On the number of negative eigenvalues of a one-dimensional Schrödinger operator with point interactions
A one-dimensional Schrödinger operator with Dirac delta interactions of intensities \(\alpha_i\), at the points of interaction \(x_i\), \(i= 1,2,\dots, n\), is defined in \(\mathbb{R}^1\) by the linear second-order operator \[ L_{x,\alpha}= -d^2/dx^2+ \sum_{i=1}^n \alpha_i\delta(x- x_i), \] \(\delta\) denoting the Dirac delta function. The state function \(\psi\) is defined on the Sobolev space \(W^2_2(\mathbb{R}^1/X)\), where \(X\) is the set on which the conjugacy conditions \(\psi(x_i+ 0)= \psi(x_i- 0)\), and \(\psi(x_i+ 0)- \psi(x_i- 0)= \alpha_i\psi(x_i)\) are true. For the Laplace operator, or the Schrödinger operator given here, it is well known that its spectrum contains the positive x-axis where it is absolutely continuous, and may contain no more than a finite number of negative eigenvalues. For a single point of interaction, the corresponding operator \(-d^2/dx^2+ \alpha\delta(x)\) has eigenvalues only if \(\alpha\) is strictly negative. In case of multiple interaction, it is difficult to find the negative eigenvalues. The authors approach this problem by finding zeros of a special solution \(\varphi\), obeying the conditions \(L_{x,\alpha}\varphi= 0\), and \(\varphi(x)= 1\) if \(x< x_1\). They prove that the number of zeros \(N(\varphi)\) of the special solution on an interval containing all interactions points is equal to the value of the signature of the sequence of values of \(\varphi\) computed at the points \(x_i\). (The signature is equal to the number of sign changes, ignoring zero values.) This preparatory work leads to an algorithm for computation of the number of negative eigenvalues.
scientific article; zbMATH DE number 2021971