Search results for "Hilbert space."

PIP-Space Valued Reproducing Pairs of Measurable Functions

2019

We analyze the notion of reproducing pairs of weakly measurable functions, a generalization of continuous frames. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space Z : = Z ( X , μ ), where ( X , μ ) is a measure space. Three cases are envisaged, with increasing generality: (i) Y and Z are both Hilbert spaces; (ii) Y is a Hilbert space, but Z is a pip-space; (iii) Y and Z are both pip-spaces. It is shown, in particular, that the requirement that a pair of measurable functions be reproducing strongly constrains the structure of the initial space Y. Examples are presented for each case.

Normed Quasi *-Algebras: Bounded Elements and Spectrum

2020

Bounded elements of a Banach quasi *-algebra are intended to be those, whose images under every *-representation are bounded operators in a Hilbert space. This rough idea can be developed in several ways, as we shall see in the present chapter. These notions lead us to discuss a convenient concept of spectrum of an element in this context.

THE BISHOP-PHELPS-BOLLOBAS PROPERTY FOR HERMITIAN FORMS ON HILBERT SPACES

2013

On non-self-adjoint operators defined by Riesz bases in Hilbert and rigged Hilbert spaces

2018

In this paper we discuss some results on non self-adjoint Hamiltonians with real discrete simple spectrum under the assumption that their eigenvectors form Riesz bases of a certain Hilbert space. Also, we exhibit a generalization of those results to the case of rigged Hilbert spaces, and we also consider the problem of the factorization of the aforementioned Hamiltonians in terms of generalized lowering and raising operators.

Linear optical implementation of nonlocal product states and their indistinguishability

2001

In a recent paper Bennett et al.[Phys. Rev.A 59, 1070 (1999)] have shown the existence of a basis of product states of a bipartite system with manifest non-local properties. In particular these states cannot be completely discriminated by means of bilocal measurements. In this paper we propose an optical realization of these states and we will show that they cannot be completely discriminate by means of a global measurement using only optical linear elements, conditional transformation and auxiliary photons.

Dynamics of a quantum particle interacting with a thermal bath and subject to an oscillating asymmetric bistable potential

2012

Exploiting the approach of the Feynman-Vernon influence functional [1] within the framework of the discrete variable representation (DVR) [2], we consider a quantum particle described by the Caldeira-Leggett model [3]. The particle, “moving” in an asymmetric bistable potential and subject to a periodical driving, interacts with a thermal bath of harmonic oscillators. In this conditions we study the dynamics of the particle by analyzing the time evolution of the populations in the DVR. Specifically we focalize on the position eigenstate located in the shallower well, i.e. metastable state, finding a non-monotonic behaviour of the corresponding population as a function of the frequency. Moreo…

Dynamics of a Driven Dissipative Quantum System

2013

We investigate the dynamics of a driven multilevel system, consisting of a particle in an asymmetric bistable potential, strongly interacting with a thermal bath according to the Caldeira-Leggett model. The populations in the discrete (position) variable representation (DVR), are obtained as solution of a Markovian approximated master equation, which is derived from a discretized path integral approach based on the Feynman-Vernon influence functional. By varying the environmental parameters (temperature and coupling strength) as well as the driving frequency and amplitude, we study the transient dynamics and stationary configuration of the system. In particular, we analyze the population of…

Transient Dynamics of a Driven Quantum Bistable System

2013

We study the transient dynamics and the asymptotic behaviour of a multilevel system in the strong dissipation regime. The system is modeled as a periodically driven quantum particle in an asymmetric double well potential, interacting with the bosonic heat bath of the Caldeira-Leggett model. The analytical approach used is non- perturbative in the particle-bath coupling and is based on a space-discretized path integral expression for the particle’s reduced density matrix. By a suitable approximation on the Feynman-Vernon influence functional a Markov-approximated master equation is obtained for the populations in the Discrete Variable Representation (DVR).

L p-Spaces and the Radon–Nikodym Theorem

2020

In this chapter, we study the spaces of functions whose pth power is integrable. In Section 7.2, we first derive some of the important inequalities (Holder, Minkowski, Jensen) and then in Section 7.3 investigate the case p=2 in more detail.

Kernel Spectral Angle Mapper

2016

This communication introduces a very simple generalization of the familiar spectral angle mapper (SAM) distance. SAM is perhaps the most widely used distance in chemometrics, hyperspectral imaging, and remote sensing applications. We show that a nonlinear version of SAM can be readily obtained by measuring the angle between pairs of vectors in a reproducing kernel Hilbert spaces. The kernel SAM generalizes the angle measure to higher-order statistics, it is a valid reproducing kernel, it is universal, and it has consistent geometrical properties that permit deriving a metric easily. We illustrate its performance in a target detection problem using very high resolution imagery. Excellent re…