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 Hilbert space $\mathcal{H}$, the corresponding scale of spaces ${\mathcal{H}}_{+1}(H_0)\subset \mathcal{H}\subset {\mathcal{H}}_{-1}(H_0)=[{\mathcal{H}}_{+1}(H_0){]}^{\ast }$, 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 $$A_V(z):=-(H_0-z{I}_{\mathcal{H}}{)}^{-1\mathrm{/}2}V(H_0-z{I}_{\mathcal{H}}{)}^{-1\mathrm{/}2}\in \mathcal{B}(\mathcal{H}),z\in \rho (H_0),$$ 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 $z=E_0\in (-\mathrm{\infty },0)$, 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.