The construction of a self-adjoint extension of the Schrödinger operator with potential concentrated on a pencil of planes. I
The author considers the Laplacian on \(\text{ L}^2(\mathbb R^6)\) with boundary conditions on a pencil of two planes that is on the set \(L=P_{1}\cup P_{2}\) where \(P_{1}\), \(P_{2}\) are three-dimensional planes intersecting only at the origin. The semiboundedness of the operator is investigated and the self-adjoint extension is given. The investigation is motivated by quantum mechanical many-body problems with potentials that are concentrated on submanifolds; examples are delta-like potentials. As was shown by \textit{F. A. Berezin} and \textit{L. D. Faddeev} [Sov. Math., Dokl. 2, 372-375 (1961); translation from Dokl. Akad. Nauk SSSR 137, 1011-1014 (1961; Zbl 0117.06601)], it is sufficient to begin with the Laplacian on a domain with sufficiently smooth functions vanishing on some neighbourhood of the submanifold to construct self-adjoint operators. For the Laplacian on \( D_{\Delta_{L}} = \{ u(x,t) \in H^2(\mathbb R^6) \mid u(x,t)=0\) in some neighbourhood of \(L\} \) it is known from the work of \textit{K. A. Makarov} [Algebra Anal. 4, 155-171 (1992; Zbl 0779.35097)] that the elements \(u\) of the adjoint of \(\Delta_{L} = (\Delta, D_{\Delta_{L}})\) can be represented as \[ u(x,t) = u_{0}(x,t) + \chi(r_{t})u_{1}(x)/r_{t} + \chi(r_{x}) u_{2}(t)/r_{x}, \quad (x,t)\in \mathbb R^3 \times \mathbb R^3 \tag{1} \] where \((r_{x},\omega_{x})\) and \((r_{t},\omega_{t})\) are the spherical coordinates of \(x\) and \(t\) respectively and \(\chi(r) = e^{-\lambda r}\), \(\lambda \in \mathbb R\). Furthermore, \(u_{0}\in H^{-1/2}(\mathbb R^6)\) and \(u_{j}\in H^{-1/2}(P_{j})\), \(j=1,2\). Adding additional conditions to the functions \(u\) of the form (1) a symmetric and semibounded operator is explicitly constructed. Eventually, the author formulates conditions for a domain that are sufficient and necessary for the self-adjointness of the Laplacian on this domain. In the last part of the article the author constructs the self-adjoint extension of the Laplacian on \(\text{ L}^2(\mathbb R^6)\) with boundary on the pencil of three planes \(L= L_{1}\cup L_{2}\cup L_{3}\) is constructed, where \(P_{j}\), \(j=1,2,3,\) are three-dimensional planes that intersect only at the origin in \(\mathbb R^6\).
scientific article; zbMATH DE number 1916105