A discreteness criterion for the spectrum of a one-dimensional Schrödinger operator with \(\delta\)-interactions
In the space \(W^2_2 \cap W^1_2\) the author introduces the Hamiltonian \(- d^2/dx^2 + \sum a_n \delta^1 (x - x_n) = H_{x,A}\), \(x_n \in X\), and \(H_0 = - d^2/dx^2\). It is well-known that if \(x\) is finite or \(a_n\) has at most finitely many nonzero elements, then \(\sigma_{\text{ess}} (H_{X,A}) = \sigma_{\text{ess}} (H_0) = [0, \infty)\). The unsolved questions involve various cases when \(X\) is infinite. The mapping \(A : X \to \mathbb{R}\) is given by \(A(x_n) = a_n\). The author orders elements of \(X\) in increasing order. Then \(\Delta_n : = x_{n + 1} - x_n > 0\). A tori-diagonal matrix \(J_{X,A}\) has elements: \(\{- {1 \over \Delta_{n - 1}},\;a_n + {1 \over \Delta_n} - {1 \over \Delta_{n - 1}}, - {1 \over \Delta_n}\}\) in the \(n\)-th row. He proves the following: Theorem: The spectrum of \(H_{X,A}\) is purely discrete if and only if (A) \(\Delta_n \to \infty\) as \(n \to \infty\); (B) the spectrum of \(J_{X,A}\) is purely discrete on the space \(\ell^2 (\mathbb{Z})\). A corollary to this theorem states that the spectrum of \(H_{X,A}\) is not purely discrete if the set \(X\) is uniformly discrete, that is if \(\inf \{\Delta_n : n \in \mathbb{Z}\} > 0\).
scientific article; zbMATH DE number 846556