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Existence, uniqueness and some properties of Schrödinger propagators

  • Kenji Yajima
Chapter
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Part of the Mathematical Physics Studies book series (MPST, volume 12)

Abstract

We consider a time dependent Schrödinger equation for quantum particles in an electro-magnetic field:
$$ih\frac{{\partial u}}{{\partial t}}\; = H\left( t \right)u = \left( {{H_0}\left( t \right) + V\left( {t,s} \right)} \right)u,\;t \in {R^1},\;x \in {R^n}$$
(1.1)
,
$$ {H_0}\left( t \right) = \sum\limits_{j = 1}^n {\frac{1}{2}} {\left( {-i\frac{\partial }{{\partial {x_j}}}-{A_j}\left( {t,x} \right)} \right)^2}$$
(1.2)
where V(t, x) and A(t, x) = (A 1(t,x), A 2(t, x),…, A n(t, x)) are the electric scalar and magnetic vector potentials of the field. The purpose of this paper is to report, without proofs, some of the author’s recent results on
  1. (1)

    the existence and uniqueness of a unitary propagator, or fundamental solution, {U (t,s), t,s ∈ R 1} in H = L 2 (R n) for (1.1) under most general conditions on A (t, x) and V (t, x); and

     
  2. (2)

    the regurality and the smoothing properties of the propagator, under various conditions on the potentials.

     

Keywords

Fundamental Solution Invariant Subspace Schrodinger Equation Smoothing Effect Weighted Sobolev Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer Science+Business Media Dordrecht 1991

Authors and Affiliations

  • Kenji Yajima
    • 1
  1. 1.Department of Pure and Applied SciencesUniversity of TokyoMeguroku, Tokyo 153Japan

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