The approximation of the Schrödinger operators with penetrable wall potentials in terms of short range Hamiltonians
The author treats the approximation problem of the formally given Schrödinger operator \[ -\Delta+q(x)\cdot\delta(| x|- a),x\in\mathbb{R}^ 3, \] found in the work by \textit{J. P. Antoine}, \textit{F. Gesztesy} and \textit{J. Shabani} [J. Phys. A, Math. Gen. 20, 3687-3712 (1987)]. \textit{T. Ikebe} and the author [J. Math. Kyoto Univ. 31, No. 1, 219-258 (1991; Zbl 0736.47007)] gave a self-adjoint operator \(H\) determined by a quadratic form \(h[u,v]\) as a rigorous realization of the formal operator. The author adopts the realization and approximates the operator \(H\) by short range Hamiltonians \(H_ \varepsilon=-\Delta+Q_ \varepsilon\) in the norm resolvent sense, where \[ Q_ \varepsilon(x)=(1/\varepsilon)\rho((| x|- a)/\varepsilon)q(ax/| x|),\quad\text{supp} \rho\subset[-1,1]. \] He gives two estimations for \((-\Delta-z)^{-1}\cdot Q_ \varepsilon\) and proves the following Theorem: \(R_ \varepsilon(z)=(H_ \varepsilon- z)^{-1}\) converges to \(R(z)=(H-z)^{-1}\) as \(\varepsilon\downarrow 0\) in \(B(L_ 2(\mathbb{R}^ 3)\), \(H^ 1(\mathbb{R}^ 3))\) uniformly. Here \(z\in\mathbb{C}\), \(\text{Im} z\neq 0\), have sufficiently large absolute values.
scientific article; zbMATH DE number 169033