A generalized Birman-Schwinger principle and applications to one-dimensional Schr\"odinger operators with distributional potentials

Abstract

Given a self-adjoint operator $H_0$ bounded from below in a complex 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 $A_V(\,\cdot\,):\rho(H_0)\to \mathcal{B}(\mathcal{H})$ by \[ A_V(z):=-\big(H_0-zI_{\mathcal{H}} \big)^{-1/2}V\big(H_0-zI_{\mathcal{H}} \big)^{-1/2}\in \mathcal{B}(\mathcal{H}),\quad z\in \rho(H_0), \] 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 Schr\"odinger operators with $H^{-1}(\mathbb{R})$ distributional potentials.