 
Summary: Physica D 123 (1998) 513524
Modal expansions and completeness relations for some
timedependent Schršodinger equations
Peter D. Miller
, N.N. Akhmediev 1
Australian Photonics Cooperative Research Centre, Optical Sciences Centre,
The Australian National University, Canberra, ACT 0200, Australia
Abstract
With the use of a variant of the method of separation of variables, the initial value problem for the timedependent linear
Schršodinger equation is solved exactly for a large class of potential functions related to multisoliton interactions in the vector
nonlinear Schršodinger equation. Completeness of states is proved for absolutely continuous initial data in L1. Copyright ©
1998 Elsevier Science B.V.
PACS: 02.30.Jr; 03.40.Kf; 03.65.Ge
Keywords: Timedependent Schršodinger equations; Exact solvability; Completeness relations
1. Introduction
The nonperturbative solution of the initial value problem for the linear Schršodinger equation
it f + 1
2 2
x f  V (x, t)f = 0 (1)
subject to the initial condition f (x, 0) = f0(x) is a central problem of quantum mechanics in one space dimension,
