Nonhomogeneity of Picard dimensions of rotation free hyperbolic densities (Q1900527)

Consider the time-independent Schrödinger equation \((- \Delta+ P(x)) u(x)= 0\) on the punctured disc \(\Omega= \{z\in C; |z|< 1\}\), for some potential \(P\) which is assumed locally Hölder continuous on \(0< |z|\leq 1\). We denote the set \(\{z; |z|= 1\}\) by \(\Gamma\) and consider the convex set \(PP_1(\Omega; \Gamma)\) of nonnegative solutions of the equations which satisfy \((- 1/2\pi) \int^{2\pi}_0 [(\partial/\partial r)u(r\exp [i\theta])]_{r= 1} d\theta= 1\) and call ``Picard dimension'' of \(P\) at \(0\) the cardinal number of the set of extremal points of \(PP_1(\Omega; \Gamma)\). A potential is called hyperbolic if the associated Picard dimension is \(\geq 1\) and if the Schrödinger equation on \(\Omega\) admits a Green function. The author shows that there are radial hyperbolic potentials \(P\) so that the Picard dimension of \(P\) is 1, whereas the Picard dimension of \(P/4\) is the cardinal number of the continuum.