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Abstract The classical time of arrival in the interacting case is quantized by way of quantizing its expansion about the free time of arrival. The quantization is formulated in coordinate representation which represents ordering rules in terms of two variable polynomial functions. This leads to representations of the quantized time of arrival operators as integral operators whose kernels are determined by the chosen ordering rule. The formulation lends itself to generalization which allows construction of time of arrival operators that cannot be obtained by direct quantization using particular ordering rules. Weyl, symmetric and Born–Jordan quantizations are specifically studied. The dynamics of the eigenfunctions of the different time of arrival operators are investigated. The eigenfunctions exhibit unitary arrival at the intended arrival point at their respective eigenvalues. [ABSTRACT FROM AUTHOR]

Dissipation in Quantum Mechanics took some time to become a robust field of investigation after the birth of the field. The main issue hindering developments in the field is that the Quantization process was always tightly connected to the Hamiltonian formulation of Classical Mechanics. In this paper we present a quantization process that does not depend upon the Hamiltonian formulation of Classical Mechanics (although still departs from Classical Mechanics) and thus overcome the problem of finding, from first principles, a completely general Schrödinger equation encompassing dissipation. This generalized process of quantization is shown to be nothing but an extension of a more restricted version that is shown to produce the Schrödinger equation for Hamiltonian systems from first principles (even for Hamiltonian velocity dependent potential). [ABSTRACT FROM AUTHOR]

In this article we prove that many Hamiltonian systems that cannot be separably quantized in the classical approach of Robertson and Eisenhart can be separably quantized if we extend the class of admissible quantizations through a suitable choice of Riemann space adapted to the Poisson geometry of the system. Actually, in this article we prove that for every quadratic in momenta Stäckel system (defined on 2 n dimensional Poisson manifold) for which Stäckel matrix consists of monomials in position coordinates there exist infinitely many quantizations–parametrized by n arbitrary functions–that turn this system into a quantum separable Stäckel system. [ABSTRACT FROM AUTHOR]

In this paper we quantize the N -dimensional classical Hamiltonian system H = | q | 2 ( η + | q | ) p 2 − k η + | q | , that can be regarded as a deformation of the Coulomb problem with coupling constant k , that it is smoothly recovered in the limit η → 0 . Moreover, the kinetic energy term in H is just the one corresponding to an N -dimensional Taub–NUT space, a fact that makes this system relevant from a geometric viewpoint. Since the Hamiltonian H is known to be maximally superintegrable, we propose a quantization prescription that preserves such superintegrability in the quantum mechanical setting. We show that, to this end, one must choose as the kinetic part of the Hamiltonian the conformal Laplacian of the underlying Riemannian manifold, which combines the usual Laplace–Beltrami operator on the Taub–NUT manifold and a multiple of its scalar curvature. As a consequence, we obtain a novel exactly solvable deformation of the quantum Coulomb problem, whose spectrum is computed in closed form for positive values of η and k , and showing that the well-known maximal degeneracy of the flat system is preserved in the deformed case. Several interesting algebraic and physical features of this new exactly solvable quantum system are analyzed, and the quantization problem for negative values of η and/or k is also sketched. [ABSTRACT FROM AUTHOR]

Abstract: The paper concerns integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also insist on the inherent probabilistic aspects of this classical–quantum map. The approach includes and generalizes coherent state quantization. Two applications based on group representation are carried out. The first one concerns the Weyl–Heisenberg group and the euclidean plane viewed as the corresponding phase space. We show that a world of quantizations exist, which yield the canonical commutation rule and the usual quantum spectrum of the harmonic oscillator. The second one concerns the affine group of the real line and gives rise to an interesting regularization of the dilation origin in the half-plane viewed as the corresponding phase space. [Copyright &y& Elsevier]

Abstract: The purpose of this work is to explore the existence and properties of reproducing kernel Hilbert subspaces of based on subsets of complex Hermite polynomials. The resulting coherent states (CS) form a family depending on a nonnegative parameter . We examine some interesting issues, mainly related to CS quantization, like the existence of the usual harmonic oscillator spectrum despite the absence of canonical commutation rules. The question of mathematical and physical equivalences between the -dependent quantizations is also considered. [Copyright &y& Elsevier]

Abstract: We revisit the quantized version of the harmonic oscillator obtained through a -dependent family of coherent states. For each , , these normalized states form an overcomplete set that resolves the unity with respect to an explicit measure. We restrict our study to the case in which is a quadratic unit Pisot number, since then the -deformed integers form Fibonacci-like sequences of integers. We then examine the main characteristics of the corresponding quantum oscillator: localization in the configuration and in the phase spaces, angle operator, probability distributions and related statistical features, time evolution and semi-classical phase space trajectories. [Copyright &y& Elsevier]