ for $n$ large enough.
dc.identifierhttps://arxiv.org/abs/0710.2832
dc.identifierhttp://arxiv.org/abs/0710.2832
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/229448
dc.subjectMathematical Physics
dc.titleSchrödinger operator with periodic plus compactly supported potentials on the half-line
dc.typetext

Files

Collections

arXiv