טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentRachel Hess Green
SubjectA Probabilistic Approach to Bounded Solutions of the
Schrodinger Equation
DepartmentDepartment of Mathematics
Supervisor Full Professor Pinsky Ross
Full Thesis textFull thesis text - English Version


Abstract


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.