Continued resolvent matrix elements and resonance poles

Let $T$ be self-adjoint on a complex Hilbert space $\mathcal{H}$ whose inner product is linear in the first variable, use the resolvent convention $R_T(z)=(T-zI{)}^{-1}$, and let $X\subseteq \mathcal{H}$ be a specified vector subspace. For $f,h\in X$, the scalar resolvent matrix element is

$r_{f,h}(z):=⟨R_T(z)f,h⟩,\mathrm{Im}z>0.$

Given an open real interval $I$, a point $E_0\in I$, and a disk $D$ centered at $E_0$, a local analytic continuation of these matrix elements through $E_0$ is a specified family of holomorphic functions ${\tilde{r}}_{f,h}:D\to \mathbb{C}$ agreeing with $r_{f,h}$ on $D\cap \{\mathrm{Im}z>0\}$, for the stated $f,h\in X$. A meromorphic continuation on $D$ is instead a specified family ${\tilde{r}}_{f,h}$ agreeing with $r_{f,h}$ on $D\cap \{\mathrm{Im}z>0\}$, holomorphic on $D$ except for isolated poles. At every crossing point where no pole occurs it is a regular analytic continuation on a smaller neighborhood. For such a meromorphic continuation, if $z_0\in D$ is nonreal and for some $f,h\in X$ the continued function satisfies

${\tilde{r}}_{f,h}(z)=(z-z_0{)}^{-k}q(z)$

on a punctured neighborhood of $z_0$, for an integer $k\ge 1$ and a holomorphic $q$ with $q(z_0)\mathrm{\ne }0$, then $z_0$ is a pole of order $k$ of that continued matrix element. Such a pole is called a resonance matrix-element pole for the specified pair $(T,X)$ and continuation. This definition asserts neither that a continuation exists nor that a pole is an eigenvalue of $T$; the continuation and the chosen test vectors are part of the data.

Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions

Resolvent and spectrum of an unbounded operator

Symmetric, self-adjoint and essentially self-adjoint operators

Resolvent of a self-adjoint operator: nonreal resolvents and the estimate