Positive definite functions and spectral properties of the Schrödinger operator with point interactions
The principal object under consideration in the present paper is the differential expression \[ -\Delta+\sum_{j=1}^m \alpha_j\delta(\cdot -x_j), \qquad x_j\in\mathbb{R}^d, \quad \alpha_j\in\mathbb{R}, \quad m\in\mathbb{N}, \] and the corresponding minimal operator \(H_d\) in \(L^2(\mathbb{R})\) \[ H_d:=-\Delta, \qquad D(H_d):=\{f\in W_2^2(\mathbb{R}^d): f(x_j)=0, \;\;j=1,2,\dots,m\}, \] with deficiency indices \(n_{\pm}(H_d)=m\). The authors study the spectral properties of self-adjoint extensions of \(H_d\). As the first step, they obtain an explicit description of the domain of definition and boundary triples for \(H_d^*\) with \(d=2,3\). The main result of the paper states that the nonnegative spectrum of each self-adjoint extension \(H_{d,\theta}\) is absolutely continuous and has infinite multiplicity. The point spectrum \(\sigma_p (H_{d,\theta})\) consists of at most \(m\) negative eigenvalues, counting multiplicity, and for \(z\in\sigma_p (H_{d,\theta})\) the corresponding eigenfunction (for \(d=3\)) is of the form \[ \psi(x,z)=\sum_{j=1}^m c_j\,\frac{e^{i\sqrt{z}|x-x_j|}}{4\pi|x-x_j|}\,. \] The key ingredient is the positive-definiteness of certain matrices, which provides a link to the theory of positive definite functions in \(\mathbb{R}^d\).
scientific article; zbMATH DE number 6013935