6533b85dfe1ef96bd12be746
RESEARCH PRODUCT
Hamiltonians Generated by Parseval Frames
Fabio BagarelloSergiusz Kużelsubject
Pure mathematicsBasis (linear algebra)Applied MathematicsFrames Hamiltonian operators Orthonormal basesSpectrum (functional analysis)Hilbert spacePhysical systemObservableComputer Science::Digital LibrariesParseval's theoremsymbols.namesakeComputer Science::Mathematical SoftwaresymbolsOrthonormal basisSettore MAT/07 - Fisica MatematicaEigenvalues and eigenvectorsMathematicsdescription
AbstractIt is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by the eigenvectors of the Hamiltonians. In some recent papers, this expansion has been extended to the case in which these eigenvectors form a Riesz basis or, more recently, a ${\mathcal{D}}$ D -quasi basis (Bagarello and Bellomonte in J. Phys. A 50:145203, 2017, Bagarello et al. in J. Math. Phys. 59:033506, 2018), rather than an orthonormal basis. Here we discuss what can be done when these sets are replaced by Parseval frames. This interest is motivated by physical reasons, and in particular by the fact that the mathematical Hilbert space where the physical system is originally defined, contains sometimes also states which cannot really be occupied by the physical system itself. In particular, we show what changes in the spectrum of the observables, when going from orthonormal bases to Parseval frames. In this perspective we propose the notion of $E$ E -connection for observables. Several examples are discussed.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2021-02-01 |