arXiv 0710.28322007-10math-phmath.MP

半直线上具有周期加紧支撑势的Schr dinger算子

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

Evgeny Korotyaev

摘要：

我们考虑薛定谔算子$H$，其周期势$p$加上半直线上的紧支势$q$。我们证明了以下结果：1）规定了共振的禁域，2）确定了共振计数函数的渐近性，3）对于足够大的$n$，在每个非简并能隙$\g_n$中恰好存在本征值或反束缚态，4）在高能下确定了本征值和反束缚态的渐近性，5）在每个间隙中本征值加上反束缚态的数目是奇数$\ge 1$，6）在任何两个本征值之间存在奇数个$\ge 1$的反束缚态，7）对于任何潜在的$q$以及对于任何序列$(\s_n)_{1}^\iy, \s_n\in \{0,1\}$和$(\vk_n)_1^\iy\in \ell^2, \vk_n\ge 0$，存在一个潜在的$p$，使得对于足够大的$n$，每个间隙长度$|\g_n|=\vk_n, n\ge 1$和$H$在每个间隙$\g_n\ne \es$中恰好具有$\s_n$本征值和$1-\s_n$反束缚态，8）如果未扰动算子（在$q=0$）具有无穷多个虚状态，则对于任何序列$(\s)_1^\iy, \s_n\in \{0,1\}$，存在潜在的$q$，使得对于足够大的$n$，$H$在每个空位开放$\g_n$中具有$\s_n$结合态和$1-\s_n$反结合态。

英文摘要：

We 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.

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摘要：

英文摘要：

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