An algebraic treatment of shape-invariant potentials is discussed. By introducing an operator which reparametrizes wave functions, the shape-invariance condition can be related to a generalized Heisenberg-Weyl algebra. It is shown that this makes it possible to define a coherent state associated with the shape-invariant potentials.

We investigate a class of algebras that provides multiparameter versions of both quantum symplectic space and quantum Euclidean 2n-space. These algebras encompass the graded quantized Weyl algebras, the quantized Heisenberg space, and a class of algebras introduced by Oh. We describe the structure of the prime and primitive ideals of these algebras. Other structural results include normal separ...

The aim of this article is to construct à la Perelomov and à la Barut–Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This generalized Weyl-Heisenberg algebra, noted A{κ}, depends on r real parameters and is an extension of the Aκ one-parameter algebra (Daoud M and Kibler M R 2010 J. Phys. A: Math. Theor. 43 115303) which covers the cases of the su(1, 1) algebra (for κ > 0)...

Deep convolutional neural networks (CNNs) used in practice employ potentially hundreds of layers and 10,000s of nodes. Such network sizes entail significant computational complexity due to the large number of convolutions that need to be carried out; in addition, a large number of parameters needs to be learned and stored. Very deep and wide CNNs may therefore not be well suited to applications...

We propose a scheme to implement the quantum walk for SU(1,1) in phase space, which generalizes those associated with Heisenberg-Weyl group. The movement of walker described by coherent states can be visualized on hyperboloid or Poincar\'{e} disk. In both one-mode and two-mode realizations, we introduce corresponding coin-flip conditional-shift operators group, whose relations group are analyze...

We discuss stable, overcomplete, non-orthogonal phase-space expansions with respect to generalized coherent states. The key concepts are Weyl-Heisenberg frames and quantitative measures of phase-space localization. On an abstract level, we introduce and investigate the localization of general frames.

We prove Hausdorff-Young inequality for the Fourier transform connected with Riemann-Liouville operator. We use this inequality to establish the uncertainty principle in terms of entropy. Next, we show that we can derive the Heisenberg-Pauli-Weyl inequality for the precedent Fourier transform.

In this paper a new quasi-triangular Hopf algebra as the quantum double of the Heisenberg-Weyl algebra is presented.Its universal R-matrix is built and the corresponding representation theory are studied with the explict construction for the representations of this quantum double. 1Permanet address:Physics Dpartment,Northeast Normal University, Changchun 130024,P.R.China 2Permanet address:Theor...

The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair of Hopf algebras. Moreover, given a dual pair of Hopf algebras, one can construct an algebra called the Heisenberg double, which is a generalization of the c...

In this paper we extend the Balian–Low theorem, which is a version of the uncertainty principle for Gabor (Weyl–Heisenberg) systems, to functions of several variables. In particular, we first prove the Balian–Low theorem for arbitrary quadratic forms. Then we generalize further and prove the Balian– Low theorem for differential operators associated with a symplectic basis for the symplectic for...