Point interaction potential approximation for \((-\Delta +U)^{-1}\) and eigenvalues of the Laplacian on wildly perturbed domain (Q1061314)

Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ 3\) with smooth boundary \(\gamma\), w(m) be a set of m points \(w_ 1^{(m)},...,w_ m^{(m)}\in \Omega\) such that: (i) for some constant C independet of f, m and \(m\to \infty\) \[ | w_ 2^{(m)}-w_ s^{(m)}| \geq Cm^{-1/3},\quad r\neq s,\quad dist(w_ r^{(m)},\gamma)\geq Cm^{-1/3},\quad 1\leq r\leq m; \] \[ (ii)\quad | (1/m)\sum^{m}_{1}f(w_ i^{(m)})- \int_{\Omega}f(x)V(x)dx| \leq C_ p m^{-p/3}\| f\|_{C_ p({\bar \Omega})},\quad 0\leq p\leq 1 \] and moreover \[ \max_{j}(1/m)\sum^{m}_{i\neq j,i=1}G(w_ i^{(m)}-w_ j^{(m)})f(w_ j^{(m)})-\int_{\Omega}G(y,w_ i^{(m)})f(y)dy| \leq C_ p m^{-p/3}\| f\|_{C_ p({\bar \Omega})} \] where G(x,y) is the Green function of the Laplacian in \(\Omega\) under Dirichlet conditions on \(\gamma\). Denote by \(0\leq \mu_ 1(\epsilon,w(m))\leq..\). the eigenvalues of the Laplacian in \(\Omega_{\epsilon,w(m)}=\Omega \setminus \cup^{m}_{1}\{x\in {\mathbb{R}}^ 3:\quad | x-w_ i^{(m)})| \leq \epsilon \}\) under the Dirichlet condition on \(\partial \Omega_{\epsilon,w(m)}.\) Theorem. Suppose that \(\alpha >0\) and the domain \(\Omega_{\alpha /m,w(m)}\) is connected. Then \[ (1)\quad \mu_ k(\alpha /m,w(m))=\mu^ V_ k+O(m^{-1/6+\delta}),\quad \delta >0,\quad m\to \infty \] where \(\mu^ V_ k\) are the eigenvalues of \(-\Delta +4\pi \alpha V(x)\), \(x\in \Omega\) under the Dirichlet conditions on \(\gamma\). This result is closely related with those of \textit{M. Kac} [Rocky Mt. J. Math. 4, 511-537 (1974; Zbl 0314.47006)] and \textit{E. Ya. Khruslov} and \textit{V. Marchenko} [Mat. Sb. 65(107), 458-472 (1964; Zbl 0171.087); see also their book Boundary value problems in the regions with fine-gained boundary (Russian) (1974; Zbl 0289.35002)]. In particular, the convergence of eigenvalues can be extracted by standard perturbation arguments from the mentiond paper of E. Khruslov and V. Marchenko. But the error estimates in (1) as well as the author's method, basing on a perturbation series for the Green function and his results on control of the Green function under singular perturbations of the boundary seems to be new.