On the Negative Spectrum of One-Dimensional Schrödinger Operators on Quantum Trees with Point Interactions

We present an explicit algorithm to determine the number of negative eigenvalues of Schrödinger operators on rooted quantum trees equipped with delta or delta-prime vertex interactions. We employ the methods of [Behrndt and Luger [5]], and the structure of trees to generate a sequence which has the same number of negative elements as the original Laplace operator. We show that the number of negative eigenvalues of the Schrödinger operators with delta interactions equals the number of negative terms in this sequence, while for delta-prime interactions, it reduces to the number of negative interaction strengths.