Search results for "biorthogonal set"
showing 6 items of 6 documents
Some invariant biorthogonal sets with an application to coherent states
2014
We show how to construct, out of a certain basis invariant under the action of one or more unitary operators, a second biorthogonal set with similar properties. In particular, we discuss conditions for this new set to be also a basis of the Hilbert space, and we apply the procedure to coherent states. We conclude the paper considering a simple application of our construction to pseudo-hermitian quantum mechanics.
Gibbs states, algebraic dynamics and generalized Riesz systems
2020
In PT-quantum mechanics the generator of the dynamics of a physical system is not necessarily a self-adjoint Hamiltonian. It is now clear that this choice does not prevent to get a unitary time evolution and a real spectrum of the Hamiltonian, even if, most of the times, one is forced to deal with biorthogonal sets rather than with on orthonormal basis of eigenvectors. In this paper we consider some extended versions of the Heisenberg algebraic dynamics and we relate this analysis to some generalized version of Gibbs states and to their related KMS-like conditions. We also discuss some preliminary aspects of the Tomita-Takesaki theory in our context.
A Note on States and Traces from Biorthogonal Sets
2019
In this paper, following Bagarello, Trapani, and myself, we generalize the Gibbs states and their related KMS-like conditions. We have assumed that H 0 , H are closed and, at least, densely defined, without giving information on the domain of these operators. The problem we address in this paper is therefore to find a dense domain D that allows us to generalize the states of Gibbs and take them in their natural environment i.e., defined in L &dagger
Weak pseudo-bosons
2020
We show how the notion of {\em pseudo-bosons}, originally introduced as operators acting on some Hilbert space, can be extended to a distributional settings. In doing so, we are able to construct a rather general framework to deal with generalized eigenvectors of the multiplication and of the derivation operators. Connections with the quantum damped harmonic oscillator are also briefly considered.
Gibbs states defined by biorthogonal sequences
2016
Motivated by the growing interest on PT-quantum mechanics, in this paper we discuss some facts on generalized Gibbs states and on their related KMS-like conditions. To achieve this, we first consider some useful connections between similar (Hamiltonian) operators and we propose some extended version of the Heisenberg algebraic dynamics, deducing some of their properties, useful for our purposes.
Hamiltonians defined by biorthogonal sets
2017
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 $\mat…