Discrete and Embedded Trapped Modes in a Plane Quantum Waveguide with a Small Obstacle: Exact Solutions

IF 1.2 4区 数学 Q2 MATHEMATICS, APPLIED
P. Zhevandrov, A. Merzon, M. I. Romero Rodríguez, J. E. De la Paz Méndez
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Abstract

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.

Abstract Image

带有小障碍物的平面量子波导中的离散和嵌入陷波模式:精确解法
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来源期刊
Acta Applicandae Mathematicae
Acta Applicandae Mathematicae 数学-应用数学
CiteScore
2.80
自引率
6.20%
发文量
77
审稿时长
16.2 months
期刊介绍: Acta Applicandae Mathematicae is devoted to the art and techniques of applying mathematics and the development of new, applicable mathematical methods. Covering a large spectrum from modeling to qualitative analysis and computational methods, Acta Applicandae Mathematicae contains papers on different aspects of the relationship between theory and applications, ranging from descriptive papers on actual applications meeting contemporary mathematical standards to proofs of new and deep theorems in applied mathematics.
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