On the number of negative eigenvalues of a multi-dimensional Schrodinger operator with point interactions

Abstract

We prove that the number $N$ of negative eigenvalues of a Schr\"odinger operator $L$ with finitely many points of $\delta$-interactions on $\mathbb R^{d}$ (${d}\le3$) is equal to the number of negative eigenvalues of a certain class of matrix $M$ up to a constant. This $M$ is expressed in terms of distances between the interaction points and the intensities. As applications, we obtain sufficient and necessary conditions for $L$ to satisfy $N=m,n,n$ for ${d}=1,2,3$, respectively, and some estimates of the minimum and maximum of $N$ for fixed intensities. Here, we denote by $n$ and $m$ the numbers of interaction points and negative intensities, respectively.