Spectroscopy of drums and quantum billiards: perturbative and non-perturbative results

Abstract

We develop powerful numerical and analytical techniques for the solution of the Helmholtz equation on general domains. We prove two theorems: the first theorem provides an exact formula for the ground state of an arbirtrary membrane, while the second theorem generalizes this result to any excited state of the membrane. We also develop a systematic perturbative scheme which can be used to study the small deformations of a membrane of circular or square shapes. We discuss several applications, obtaining numerical and analytical results.

29 pages, 12 figures, 7 table

References in corpus (4)

• Solving the Helmholtz equation for membranes of arbitrary shape
• Variational collocation on finite intervals
• A new method for studying the vibration of non-homogeneous membranes
• Computing the Casimir energy using the point-matching method

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• Weakly bound states in heterogeneous waveguides
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• $\mathcal{P}\mathcal{T}$-symmetric strings
• Exact sum rules for quantum billiards of arbitrary shape
• Casimir force between integrable and chaotic pistons
• Statistical problem of ideal gas in general 2-dimensional regions
• Boundary perturbations and the Helmholtz equation in three dimensions