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Fourier Type Analysis and Applications to Quantum Mechanics

Shuji Watanabe

Abstract


We discuss Fourier type analysis originating from quantum mechanics. The usual Fourier transform is an example of our Fourier type analysis. Our Fourier type analysis is suitable for differential operators in bounded or unbounded domains with variable coefficients. Here some variable coefficients are singular. We construct an integral transform U, which is a generalized Fourier transform. We define spaces of Sobolev type using our transform, and show an embedding theorem for each space. Our embedding theorem is a generalization of the Sobolev embedding theorem. We apply our results both to partial differential equations in bounded or unbounded domains with singular variable coefficients and to quantum mechanics.

Keywords


Fourier type analysis, integral transform, a generalized Fourier transform, space of Sobolev type, embedding theorem, partial differential equation in bounded or unbounded domains with singular variable coefficients, quantum mechanics.

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