We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wick's theorem. The methodology can be straightforwardly generalized from the simple example given herein to a wide class of operators.

We solve the boson normal ordering problem for (q(a † )a + v(a † )) n with arbitrary functions q and v and integer n, where a and a † are boson annihilation and creation operators, satisfying [a, a † ] = 1. This leads to exponential operators generalizing the shift operator and we show that their action can be expressed in terms of substitutions. Our solution is naturally related through the coherent state representation to the exponential generating functions of Sheffer-type polynomials. This in turn opens a vast arena of combinatorial methodology which is applied to boson normal ordering and illustrated by a few examples.

The use of operator methods of algebraic nature is shown to be a very powerful tool to deal with different forms of relativistic wave equations. The methods provide either exact or approximate solutions for various forms of differential equations, such as relativistic Schrödinger, Klein-Gordon and Dirac. We discuss the free particle hypotheses and those relevant to particles subject to non-trivial potentials. In the latter case we will show how the proposed method leads to easily implementable numerical algorithms.

1 October 27, 2018 11:23 WSPC/Trim Size: 9in x 6in for Proceedings 1param2 2In this communication, we consider the normal ordering of operators of the typewhere a (resp. a + )is a boson annihilation (resp. creation) operator; these satisfy [a, a + ] ≡ aa + − a + a = 1, and for the purposes of this note may be thought of as a ≡ d/dx and a + ≡ x. We discuss the integration of the one-parameter groups e λΩ and their combinatorial by-products. In particular we show how these groups can be realized as groups of substitutions with prefunctions.

We study the higher-order heat-type equation with first time and M -th spatial partial derivatives, M = 2, 3, . . .. We demonstrate that its exact solutions for M even can be constructed with the help of signed Lévy stable functions. For M odd the same role is played by a special generalization of Airy Ai function that we introduce and study. This permits one to generate the exact and explicit heat kernels pertaining to these equations. We examine analytically and graphically the spacial and temporary evolution of particular solutions for simple initial conditions.