Institute of Applied Mathematics,

Odesa University OSEU,

P.O.Box 24A,

65009 Odesa,

UKRAINE

E-mail: glushkov@paco.net

**Quantization of quasi-stationary states for the Schrodinger
equation: Stark task**

**Abstract:**

We consider a new principle for quantization of the quasi-stationary
states for Schrodinger equation with singular potential (the Stark task)
and develop a numerical procedure for its realization. The principal tasks,
where everybody needs a correct realization of the quasi-stationary and
continuum states of the Schrodinger equation, are the scattering and collision
problem, Stark task, etc. The standard procedure relates complex eigen-energies
(EE) *E *= *Er*+0,5i*G* and complex eigen-functions (EF)
to resonances in the spectrum of eigen-values of the Schrodinger equation.
The calculation difficulties in the standard methods are well known. The
WKB approximation overcomes these difficulties for the states, lying far
from "new continuum" boundary and, as rule, is applied in the case of a
relatively weak field. Quite another calculation procedures are used in
the Borel summation of the divergent perturbation theory (PT) series and
in numerical solution of the difference equations following from expansion
of the wave-function over finite basis. We developed a consistent approach
to the non-stationary state problem solution including also scattering
problems (operator perturbation theory). The essence of the method is the
inclusion of the well-known method of "distorted waves approximation" in

the frame of the formally exact PT [1]. The zeroth order Hamiltonian
*H*_{0}
of this PT possesses only stationary bound and
scattering states. To overcome formal difficulties, the zeroth order
Hamiltonian was defined by the set of the orthogonal EF and EE without
specifying the explicit form of the corresponding zeroth order potential.
The Shrodinger equation for the wave function with taking into account
the uniform electric field has the standard form. After separation of variables
in parabolic co-ordinates it transformed to the system of two equations
for the functions *f,* *g*, coupled through the constraint on
the separation constants: *b*_{1}+*b*_{2}=1.
Potential energy has the barrier and two turning points for the classical
motion. Further we substitute the external field by the model 1-parameter
function, which satisfies to necessary asymptotic conditions. The final
results do not depend on parameter of the function. To calculate the width
*G*
of the concrete quasi-stationary state in the lowest PT order it is necessary
to know two zeroth order EF of *H*_{0}: bound state function
and scattering state function with the same EE. First, one has to define
the EE of the expected bound state. It is the well-known problem of states
quantization in the case of the penetrable barrier. We solve the system
with total Hamiltonian *H* under the two conditions. These two conditions
quantify the bound energy *E*, separation constant
*b*_{1}.
The further procedure for this 2D eigen-value problem is resulted in solving
the system of the ordinary differential equations with probe pairs of *E*,
*b*_{1}. The bound state EE, eigen-value *b*_{1}
and EF for the zero order Hamiltonian
*H*_{0} coincide with
those for the total Hamiltonian *H* at field strength approaching
to 0. The scattering state functions must be orthogonal to the above-defined
bound state function and to each other. It may be written as: *g*(*t*)=g_{1}(*t*)-*zg*_{2}(*t*)
with *g*_{1} satisfying the starting differential equation;
the function *g*_{2} satisfies the non-homogeneous differential
equation, which differs from starting one only by the right hand term,
disappearing at t approaching to infiniteness. The

coefficient z ensures the orthogonality condition. The imaginary energy
part and resonance width *G* in lowest PT order are connected directly.
The calculation procedure at known energy *E* and parameter b has
been reduced to solution of system of the ordinary differential equations.

- A.Glushkov and L.N.Ivanov, J. Phys. B. 26, 379 (1993).