Mobius structure of the spectral space of Schrodinger operators with pointinteraction

The Schrodinger operator with point interaction in one dimension has a U(2) family of self-adjoint extensions. We study the spectrum of the operator a nd show that (i) the spectrum is uniquely determined by the eigenvalues of the matrix U is an element ofU(2) that characterizes the extension, and tha t (ii) the space of distinct spectra is given by the orbifold T-2/Z(2) whic h is a Mobius strip with boundary. We employ a parametrization of U(2) that admits a direct physical interpretation and furnishes a coherent framework to realize the spectral duality and anholonomy recently found. This allows us to find that (iii) physically distinct point interactions form a three- parameter quotient space of the U(2) family. (C) 2001 American Institute of Physics.