Asymptotic behaviour of the eigenvalues of a Schrödinger operator arising from a simple model of predissociation

Dowling, Dónal
(2003)
Asymptotic behaviour of the eigenvalues of a Schrödinger operator arising from a simple model of predissociation.
Master of Science thesis, Dublin City University.

With in the perturbation theory of linear differential equations there has been considerable interest in recent years in calculating the imaginary part of an eigenvalue E which moves off the real axis when a small positive perturbation e is switched on. Typically the perturbation in Re E is algebraic in e, while that in Im E is exponentially small as e —> 0. This phenomenon occurs in several physical applications including resonance theory in quantum mechanics, wave trapping by small islands, viscous fingering in fluid dynamics, and in energy losses at bends in optical fibres. In this thesis the problem arises from a model of molecular predissociation in quantum chemistry. I t is more complicated than the above examples, firs tly because there are two Schrödinger equations in the system and secondly because the small parameter appears in the coupling term.
In 1995 operator theoretic methods were used by Duclos and Meller [5] to obtain bounds on both the real and imaginary parts of the eigenvalue for such a problem, but gave no information about the associated eigenfunction. Here we consider a similar model proposed by Asch [2] and also use operator theoretic methods to get a bound on the resonances. We then improve on this bound by Fourier transforming the 2 x 2 system to a single second order equation whose solutions we approximate asymptotically by the classical analysis methods of Olver [13] as found in the paper of Dunster [6). We then substitute the approximate solution plus its error term into the boundary condition at the origin to obtain an eigenvalue relation which yields another estimate for the perturbation in E. In the final chapter we report on other approaches which have been tried on this problem, outline the difficulties associated w ith each of them and make some suggestions for extending our results.