Strong Coupling Asymptotics for Schrödinger Operators with an Interaction Supported by an Open Arc in three Dimensions

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• 作者：Pavel Exner ^{exner@ujf.cas.cz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Primary 81Q10 ; Secondary 35J10 ; 35P15
• 刊名：Reports on Mathematical Physics
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：77
• 期：1
• 页码：1-17
• 全文大小：156 K

文摘

We consider Schrödinger operators with a strongly attractive singular interaction supported by a finite curve Γ of length L in ℝ³. We show that if Γ is C⁴-smooth and has regular endpoints, the j  -th eigenvalue of such an operator has the asymptotic expansionormulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0034487716000057&_mathId=si1.gif&_user=111111111&_pii=S0034487716000057&_rdoc=1&_issn=00344877&md5=fc320f2eb4ee633f13c411039f998ef6" title="Click to view the MathML source">λ_j(H_α,Γ)=ξ_α+λ_j(S)+O(eπα)ontainer hidden">ode">ow="scroll">ow>λjow>o>(o>ow>How>αo>,o>Γow>ow>o>)o>ow>o>=o>ξαo>+o>λjow>o>(o>So>)o>ow>o>+o>Oow>o>(o>ow>eow>παow>ow>o>)o>ow>ow> as the coupling parameter α → -∞, where ξ_α = −4e^{2(−2πα+ψ(1))} and λ_j(S) is the j  -th eigenvalue of the Schrödinger operatorource" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0034487716000057&_mathId=si2.gif&_user=111111111&_pii=S0034487716000057&_rdoc=1&_issn=00344877&md5=c71bfc2fbc28707c73b6f103a196bd9a">ource" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0034487716000057-si2.gif">oscript>order="0" style="vertical-align:bottom" width="124" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0034487716000057-si2.gif">oscript>ontainer hidden">ode">ow="scroll">So>=o>o>−o>ow>d2ow>ow>ds2ow>o>−o>14γ2ow>o>(o>so>)o>ow>on L²(0, L) with Dirichlet condition at the interval endpoints in which γ is the curvature of Γ.