|M.Sc Student||Hess Green Rachel|
|Subject||A Probabilistic Approach to Bounded Solutions of the|
|Department||Department of Mathematics||Supervisor||Professor Ross Pinsky|
|Full Thesis text|
In this work we present three results. The first two concern the operator on with a potential V : [0;1) -> R which is non-negative, piecewise continuous and compactly supported. Extend V to all of R by V (x) = 0, x < 0.
We define where , and we investigate the behavior of the critical value , i.e. the value at which becomes a critical operator.
The second result, called the localization of binding, determines for what values of t; s > 0 the operator will be subcritical where the potentials V1, V2 are non-negative, piecewise continuous, compactly supported and not identically zero.
In the third result we consider the equation (1) where . It is known that there exists a solution to (1) if and only if (2) a.s. where X(s) is a Brownian motion. One would like to have an analytic condition for the existence of solutions to (1) rather than just a probabilistic one. We show that under certain restrictions, a necessary and suffcient analytic condition can be given.