Higher-level eigenvalues of Q-operators and Schroedinger equation

TL;DR

The paper addresses how higher-level eigenvalues of Q-operators in conformal field theory relate to Schrödinger operators with specially engineered Q-potentials. It constructs Q-potentials with poles determined by algebraic equations in the level parameter $L$ and shows that Schrödinger spectral determinants $D_\pm(E)$ match Q-operator data via $D_\pm(\nu s)=A_\pm(s)$, linking level $L$ to both the pole structure and the partition count ${\tt p}(L)$. The large-$s$ expansion of $\log Q_\pm(s)$ is organized by local integrals of motion $I_{2n-1}$ and dual nonlocal IMs, yielding level-dependent shifts such as $I_1=I_1^{(vac)}(\Delta+L)$ and aligning with a WKB interpretation of the Schrödinger problem. By establishing this correspondence, the work strengthens the bridge between integrable quantum field theory and quantum-mechanical spectral theory, with potential implications for boundary flows and related conformal boundary states.

Abstract

Relation between one-dimensional Schroedinger equation and the vacuum eigenvalues of the Q-operators is extended to their higher-level eigenvalues.