Vinogradov series in the Cauchy problem for equations of Schrödinger type. (English. Russian original) Zbl 0831.35038

Oskolkov, K. I.

Proc. Steklov Inst. Math. 200, 291-315 (1993); translation from Tr. Mat. Inst. Steklova 200, 265-288 (1991).

Let \(f(x)\) be a periodic complex valued function with period 1 of the real variable \(x\). Then the Vinogradov extensions of \(f\) is the trigonometric series \[ V(f; x_r, \ldots, x_1) = \sum^\infty_{n = - \infty} \widehat f(n) e(n^r x_r + \cdots + nx_1), \] where the \(\widehat f(n)\) are the Fourier coefficients of \(f\). Such series and their applications were studied in a previous paper by the author [Tr. Mat. Inst. Steklova 190, 186-221 (1989; Zbl 0707.11059)]. After a survey of results on the relation between a function and its Vinogradov extension under various assumptions, the author turns to the main topic of this paper: partial differential equations of the Schrödinger type in the space variable \(x\) and in the time variable \(t\), the initial condition being given by the function \(f\). In the paper mentioned above, the case of equations with constant coefficients was considered. Now the coefficients are allowed to depend on \(x\) and \(t\). The (generalized) solution can be expressed in terms of the Vinogradov extension of \(f\) even in this general case. The author discusses first the case \(f \in L^2\), and then the more restricted case where \(f\) is of bounded variation.
For the entire collection see [Zbl 0774.00012].

Reviewer: M.Jutila (Turku)

MSC:

35J10	Schrödinger operator, Schrödinger equation
42A16	Fourier coefficients, Fourier series of functions with special properties, special Fourier series
11L15	Weyl sums

Keywords:

Vinogradov extensions; equations with constant coefficients