Search results for "Hilbert space."
showing 10 items of 227 documents
Star-products, spectral analysis, and hyperfunctions
2000
We study the ⋆-exponential function U(t;X) of any element X in the affine symplectic Lie algebra of the Moyal ⋆-product on the symplectic manifold (ℝ × ℝ;ω). When X is a compact element, a natural specific candidate for U (t;X) to be the exponential function is suggested by the study we make in the non-compact case. U (t;X) has singularities in the t variable. The analytic continuation U(z;X),z = t + iy, defines two boundary values δ+ U (t;X) = limy↓0 U(z;X) and δ-(t;X) = limy↑0 U(z; X). δ+ U (t;X) is a distribution while δ- U (t;X) is a Beurling-type, Gevrey-class s — 2 ultradistribution. We compute the Fourier transforms in t of δ± U (t;X). Both Fourier spectra are discrete but different …
Hyper-Entanglement in Time and Frequency
2019
Hyper-entanglement, i.e. entanglement in more than one degree of freedom, enables a multiplicative increase in Hilbert space size. Such systems can be treated as multi-partite even though the number of state particles is not increased, making them highly attractive for applications in high-capacity quantum communications and information processing [1]. Until now, such states have been realized only using combinations of fully independent degrees of freedom, described by commuting operators, such as polarization and optical paths. Time and frequency, in turn, are linked and described by non-commuting operators. Here, using two discrete forms of energy-time entanglement we demonstrate that ti…
2012
We study the Wigner function for a quantum system with a discrete, infinite dimensional Hilbert space, such as a spinless particle moving on a one dimensional infinite lattice. We discuss the peculiarities of this scenario and of the associated phase space construction, propose a meaningful definition of the Wigner function in this case, and characterize the set of pure states for which it is non-negative. We propose a measure of non-classicality for states in this system which is consistent with the continuum limit. The prescriptions introduced here are illustrated by applying them to localized and Gaussian states, and to their superpositions.
Topologies on Partial O*-Algebras
2002
In this chapter, we introduce some basic locally convex topologies on partial O*-algebras and we establish general properties of these topologies. In Section 4.1, we compare the graph topologies induced by different O-families on the same domain (and the corresponding families of bounded subsets). In the case where the domain D M of an O-family M is a (quasi-) Frechet space, the structure of bounded subsets in D M can be described in a rather explicit way. Section 4.2 and Section 4.3 are devoted to the topologization of (partial) O*-algebras. Section 4.2 deals with locally convex topologies, the so-called uniform topologies τ u , τ u , τ * u and quasiuniform topologies τ qu , and Section 4.…
The Schur property on projective and injective tensor products
2008
The problem of whether the Schur property is passed from a Banach space to its (symmetric) projective n-fold tensor product is reformu lated in the language of polynomial ideals. As a result, a very closely related question is solved in the negative. It is also proved that the injective tensor product of infrabarrelled locally convex spaces with the Schur property has the Schur property as well.
Finite-dimensional pseudo-bosons: a non-Hermitian version of the truncated harmonic oscillator
2018
We propose a deformed version of the commutation rule introduced in 1967 by Buchdahl to describe a particular model of the truncated harmonic oscillator. The rule we consider is defined on a $N$-dimensional Hilbert space $\Hil_N$, and produces two biorhogonal bases of $\Hil_N$ which are eigenstates of the Hamiltonians $h=\frac{1}{2}(q^2+p^2)$, and of its adjoint $h^\dagger$. Here $q$ and $p$ are non-Hermitian operators obeying $[q,p]=i(\1-Nk)$, where $k$ is a suitable orthogonal projection operator. These eigenstates are connected by ladder operators constructed out of $q$, $p$, $q^\dagger$ and $p^\dagger$. Some examples are discussed.
On the structure of the similarity orbits of Jordan operators as analytic homogeneous manifolds
1989
For Jordan elementsJ in a topological algebraB with unite, an open groupB−1 of invertible elements and continuous inversion we consider the similarity orbitsS G (J)={gJg−1:g∈G} (G the groupB−1⋂{e+c:c∈I},I⊂B a bilateral continuous embedded topological ideal). We construct rational local cross sections to the conjugation mapping\(\pi ^J G \to S_G \left( J \right)\left( {\pi ^J \left( g \right) = gJg^{ - 1} } \right)\) and give to the orbitS G (J) the local structure of a rational manifold. Of particular interest is the caseB=L(H) (bounded linear operators on a separable Hilbert spaceH),I=B, for which we obtain the following: 1. If for a Hilbert space operator there exist norm continuous local…
Partial Multiplication of Operators in Rigged Hilbert Spaces
2005
The problem of the multiplication of operators acting in rigged Hilbert spaces is considered. This is done, as usual, by constructing certain intermediate spaces through which the product can be factorized. In the special case where the starting space is the set of C∞-vectors of a self-adjoint operator A, a general procedure for constructing a special family of interspaces is given. Their definition closely reminds that of the Bessel potential spaces, to which they reduce when the starting space is the Schwartz space \(\mathcal{S}(\mathbb{R}^n ).\) Some applications are considered.
An implicit non-linear time dependent equation has a solution
1991
has a solution (u, u, w). The operators &s(l) and a(t) are maximal monotone from a real Hilbert space V to its dual such that &(r) + 9?(r) are V-coercive and a(r) are not degenerate. A linear compact injection i embeds V to a real Banach space W and each d(r) is the strongly monotone subdifferential of a continuous convex function #(I, ) on W. The function f is square integrable. The functions W(r): V+ W* are Lipschitzian as V*-valued functions. Section 3 contains the theorems. The main result is Theorem 2. Theorems 3 and 4 demonstrate the smoothing effect on the initial condition. Their proofs are given in Section 4. They exploit the methods of di Benedetto and Showalter, [4], who studied …
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.