An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter

详细信息    查看全文

• 作者：Konstantin Pankrashkin^a ; ^{konstantin.pankrashkin@math.u-psud.fr" class="auth_mail" title="E-mail the corresponding author} ; Nicolas Popoff^b ; ^{nicolas.popoff@math.u-bordeaux1.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35J05 ; 49R05 ; 58C40
• 刊名：Journal de mathematiques pures et appliquees
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：106
• 期：4
• 页码：615-650
• 全文大小：593 K

文摘

We consider the Laplacian on a class of smooth domains pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416300058&_mathId=si1.gif&_user=111111111&_pii=S0021782416300058&_rdoc=1&_issn=00217824&md5=ac5882de620b3391149f2b7c9097b18e" title="Click to view the MathML source">Ω⊂R^ν$\mathrm{\Omega }\subset {\mathbb{R}}^{\nu }$, pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416300058&_mathId=si2.gif&_user=111111111&_pii=S0021782416300058&_rdoc=1&_issn=00217824&md5=9aed457c49913b77b80deae5891d5b06" title="Click to view the MathML source">ν≥2$\nu \ge 2$, with attractive Robin boundary conditions: where n   is the outer unit normal, and study the asymptotics of its eigenvalues $E_j(Q_{\alpha }^{\mathrm{\Omega }})$ as well as some other spectral properties for α   tending to +∞. We work with both compact domains and non-compact ones with a suitable behavior at infinity. For domains with compact pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416300058&_mathId=si5.gif&_user=111111111&_pii=S0021782416300058&_rdoc=1&_issn=00217824&md5=3fd7f5e24fa3df83d9db14804e42283a" title="Click to view the MathML source">C²$C^2$ boundaries we show that, for each fixed j, where pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416300058&_mathId=si7.gif&_user=111111111&_pii=S0021782416300058&_rdoc=1&_issn=00217824&md5=58d1bef3381dc5ce08e7b5c033563a96" title="Click to view the MathML source">μ_j(α)${\mu }_j(\alpha )$ is the j  th eigenvalue of the operator pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416300058&_mathId=si8.gif&_user=111111111&_pii=S0021782416300058&_rdoc=1&_issn=00217824&md5=c92562f8d4adb4805c6e48d048b531cd" title="Click to view the MathML source">−Δ_S−(ν−1)αH$-{\mathrm{\Delta }}_S-(\nu -1)\alpha H$ with pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416300058&_mathId=si9.gif&_user=111111111&_pii=S0021782416300058&_rdoc=1&_issn=00217824&md5=e283f52bd06b8bfc83b51683c2ce6e76" title="Click to view the MathML source">(−Δ_S)$(-{\mathrm{\Delta }}_S)$ and H being respectively the positive Laplace–Beltrami operator and the mean curvature on ∂Ω. Analogous results are obtained for a class of domains with non-compact boundaries. In particular, we discuss the existence of eigenvalues for non-compact domains and the existence of spectral gaps for periodic domains. We also show that the remainder estimate can be improved under stronger regularity assumptions.

The effective Hamiltonian pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416300058&_mathId=si8.gif&_user=111111111&_pii=S0021782416300058&_rdoc=1&_issn=00217824&md5=c92562f8d4adb4805c6e48d048b531cd" title="Click to view the MathML source">−Δ_S−(ν−1)αH$-{\mathrm{\Delta }}_S-(\nu -1)\alpha H$ enters the framework of semi-classical Schrödinger operators on manifolds, and we provide the asymptotics of its eigenvalues for large α   under various geometric assumptions. In particular, we describe several cases for which our asymptotics provides gaps between the eigenvalues of $Q_{\alpha }^{\mathrm{\Omega }}$ for large α.