Spatial Localization of Quantum States and Physical Meaning of the Matrix Elements of the Resolvent Operator

摘要

We investigate how the second sheet singularities of the matrix elements of the resolvent operator depend on the dense set of states on which they are evaluated. We illustrate the fact that, for a given Hamiltonian, one can choose a dense set of states such that the corresponding matrix elements of the resolvent can be continued across the spectrum into the lower half plane and exhibit therein singularities at arbitrary preassigned locations. This shows that, given a resonant dynamics, some further physical input is needed in addition to the assignment of the total Hamiltonian, in order to identify privileged dense sets of states for which the singularities of the analytic continuation of the matrix elements of the resolvent have a physical meaning as characterizing the unstable states of the system. We develop these considerations explicitly in the examples of a nonrelativistic spinless particle subject to a local central potential or to a degenerate interaction. In the above cases, we show that a privileged set can either be chosen to be the set of states which are entire functions in the representation in which the free Hamiltonian is diagonal or, equivalently, the set of spatially localized states. The close connection with the preparation procedure of an unstable state as a localization process is discussed. (Atomindex citation 13:711818)