Hamiltonians defined by biorthogonal sets

Abstract

In some recent papers, the studies on biorthogonal Riesz bases has found a renewed motivation because of their connection with pseudo-hermitian Quantum Mechanics, which deals with physical systems described by Hamiltonians which are not self-adjoint but still may have real point spectra. Also, their eigenvectors may form Riesz, not necessarily orthonormal, bases for the Hilbert space in which the model is defined. Those Riesz bases allow a decomposition of the Hamiltonian, as already discussed is some previous papers. However, in many physical models, one has to deal not with o.n. bases or with Riesz bases, but just with biorthogonal sets. Here, we consider the more general concept of Gquasi basis and we show a series of conditions under which a definition of non self-adjoint Hamiltonian with purely point real spectra is still possible.

Weak pseudo-bosons

“…which can be seen as a weak form of the resolution of the identity, restricted to G. Of course, if f ∈ G is orthogonal to all the ϕ n 's, or to all the Ψ n 's, then (2.5) implies that f = 0. Hence F ϕ and F Ψ are complete in G, [13]. The families F ϕ and F Ψ can be used to define two densely defined operators S ϕ and S Ψ via their action respectively on F Ψ and F ϕ :…”

Generalized Riesz systems and orthonormal sequences in Krein spaces

“…Hφ n = E n φ n and H * ψ n = E n ψ n , where E n = n + 1 2 . Simple computations show that H * = 1 2 − d 2 dx 2 + x d dx + 1 2 3x 2 2 + 1 , see [6]. The sets {φ n } and {ψ n } are both complete in L 2 (R).…”

“…Example 1:-A first simple example of GRS can be extracted from [6]: if we take Q = − x 2 2 , x being the position operator, it is clear that D(e Q/2 ) = L 2 (R), while…”

“…These two concepts were originally introduced in [15] and [2], respectively and then analyzed in a series of papers, see, e.g. [4,Chapter 3], [6,7], [14]. The motivation for introducing GRS and G-quasi bases was the need to put on a mathematically rigorous ground several physical models where the eigenstates of some non self-adjoint operator were usually claimed to be bases, while they were not.…”

Frames and weak frames for unbounded operators

“…Still, continuous weak A-frames clearly call to mind continuous multipliers which are the object of interest of a recent literature even though unbounded multipliers, as far the authors knows, have been little looked over. For example, some initial steps toward this direction has been done, in the discrete case, in [5,6,7,8,22] where some unbounded multipliers have been defined. Therefore this paper can spure investigation in the direction of unbounded multipliers in the continuous case.…”