{"title":"Nondassical eigenvalue distribution of one-dimensional Schrödinger operators","authors":"M. Klaus","doi":"10.1017/S030821050002624X","DOIUrl":null,"url":null,"abstract":"Synopsis We consider differential operators of the form H = −d2/dx2 + q(x) acting on u ∈ L2(0,∞) with boundary condition u(0) = 0. The potential q(x) is such that H has essential spectrum [0,∞) and an infinite sequence of negative eigenvalues converging to zero. Let n(E) denote the number of eigenvalues of H which are less than E. Under certain conditions on q(x), the well-known formula n(E)∼(2φ)−1 vol {x, p | p2 + q(x)<E}, E↑0, holds. We shall study the validity of this formula for potentials which show oscillatory behaviour as x →∞, like e.g. q(x) = −(1 + x)−α(a + b sin x) with 0<α <2, a≧0, b≠0. We shall obtain the leading-order behaviour of both n(E) and vol n(E)∼(2φ)−1 vol {x, p | p2 + q(x)<E} as E↑0 for a certain class of q's, and we shall see that the classical formula fails in most cases, but there are some noteworthy exceptions.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"6 1","pages":"149 - 158"},"PeriodicalIF":0.9000,"publicationDate":"1985-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/S030821050002624X","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Synopsis We consider differential operators of the form H = −d2/dx2 + q(x) acting on u ∈ L2(0,∞) with boundary condition u(0) = 0. The potential q(x) is such that H has essential spectrum [0,∞) and an infinite sequence of negative eigenvalues converging to zero. Let n(E) denote the number of eigenvalues of H which are less than E. Under certain conditions on q(x), the well-known formula n(E)∼(2φ)−1 vol {x, p | p2 + q(x)
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A flagship publication of The Royal Society of Edinburgh, Proceedings A is a prestigious, general mathematics journal publishing peer-reviewed papers of international standard across the whole spectrum of mathematics, but with the emphasis on applied analysis and differential equations.
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