A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains. (English) Zbl 1198.35159
Summary: We investigate the asymptotic behavior of the \(s\)-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain \(\Omega \) with smooth boundary \(\partial \Omega \). For this, we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on \(\partial \Omega \). It is shown that the resolvent difference of two generalized Robin Laplacians belongs to the Schatten-von Neumann class of any order \(p\) for which
\[ p > \frac{\dim \Omega -1}{3}. \]
Moreover, we also give a simple sufficient condition for the resolvent difference of two generalized Robin Laplacians to belong to a Schatten-von Neumann class of arbitrary small order. Our results extend and complement classical theorems due to M. Š. Birman on Schatten-von Neumann properties of the resolvent differences of Dirichlet, Neumann and self-adjoint Robin Laplacians.
\[ p > \frac{\dim \Omega -1}{3}. \]
Moreover, we also give a simple sufficient condition for the resolvent difference of two generalized Robin Laplacians to belong to a Schatten-von Neumann class of arbitrary small order. Our results extend and complement classical theorems due to M. Š. Birman on Schatten-von Neumann properties of the resolvent differences of Dirichlet, Neumann and self-adjoint Robin Laplacians.
MSC:
| 35P05 | General topics in linear spectral theory for PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 47F05 | General theory of partial differential operators |
| 47A10 | Spectrum, resolvent |
Keywords:
Laplacian; self-adjoint extension; quasi boundary triple; Weyl function; Kreĭn’s formula; non-local boundary condition; Schatten-von Neumann class; singular numbersReferences:
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