__Ionization Problems in Simple Quantum Systems Subject to Periodic
Electromagnetic Fields of Arbitrary Strength __

## Student: John McClain

Advisors: Prof. Ovidiu Costin (www.math.rutgers.edu/~costin) and Prof. Rodica Costin

I am working with the time-dependent Schrodinger equation which is
meant to be a simple description of a quantum system in which an atom (hydrogen) is bombarded with electromagnetic
radiation, say, in the form of a laser or an intense microwave field.
The solution to this equation (given initial conditions) describe
the behavior of the electron in the atom. We study the long time
dependence of the solution to this equation to see if the electron
eventually leaves the system. If it does, the atom has been ionized.
If not, this system describes a bound state of the electron, meaning
that, despite the disturbance from the radiation, the electron remains
within the atom.
The general method is to take the Laplace transform (defined later)
of the time-dependent Schrodinger Equation with respect to time. We
then study the regularity and analyticity properties of the solution
to the transformed equation. These properties reveal information
about the time dependence of the solution of the original
(untransformed) Schrodinger Equation.

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__References__

Boas, Mary. (1983) *Mathematical Methods in the Physical
Sciences.* John Wiley and Sons, Inc.

O. Costin, R. D. Costin, and J.L. Lebowitz *Regularity
of the Laplace Transform and Asymptotics of the Wave Function in
Time-periodic Potentials.*

Reed, Michael and Simon, Barry. (1972) *Methods of Modern
Mathematical Physics 1: Functional Analysis.* New York and London:
Academic Press.

Roklenko, A., Costin, O., and Lebowitz, J. L. *Decay Survival of
a Localized State Subject to Harmonic Forcing: Exact
Solutions.* J. Phys. A: Math. Gen. 35, pp 8943-8951 (2002).