2023/12/13 by P. Zhevandrov, Zhevandrov, P., Anatoli Merzon +5
Physics and Astronomy · #35P15 #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Orbital Angular Momentum in Optics #Quantum optics and atomic interactions
paper · pdf · doi:10.48550/arxiv.2312.08480
openalex publication_date 2023/12/13 · openalex created_date 2023/12/16 · openalex updated_date 2026/07/28
Exact solutions describing trapped modes in a plane quantum waveguide with a small rigid obstacle are constructed in the form of convergent series in powers of the small parameter characterizing the smallness of the obstacle. The terms of this series are expressed through the solution of the exterior Neumann problem for the Laplace equation describing the flow of unbounded fluid past the inflated obstacle. The exact solutions obtained describe discrete eigenvalues of the problem under certain geometric conditions, and, when the obstacle is symmetric, these solutions describe embedded eigenvalues. For obstacles symmetric with respect to the centerline of the waveguide, the existence of embedded trapped modes is known (due to the decomposition trick of the domain of the corresponding differential operator) even without the smallness assumption. We construct these solutions in an explicit form for small obstacles. For obstacles symmetric with respect to the vertical axis, we find embedded trapped modes for a specific vertical displacement of the obstacle.