Quantum mechanics and umbral calculus

@inproceedings{LopezSendino2008QuantumMA,
  title={Quantum mechanics and umbral calculus},
  author={E. Lopez-Sendino and J. Negro and Mariano A. del Olmo and E. Salgado},
  year={2008}
}
In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schrodinger equation substituting the continuous derivatives by discrete ones and the space-time continuous variables by well determined operators that verify some Umbral Calculus conditions. In this way we assure that some properties of integrability and symmetries of the continuous equation are preserved and also the solutions of the… Expand
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References

SHOWING 1-10 OF 47 REFERENCES
Umbral calculus, difference equations and the discrete Schrödinger equation
In this paper, we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. TheExpand
Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice
`Umbral calculus' deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models andExpand
Discrete q-derivatives and symmetries of q-difference equations
In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many properties considered for shift invariantExpand
Continuous symmetries of the lattice potential KdV equation
In this paper we present a set of results on the integration and on the symmetries of the lattice potential Korteweg–de Vries (lpKdV) equation. Using its associated spectral problem we construct theExpand
Multiscale expansion on the lattice and integrability of partial difference equations
We conjecture an integrability and linearizability test for dispersive Z^2-lattice equations by using a discrete multiscale analysis. The lowest order secularity conditions from the multiscaleExpand
The Umbral Calculus
In this chapter, we give a brief introduction to a relatively new subject, called the umbral calculus. This is an algebraic theory used to study certain types of polynomial functions that play anExpand
Discrete derivatives and symmetries of difference equations
We show with an example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable for finding the symmetries of discrete equations. InExpand
Lie symmetries of multidimensional difference equations
A method is presented for calculating the Lie point symmetries of a scalar difference equation on a two-dimensional lattice. The symmetry transformations act on the equations and on the lattice. TheyExpand
Continuous symmetries of difference equations
Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program:Expand
Lie symmetries of difference equations
The discrete heat equation is worked out to illustrate the search of symmetries of difference equations. Special attention it is paid to the Lie structure of these symmetries, as well as to theirExpand
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