A Generalized Birman–Schwinger Principle and Applications to One-Dimensional Schrödinger Operators with Distributional Potentials

Given a self-adjoint operator \(H_0\) bounded from below in a complex, separable Hilbert space \(\mathcal H\), the corresponding scale of spaces \(\mathcal H_{+1}(H_0) \subset \mathcal H \subset \mathcal H_{-1}(H_0)=[\mathcal H_{+1}(H_0)]^*\), and a fixed \(V\in \mathcal B(\mathcal H_{+1}(H_0),\mathcal H_{-1}(H_0))\), we define the operator-valued map \(A_V(\,\cdot\,)\colon \rho(H_0)\to \mathcal B(\mathcal H)\) by

where \(\rho(H_0)\) denotes the resolvent set of \(H_0\). Assuming that \(A_V(z)\) is compact for some \(z=z_0\in \rho(H_0)\) and has norm strictly less than one for some \(z=E_0\in (-\infty,0)\), we employ an abstract version of Tiktopoulos’ formula to define an operator \(H\) in \(\mathcal H\) that is formally realized as the sum of \(H_0\) and \(V\). We then establish a Birman–Schwinger principle for \(H\) in which \(A_V(\,\cdot\,)\) plays the role of the Birman–Schwinger operator: \(\lambda_0\in \rho(H_0)\) is an eigenvalue of \(H\) if and only if \(1\) is an eigenvalue of \(A_V(\lambda_0)\). Furthermore, the geometric (but not necessarily the algebraic) multiplicities of \(\lambda_0\) and \(1\) as eigenvalues of \(H\) and \(A_V(\lambda_0)\), respectively, coincide.

As a concrete application, we consider one-dimensional Schrödinger operators with \(H^{-1}(\mathbb{R})\) distributional potentials.