{"title":"Semiclassical approximation of the magnetic Schrödinger operator on a strip : dynamics and spectrum","authors":"M. Dimassi","doi":"10.2140/TUNIS.2020.2.197","DOIUrl":null,"url":null,"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 .","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/TUNIS.2020.2.197","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tunisian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/TUNIS.2020.2.197","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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 .