A generalized Fourier transform
by
Shuji Watanabe
Aichi University of Technology

Let D_{cj,  j} be the operator in L²(R^N):

Dcj,  j =  \partial  \partialxj  -  cj  xj  Rj  , Rj u(x1, ..., xj - 1, xj, xj + 1, ..., xN ) = u(x1, ..., xj - 1, - xj, xj + 1, ..., xN )  ,

where c_j > -1/2 and j = 1, 2, ..., N.

We construct a generalized Fourier transform B_{c1 ... cj ... cN}, which converts the operator D_{cj,  j} into the multiplication operator i  y_j, i.e.,

Bc1 ... cj ... cN  Dcj,  j  B * c1 ... cj ... cN = i yj  .

Here B ^* _{c1 ... cj ... cN} is the adjoint operator of B_{c1 ... cj ... cN} and i = \surd{ -1  } .

When c₁ = c₂ = ... = c_N = 0, the operator D_{cj,  j} becomes \partial/ \partialx_j and the transform B_{c1 ... cj ... cN} coincides with the Fourier transform. We can therefore consider the transform B_{c1 ... cj ... cN} as a generalized Fourier transform.

On the basis of the transform we explicitly find out the solutions of the Cauchy problems for the heat equations with strongly singular coefficients and for the Schrödinger equations with strongly singular potentials.

Moreover, we show that there is the Friedrichs extension of - \Delta+ k/(|x|²), x in R^N as long as k > - N/4.

Using the transform above we define spaces of Sobolev type. Each space is a generalized Sobolev space. We show an embedding theorem for these spaces. We see that the embedding theorem is a generalization of the Sobolev embedding theorem. We finally apply the embedding theorem to the Cauchy problem for the wave equation with a strongly singular coefficient and study some properties of its solution.

Date received: May 22, 2001

Copyright © 2001 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 # cahk-81.