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arXiv:0904.0882 · Journal of Mathematical Physics

Invariant analytic orthonormalization procedure with an application to coherent states

Fabio Bagarello, Salvatore Triolo

Published Apr 1, 2007• 6 citations
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Abstract

We discuss a general strategy which produces an orthonormal set of vectors, stable under the action of a given set of unitary operators Aj, j=1,2,…,n, starting from a fixed normalized vector in H and from a set of unitary operators. We discuss several examples of this procedure and, in particular, we show how a set of coherentlike vectors can be produced and in which condition over the lattice spacing this can be done.

DOI
10.1063/1.2711371

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We discuss a general strategy which produces an orthonormal set of vectors, stable under the action of a given set of unitary operators Aj, j=1,2,…,n, starting from a fixed normalized vector in H and from a set of unitary operators. We discuss several examples of this procedure and, in particular, we show how a set of coherentlike vectors can be produced and in which condition over the lattice spacing this can be done.
In this paper we discuss a general strategy to construct vector coherent states of the Gazeau-Klauder type and we use them to built up examples of isospectral hamiltonians. For that we use a general strategy recently proposed by the author and which extends well known facts on intertwining operators. We also discuss the possibility In a recent paper, [1], we have proposed a new procedure which gives rise, given few ingredients, to a hamiltonian h2 which has the same spectrum of a given hamiltonian h1 and whose respective eigenstates are related by a given intertwining operator. These results extend what was discussed in the previous literature on this subject, [2], and have
An orthonormal basis of modified coherent states is constructed. Each member of the basis is an infinite sum of coherent states on a von Neumann lattice. A single state is assigned to each unit cell of area h (Planck constant) in the phase plane. The uncertainties of the coordinate x and the square of the momentum p 2 for these states are shown to be similar to those for the usual coherent states. Expansions in the newly established set are discussed and it is shown that any function in the kq -representation can be written as a sum of two fixed kq -functions. Approximate commuting operators for x and p 2 are defined on a lattice in phase plane according to von Neumann's prescription.
We derive a generalization of the well-known theorem for the square integrability of a unitary irreducible representation of a locally compact group. The generalization covers the case of representations admitting vector coherent states. The result is illustrated by an example drawn from the isochronous Galilei group. The construction yields a wide variety of coherent states, labeled by phase space points, which satisfy a resolution of the identity condition, and incorporate spin degrees of freedom.
Pertinent properties of the unitary operators that create coherent states and squeezed coherent states are discussed. We show that certain generalizations of squeezed coherent states do not exist. This is accomplished by demonstrating that for the generalized squeeze operators ${U}_{k}=\mathrm{exp}(i{A}_{k})=\mathrm{exp}[{z}_{k}{({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{k}+i{h}_{k\ensuremath{-}1}\ensuremath{-}{({z}_{k})}^{*}{a}^{k}]$, 〈0∣Uk∣0〉〈0|{U}_{k}|0〉〈0∣Uk​∣0〉 diverges, k>2k>2k>2. This implies that ∣0〉|0〉∣0〉 is not an analytic vector of Ak{A}_{k}Ak​ for all k>2k>2k>2, where ${h}_{k\ensuremath{-}1}$ is a Hermitian polynomial in aaa and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ up to powers of ($k\ensuremath{-}1$).
A generalization of the Perelomov procedure for the construction of coherent states is proposed. The new procedure is used to construct systems of coherent states in the carrier spaces of unitary irreducible representations of groups G=S∅V, where V is a vector space and S⊂GL(V). The coherent states are shown to be labeled by the points in cotangent bundles T*𝒪* of orbits 𝒪* of S in V*, the dual of V; it is proven that T*𝒪* is a symplectic homogeneous space of G. The generalized procedure for the construction of coherent states presented in this paper is shown to encompass as special cases the constructions known in the literature for the coherent states of the Weyl–Heisenberg, the ‘‘ax+b,’’ and the Galilei and Poincaré groups. Moreover, completely new sets of coherent states are constructed for the Euclidean group E(n), where the Perelomov construction fails.