Spectral Analysis in Hilbert Space

Valter Moretti

Peter Bongaarts

Lajos Molnár, Peter Šemrl

Jean-Pierre Antoine

We review the different formalisms that can be used for quantum mechanics, all of them going beyond the standard Hilbert space formulation: Rigged Hilbert Spaces (RHS), partial inner product spaces (PIP spaces), etc. In particular, we consider the spectral analysis of observables and the description of symmetries.

M. O. Farrukh

Quantum mechanics is formulated using a nonstandard Hilbert space. The concept of an eigen vector of a linear operator, which applies to standard as well as nonstandard Hilbert spaces, is replaced by the more general concept of an ultra eigen vector, which applies to nonstandard Hilbert spaces alone. Ultra eigen vectors corresponding to all spectral points of internal self−adjoint operators are proved to exist. This result enables us to set up a formalism which is equally valid for the discrete, as well as the continuous spectrum. Finally, Dirac’s formalism is reproduced, in a rigorous form within the nonstandard Hilbert space structure.