Semiclassical approximation of the magnetic Schrödinger operator on a strip : dynamics and spectrum

IF 0.8 Q2 MATHEMATICS Tunisian Journal of Mathematics Pub Date : 2020-01-01 DOI:10.2140/TUNIS.2020.2.197
M. Dimassi
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Abstract

In the semiclassical regime (i.e., (cid:15) (cid:38) 0), we study the effect of a slowly varying potential V ((cid:15) t , (cid:15) z ) on the magnetic Schrödinger operator P = D 2 x + ( D z + µ x ) 2 on a strip [− a , a ] × (cid:82) z . The potential V ( t , z ) is assumed to be smooth. We derive the semiclassical dynamics and we describe the asymptotic structure of the spectrum and the resonances of the operator P + V ((cid:15) t , (cid:15) z ) for (cid:15) small enough. All our results depend on the eigenvalues corresponding to D 2 x + (µ x + k ) 2 on L 2 ( [− a , a ] ) with Dirichlet boundary condition. x ≤ a } . The Fourier transfor-mation with respect to z reduces the spectral problem of P to an analysis of the ( k depending) eigenvalues E 0 ( k ), E 1 ( k ), . . . of the Sturm-Liouville operator on the interval [− a , a ] with Dirichlet boundary condition at − a and a .
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带上磁性薛定谔算子的半经典近似:动力学和谱
在半经典区(即(cid:15) (cid:38) 0)中,我们研究了慢变电位V ((cid:15) t, (cid:15) z)对条带[- a, a] × (cid:82) z上的磁Schrödinger算符P = d2 x + (D z +µx) 2的影响。假设势能V (t, z)是光滑的。我们导出了半经典动力学,并描述了谱的渐近结构和算子P + V ((cid:15) t, (cid:15) z)在(cid:15)足够小时的共振。我们所有的结果都依赖于d2 +(µx + k) 2在l2([−a, a])上对应的特征值,并具有Dirichlet边界条件。X≤a}。关于z的傅里叶变换将P的频谱问题简化为(取决于k的)特征值e0 (k), e1 (k),…的分析。在- a和- a处具有Dirichlet边界条件的区间[- a, a]上Sturm-Liouville算子的性质。
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来源期刊
Tunisian Journal of Mathematics
Tunisian Journal of Mathematics Mathematics-Mathematics (all)
CiteScore
1.70
自引率
0.00%
发文量
12
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