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arXiv · 2607.29526

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

Benjamin Alvarez·Hugo Gouttenegre
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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 Schr\"odinger operator.

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Benjamin Alvarez, Hugo Gouttenegre. 2026-07-31. Spectral Analysis of a Soft Quantum Waveguide Coupled to a Massive Quantum Scalar Field. https://arxiv.org/abs/2607.29526

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