Discrete spectrum for Schrödinger operators with oscillating decaying potentials

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• 作者：Georgi Raikov ^{graikov@mat.puc.cl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Schrö ; dinger operators ; Oscillating decaying potentials ; Discrete spectrum
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 June 2016
• 年：2016
• 卷：438
• 期：2
• 页码：551-564
• 全文大小：379 K

文摘

We consider the Schrödinger operator class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16001293&_mathId=si1.gif&_user=111111111&_pii=S0022247X16001293&_rdoc=1&_issn=0022247X&md5=c4f1dd004d355f2f7be80ca70de6fafd" title="Click to view the MathML source">H_ηW=−Δ+ηWclass="mathContainer hidden">class="mathCode">$H_{\eta W}=-\mathrm{\Delta }+\eta W$, self-adjoint in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16001293&_mathId=si2.gif&_user=111111111&_pii=S0022247X16001293&_rdoc=1&_issn=0022247X&md5=6f4bdbf1a1356756fd7bdad164919d57" title="Click to view the MathML source">L²(R^d)class="mathContainer hidden">class="mathCode">$L^2({\mathbb{R}}^d)$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16001293&_mathId=si3.gif&_user=111111111&_pii=S0022247X16001293&_rdoc=1&_issn=0022247X&md5=256f5b69020f46ffd39ee84d6459b70b" title="Click to view the MathML source">d≥1class="mathContainer hidden">class="mathCode">$d\ge 1$. Here η is a non-constant oscillating function, while W   decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16001293&_mathId=si4.gif&_user=111111111&_pii=S0022247X16001293&_rdoc=1&_issn=0022247X&md5=45ced6f05c16714eed68ec788d4c611f" title="Click to view the MathML source">H_ηWclass="mathContainer hidden">class="mathCode">$H_{\eta W}$ near the origin, and due to the irregular decay of ηW  , we encounter some non-semiclassical phenomena. In particular, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16001293&_mathId=si4.gif&_user=111111111&_pii=S0022247X16001293&_rdoc=1&_issn=0022247X&md5=45ced6f05c16714eed68ec788d4c611f" title="Click to view the MathML source">H_ηWclass="mathContainer hidden">class="mathCode">$H_{\eta W}$ has less eigenvalues than suggested by the semiclassical intuition.