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Distant perturbations of the Laplacian in a multi-dimensional space. (English) Zbl 1143.35075

For \(i \in \{1,\dots,m\}\), let \(\Omega_{i}\) be bounded non-empty domains of \({\mathbb{R}}^{n}\) with infinitely differentiable boundary, \({\mathcal{L}}_{i} \colon W^{2}_{2}(\Omega_{i})\rightarrow L^{2}(\Omega_{i})\) linear bounded operators such that \(\langle {\mathcal{L}}_{i} u_{1}, u_{2}\rangle = \langle u_{1}, {\mathcal{L}}_{i} u_{2} \rangle\), \(\langle {\mathcal{L}}_{i} u, u \rangle \geq -c_{0} | \nabla u| ^{2} - c_{1} | u| ^{2}\) for all \(u, \, u_{1}, \, u_{2} \in W^{2}_{2}(\Omega_{i})\), where \(c_{0} < 1\), \(\langle \; , \; \rangle\) and \(| \; | \) are the scalar product and the norm in \(L^{2}(\Omega)\), respectively. Let \(S(a)u = u(\cdot +a)\), where \(a \in {\mathbb{R}}^{n}\) and \(u \in L^{2}({\mathbb{R}}^{n}), \; X=(X_1, \dots ,X_m), \; X_i \in {\mathbb{R}}^{n}, \; {\mathcal{L}}_X=\sum_{i=1}^{m} S(-X_i){\mathcal{L}}_iS(X_i) \colon W^{2}_{2}(\Omega_X)\rightarrow L^{2}(\Omega_X), \; \Omega_X = \cup_{i=1}^{m} \{x: x-X_i \in \Omega_{i}\}, \; {\mathcal{H}}_X=- \Delta + {\mathcal{L}}_X: W^{2}_{2}({\mathbb{R}}^{n})\rightarrow L^{2}({\mathbb{R}}^{n})\). The author constructs the asymptotic expansions for the isolated eigenvalues and the associated eigenfunctions of \({\mathcal{H}}_X\) as \(\min \{| X_i-X_j| : i \neq j\} \rightarrow + \infty\).

MSC:

35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
47A75 Eigenvalue problems for linear operators
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