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Fourier type analysis and quantum mechanics
by
Shuji Watanabe
Dept. of Math., Faculty of Engr., Gunma Univ., 4-2 Aramaki-machi, Maebashi 371-8510, Japan
Coauthors: Yoshio Ohnuki
We formulate Fourier type analysis originating from quantum mechanics. The usual Fourier transform is an example of our Fourier type analysis. For simplicity we let x Î R instead of x Î Rn throughout this talk. Our Fourier type analysis is suitable for differential operators in bounded or unbounded open intervals with variable coefficients. Here some variable coefficients are singular. We construct an integral transform U that transforms a certain differential operator D with a singular variable coefficient into the multiplication by iy (i=Ö{-1}, y Î R). We find that our transform is a generalized Fourier transform. We then 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 both our transform and our embedding theorem to partial differential equations in bounded or unbounded open intervals with singular coefficients so as to study properties of the solutions. We write some solutions in explicit forms.
Date received: May 23, 2005
Copyright © 2005 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caqw-71.