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 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 $A_V(\,\cdot\,)\colon \rho(H_0)\to \mathcal B(\mathcal H)$ by
$$ A_V(z):=-(H_0-zI_{\mathcal H} )^{-1/2}V(H_0-zI_{\mathcal H} )^{-1/2}\in \mathcal B(\mathcal H), \qquad z\in \rho(H_0), $$
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 Schrödinger operators with $H^{-1}(\mathbb{R})$ distributional potentials.