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arXiv:1509.00340 · Journal of Mathematical Physics

The Bender-Dunne basis operators as Hilbert space operators

Joseph Raphael R. Bunao, Eric A. Galapon

Published Feb 1, 2014• 10 citations
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Abstract

The Bender-Dunne basis operators, $\mathsf {T}_{-m,n}=2^{-n}\sum _{k=0}^n { n \choose k} \mathsf {q}^k \mathsf {p}^{-m} \mathsf {q}^{n-k}$T−m,n=2−n∑k=0nnkqkp−mqn−k where $\mathsf {q}$q and $\mathsf {p}$p are the position and momentum operators, respectively, are formal integral operators in position representation in the entire real line $\mathbb {R}$R for positive integers n and m. We show, by explicit construction of a dense domain, that the operators $\mathsf {T}_{-m,n}$T−m,n's are densely defined operators in the Hilbert space $L^2(\mathbb {R})$L2(R).

DOI
10.1063/1.4863901

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The Bender-Dunne basis operators, T−m,n=2−n∑k=0n(nk)qkp−mqn−k\mathsf {T}_{-m,n}=2^{-n}\sum _{k=0}^n { n \choose k} \mathsf {q}^k \mathsf {p}^{-m} \mathsf {q}^{n-k}T−m,n​=2−n∑k=0n​(kn​)qkp−mqn−kT−m,n=2−n∑k=0nnkqkp−mqn−k where q\mathsf {q}qq and p\mathsf {p}pp are the position and momentum operators, respectively, are formal integral operators in position representation in the entire real line R\mathbb {R}RR for positive integers n and m. We show, by explicit construction of a dense domain, that the operators T−m,n\mathsf {T}_{-m,n}T−m,n​T−m,n's are densely defined operators in the Hilbert space L2(R)L^2(\mathbb {R})L2(R)L2(R).
This paper undertakes the problem of constructing first time-of-arrival operators for arbitrary everywhere analytic potentials in the space of operators spanned by the Bender–Dunne basis (Bender and Dunne 1989 Phys. Rev. D 40 3504). The operators are conjugate with their respective system Hamiltonians and reduce to their corresponding classical time-of-arrival expressions for infinitesimal ℏ. The supraquantized time-of-arrival operators (Galapon 2004 J. Math. Phys. 45 3180) and the confined time-of-arrival operators (Galapon et al 2004 Phys. Rev. Lett. 93 180406, Galapon 2006 J. Mod. Phys. A 21 6351) are then shown to be two distinct representations of the time-of-arrival operators constructed in the Bender–Dunne space.
Of crucial importance to the development of quantum computing and information has been the construction of a quantum operations formalism that admits a description of quantum noise while simultaneously revealing the behavior of an open quantum system. The operator-sum representation is such a formalism and has provided a succinct description for set of bounded operators that act on a finite dimensional quantum system. In this paper we derive a basis for the set of bounded operators that act on a ddd-dimensional Hilbert space and we illustrate how this basis set may be extended and identified with a set of elements upon which the operator-sum representation rests.
In this chapter we assume that H is a Gaussian Hilbert space. We will use H to define and study several operators on L 2 (Ω, F(H) , P). These operators are important in quantum physics, but we will not go into any such applications here; cf. for example Segal (1956), Glimm and Jaffe (1981, Chapter 6) and Meyer (1993). There are also other applications of these operators, and we will use some of the results below in Chapter 15. Some of the operators will be studied again in Chapters 15 and 16, where we also consider actions on L p for p ≠ 2. The reader may note that the operators treated here can be defined for any abstract Fock space based on an arbitrary Hilbert space, and that many of the results make sense in this generality, cf. for example Baez, Segal and Zhou (1992), Meyer (1993) and Parthasaraty (1992). Nevertheless, we will exclusively consider the Gaussian case here, where the extra structure is both helpful and, we hope, illuminating. (Of course, any result that can be stated for an abstract Fock space is valid in general as long as it is valid for the concrete realization treated here for Gaussian spaces.) We warn the reader that several of the operators defined below will be unbounded and defined only on a dense subset of L 2 .
The ability of the Rigged Hilbert Space formalism to deal with continuous spectrum is demonstrated within the example of the square barrier potential. The non-square integrable solutions of the time-independent Schrödinger equation are used to define Dirac kets, which are (generalized) eigenvectors of the Hamiltonian. These Dirac kets are antilinear functionals over the space of physical wave functions. They are also basis vectors that expand any physical wave function in a Dirac basis vector expansion. It is shown that an acceptable physical wave function must fulfill stronger conditions than just square integrability — the space of physical wave function is not the whole Hilbert space but rather a dense subspace of the Hilbert space. We construct the position and energy representations of the Rigged Hilbert Space generated by the square barrier potential Hamiltonian. We shall also construct the unitary operator that transforms from the position into the energy representation. We shall see that in the energy representation the Dirac kets act as the antilinear Schwartz delta functional. In constructing the Rigged Hilbert Space of the square barrier potential, we will find a systematic procedure to construct the Rigged Hilbert Space of a large class of spherically symmetric potentials. The example of the square barrier potential will also make apparent that the natural framework for the solutions of a Schrödinger operator with continuous spectrum is the Rigged Hilbert Space rather than just the Hilbert space.
We solve the problem of integrating operator equations for the dynamics of nonautonomous quantum systems by using time-dependent canonical transformations. The studied operator equations essentially reproduce the classical integrability conditions at the quantum level in the basic cases of one-dimensional nonautonomous dynamical systems. We seek solutions in the form of operator series in the Bender–Dunne basis of pseudodifferential operators. Together with this problem, we consider quantum canonical transformations. The minimal solution of the operator equation in the representation of the basis at a fixed time corresponds to the lowest-order contribution of the solution obtained as a result of applying a canonical linear transformation to the basis elements.