The discrete spectrum of Schrödinger operators with \(\delta\)-type conditions on regular metric trees (Q1650224)

The paper proves the discreteness of the spectra of the Schrödinger operators on a regular metric tree graphs with \(\delta\)-coupling condition at the vertices. A rooted tree metric graph (a graph without cycles and multiple edges between the given two vertices) is considered. The graph is called regular if its branching numbers and lengths of the edges depend only on the distance from the root. The graph can be in general infinite, which is the case of interest. The graph is equipped with the Schrödinger operator acting as negative second derivative plus potential on the edges on the graph. There is a Dirichlet coupling condition (functional value vanishes) at the root and \(\delta\)-conditions at the internal vertices (function is continuous at the vertex and the sum of the outgoing derivatives is equal to \(\alpha_v\)-multiple of the functional value). There is a known construction (see e.g. [\textit{K. Naimark} and \textit{M. Solomyak}, Proc. Lond. Math. Soc. (3) 80, No. 3, 690--724 (2000; Zbl 1046.34092)]) which states a unitary equivalence between the Hamiltonian on the tree and the direct sum of the operators on the half-lines. This construction is used in the current paper. First, the authors show the selfadjointness of their Hamiltonian. Then they find the quadratic forms of the selfadjoint operators on the half-lines. The main results of the paper involve discreteness of the spectra. They prove that the Hamiltonian on the tree has discrete spectrum if and only if the two conditions are satisfied: the spectrum of the first half-line operator is discrete and the minimum of the spectra of other half-line operators goes to infinity as their index goes to infinity. Then they prove discreteness of the spectra of the first half-line operator under conditions on the potential and the strengths of the \(\delta\)-conditions. Using these results they prove (again under the conditions on the potential and strengths of the \(\delta\)-couplings) necessary and sufficient condition on the potential giving the discreteness of the spectra of the tree operator.