Higher-Rank Connections and Deformed Schr\"odinger Operators

TL;DR

This paper investigates the boundary value problems and quantization conditions for a class of linear differential equations related to the quantum Toda chain, linking spectral theory and topological string duality.

Contribution

It derives the weakest quantization conditions for these equations, connecting monodromy data with spectral problems and topological string duality predictions.

Findings

Proves quantization conditions predicted by topological string/spectral theory duality.

Establishes a hierarchy of spectral problems interpolating boundary conditions.

Identifies rich boundary value problem structures for higher-order differential equations.

Abstract

We study the connection problem for a class of linear differential equations of order $N$ closely related to the Baxter equation of the quantum Toda chain. The space of solutions is $N$-dimensional and several linearly independent solutions decay at each singularity, leading to a rich structure of boundary value problems. We derive the weakest quantization conditions compatible with decaying behavior at both singularities, and formulate these conditions in terms of the associated monodromy data. In doing so, we prove the quantization conditions predicted by the topological string/spectral theory duality for a family of deformed Schr\"odinger equations. More generally, our results point to a hierarchy of spectral problems interpolating between the minimal conditions studied here and the maximally decaying boundary conditions of the $N$-particle quantum Toda chain.