×

Singularities of the Green function of the nonstationary Schrödinger equation. (English. Russian original) Zbl 0920.35008

Funct. Anal. Appl. 32, No. 2, 132-134 (1998); translation from Funkts. Anal. Prilozh. 32, No. 2, 80-83 (1998).
The Green function \(\psi= \psi(x,y,t)\) of the nonstationary Schrödinger equation is determined by the relations \[ ih\psi_t= -\textstyle{{1\over 2}} h^2\Delta\psi+ v(x)\psi,\quad \psi(x,y,0)= \delta(x- y),\;t\geq 0,\;y\in\mathbb{R}^d. \] In what follows, we assume that \(v\) is a smooth real function. For a chosen \(t>0\), the function \(\psi\) can have singularities with respect to \(x\). These singularities depend on the rate of growth of \(v(x)\) as \(x\to\infty\). If \(v\) satisfies the condition \[ | v(x)|\leq c| x|^{2\gamma},\quad \gamma<1, \] for sufficiently large \(x\), then \(\psi\) is smooth. This assertion, as well as several subsequent assertions, is given here in a simplified form.
The main objective of the present paper is to describe the formula mechanism responsible for the form of the singularities of \(\psi\) for \(\gamma>1\) in the case \(d=1\). Under certain restrictions, we shall be able to say something about the character of the singularities of \(\psi\) for \(d>1\).

MSC:

35A20 Analyticity in context of PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] S. Zeldich, Commun. Math. Phys.,90, 1–26 (1983). · Zbl 0554.35031
[2] W. Craig, T. Kapeller, and W. Strauss, Microlocal Dispersive Smoothing for the Schrödinger Equation, Preprint, 1996.
[3] K. Yajima, Smoothness and Nonsmoothness of the Fundamental Solution for Initial Value Problem for Time-Dependent Schrödinger Equations, Preprint, 1995.
[4] L. Kapitanski and I. Rodnianski, IMRN, No. 2, 41–54, 1996. · Zbl 0859.35146
[5] L. Kapitanski, I. Rodnianski, and K. Yajima, On the Fundamental Solution of a Perturbed Harmonic Oscillator, Preprint, 1996. · Zbl 0892.35035
[6] V. P. Maslov and M. V. Fedoryuk, Semiclassical Approximation for Equations of Quantum Mechanics [in Russian], Nauka, Moscow, 1976. · Zbl 0449.58002
[7] V. S. Buslaev, Funkts. Anal. Prilozhen.,3, No. 3, 17–31 (1969).
[8] H. L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS, Regional Conference Series in Math., No. 84, Amer. Math. Soc. (1994).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.