The quantum phase 2-form near degeneracies: two numerical studies

The phase of a quantum state changes rapidly as parameters X = (X$_1$, X$_2$,...) are varied near a degeneracy X$^*$, reflecting the monopole singularity of the underlying phase 2-form V(X) at X$^*$. The singularities may be sources or sinks of V. We study them numerically and display them graphically for two families of hamiltonians whose degeneracy structure is typical. First is a particle moving along a line segment with kinetic energy quartic in the momentum ('quartic-momentum square well'); the X are incorporated into the boundary conditions. Second is a charged particle moving in a domain D of the plane which is threaded by a magnetic flux line of strength $\alpha$, with wavefunction vanishing on the boundary $\partial$D ('Aharonov-Bohm billiards'); the X are $\alpha$ and parameters specifying $\partial$D; V is not invariant under gauge transformations of the vector potential generating the flux. For Aharonov-Bohm billiards we study how the spatial patterns of phase of wavefunctions change round circuits near degeneracies; these patterns also have singularities (wavefront dislocations) that appear and disappear by colliding with each other and with $\partial$D.