Spectral Analysis of a Soft Quantum Waveguide Coupled to a Massive Quantum Scalar Field

Abstract

We study a Nelson-type Hamiltonian describing a massive spinless non-relativistic particle moving in a planar soft quantum waveguide and linearly coupled to a massive scalar bosonic field. Our spectral analysis relies on comparing the curved waveguide with its straight counterpart. For the straight Hamiltonian, longitudinal translation invariance yields a fiber decomposition, and we prove that its spectrum is purely essential and gapless. For the curved Hamiltonian, we establish an HVZ-type formula for the bottom of the essential spectrum, involving that of the straight system and the one-boson threshold. We also derive a sufficient condition for the existence of non-empty discrete spectrum in the curved case, expressed in terms of an effective one-dimensional Schrödinger operator.