How to trap a particle without using potential field which is infinity at some point? (quantum physics) If impossible, how to prove it? As we all know, the wave function of the stationary state a quantum particle trapped in a rigid box (with infinite potential outside the box) cannot have a non-zero value outside the box. So can we achieve the same thing (constraining the particle in a certain area) in a finite potential field?
 A: Trapping means that an eigenfunction has compact support. For locally bounded potential this is impossible, because of uniqueness theorem. Suppose for simplicity that our system is 1-dimensional. Then our wave function is a solution of the 1-dimensional stationary Schrodinger equation
$$y''=p(x)y.$$
If the potential $p$ is bounded, then the RHS satisfies Lipschitz condition, as a function of $y$, and we have uniqueness of solutions by the classical uniqueness theorem for ODE.
(If a solution is zero on an open set then it coincides everywhere with the zero solution).
For higher dimension, the corresponding property is called the unique continuation property, see
T. Carleman, Sur les systèmes linéaires aux dérivées partielles du premier ordre à deux variables, C. R. Acad. Sc. 197 (1933), p. 471-474, and
C. Muller, On the behavior of the solutions of the differential equation $\Delta u = F(x, u)$ in the
neighborhood of a point, Comm. Pure Appl. Math. vol. 7 (1954) pp. 505-551.
Ph. Hartman, A. Wintner,
On the local behavior of solutions of non-parabolic partial differential equations. III. Approximations by spherical harmonics.
Amer. J. Math. 77 (1955), 453–474.
These results were very much generalized, look under ``unique continuation property''.
