The local structure of the spectrum of the one-dimensional Schrödinger operator
The local structure of the spectrum of the one-dimensional Schrödinger operator
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一维薛定谔算子谱的局部结构
DOI：
10.1007/bf01942333
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发表时间：
1981
影响因子：
2.4
通讯作者：
S. Molčanov
中科院分区：
文献类型：
--
作者：
S. Molčanov
AbstractLet
$$H_V = - \frac{{d^{\text{2}} }}{{dt^{\text{2}} }} + q(t,\omega )$$
be an one-dimensional random Schrödinger operator in ℒ2(−V,V) with the classical boundary conditions. The random potentialq(t, ω) has a formq(t, ω)=F(xt), wherext is a Brownian motion on the compact Riemannian manifoldK andF:K→R1 is a smooth Morse function,
$$\mathop {\min }\limits_K F = 0$$
. Let
$$N_V (\Delta ) = \sum\limits_{Ei(V) \in \Delta } 1 $$
, where Δ∈(0, ∞),Ei(V) are the eigenvalues ofHV. The main result (Theorem 1) of this paper is the following. IfV→∞,E0>0,k∈Z+ anda>0 (a is a fixed constant) then
$$P\left\{ {N_V \left( {E_0 - \frac{a}{{2V}},E_0 + \frac{a}{{2V}}} \right) = k} \right\}\xrightarrow[{V \to \infty }]{}e^{ - an(E_0 )} (an(E_0 ))^k |k!,$$
wheren(E0) is a limit state density ofHV,V→∞. This theorem mean that there is no repulsion between energy levels of the operatorHV,V→∞.The second result (Theorem 2) describes the phenomen of the repulsion of the corresponding wave functions.
AbstractLet
$$H_V = - \frac{{d^{\text{2}} }}{{dt^{\text{2}} }} + q(t,\omega )$$
be an one-dimensional random Schrödinger operator in ℒ2(−V,V) with the classical boundary conditions. The random potentialq(t, ω) has a formq(t, ω)=F(xt), wherext is a Brownian motion on the compact Riemannian manifoldK andF:K→R1 is a smooth Morse function,
$$\mathop {\min }\limits_K F = 0$$
. Let
$$N_V (\Delta ) = \sum\limits_{Ei(V) \in \Delta } 1 $$
, where Δ∈(0, ∞),Ei(V) are the eigenvalues ofHV. The main result (Theorem 1) of this paper is the following. IfV→∞,E0>0,k∈Z+ anda>0 (a is a fixed constant) then
$$P\left\{ {N_V \left( {E_0 - \frac{a}{{2V}},E_0 + \frac{a}{{2V}}} \right) = k} \right\}\xrightarrow[{V \to \infty }]{}e^{ - an(E_0 )} (an(E_0 ))^k |k!,$$
wheren(E0) is a limit state density ofHV,V→∞. This theorem mean that there is no repulsion between energy levels of the operatorHV,V→∞.The second result (Theorem 2) describes the phenomen of the repulsion of the corresponding wave functions.