Reports on Mathematical Physics最新文献

In 2021, Fassari et al. introduced a remarkable summation involving the even-indexed eigenfunctions of the quantum harmonic oscillator, and introduced a proof, based on manipulations of multiple elliptic integrals, for an evaluation involving Catalan's constant for the aforementioned summation. This motivates the development of generalizations and extensions of this evaluation. In this article, we show, using the Wilf–Zeilberger method, how this series evaluation may be extended to infinite families of hypergeometric expressions that have free parameters and that are explicitly evaluable in terms of the digamma function. We also consider how an identity related to a nonterminating q-Pfaff–Saalschütz sum due to Krattenthaler and Srivastava may be applied to further generalize the main result due to Fassari et al.

In this paper we shall continue the studies of T^†-algebras done in [8, 9], and above all we investigate decompositions of the vector representation part of a T^†-algebra and apply the results to invariant positive sesquilinear forms on *-algebras.

This work is dedicated to quantization of the light-front Yukawa model in D = 1 + 3 dimensions with higher-order derivatives of a scalar field. The problem of computing Dirac brackets and the (anti-)commutator algebra of interacting fields in the presence of constraints is discussed. The Dirac method and the Ostrogradski formalism of the higher-order derivatives are exploited. The systematic method of obtaining the inverse of the functional Dirac–Bergmann matrix with interactions and higher-order derivatives is introduced in two variants. The discussion of applications and details of these two variants are conducted. The results of quantization in the form of the (anti-)commutator algebra are presented and analyzed. There is a particular emphasis on the structure of interactions for the light-front Yukawa model with higher-order derivatives.

In the standard approach to Finsler geometry the metric is defined as a vertical Hessian and the Chern or Cartan connections appear as just two among many possible natural linear connections on the pullback tangent bundle. Here it is shown that the Hessian nature of the metric, the nonlinear connection and the Chern or Cartan connections can be derived from a few compatibility axioms between metric and Finsler connection. This result provides a metrical formulation of Finsler geometry which is well adapted to field theory, and which has proved useful in Einstein–Cartan-like approaches to Finsler gravity.

We consider a problem occurring in a magnetostatic levitation. The problem leads to a linear PDE in a strip. In engineering literature a particular solution is obtained. Such a solution enables one to compute lift and drag forces of the levitating object. It is in agreement with the experiment. We show that such a solution is unique in a class of bounded regular functions. Moreover, as a byproduct, we obtain nonstandard uniqueness results in two linear PDEs in unbounded domains. One of them is an eigenvalue problem for the Laplacian in the strip in the nonstandard class of functions.

According to the ‘folk knowledge', the Hartree–Fock (H-F) approximation applied to the Hubbard model becomes exact in the limit of small coupling U (the smaller |U|, the better is the H-F approximation). In [1] Bach and Poelchau have substantiated a certain version of this assertion by providing a rigorous estimate of the difference between the true ground-state energy of the simplest version of the Hubbard model and the H-F approximation to this quantity. In this paper we extend their result in two directions: (i) we show how to apply it to the system the hopping matrix of which has period greater than 1 (the case without strict translational invariance) — this allows us to consider systems the dispersion function of which is not (as assumed in [1]) a Morse function; (ii) we point out that the same technique allows to establish analogous estimates for a class of multiband Hubbard models.

Dynamical symmetry algebra for a semiconfined harmonic oscillator model with a position-dependent effective mass is constructed. Selecting the starting point as a well-known factorization method of the Hamiltonian under consideration, we have found three basis elements of this algebra. The algebra defined through those basis elements is an su (1,1) Heisenberg–Lie algebra. Different special cases and the limit relations from the basis elements to the Heisenberg–Weyl algebra of the nonrelativistic quantum harmonic oscillator are discussed, too.

According to the basic idea of category theory, any Einstein algebra, essentially an algebraic formulation of general relativity, can be considered from the point of view of any object of the category of C^∞-algebras; such an object is then called a stage. If we contemplate a given Einstein algebra from the point of view of the stage, which we choose to be an “algebra with infinitesimals” (Weil algebra), then we can suppose it penetrates a submicroscopic level, on which quantum gravity might function. We apply Vinogradov's notion of geometricity (adapted to this situation), and show that the corresponding algebra is geometric, but then the infinitesimal level is unobservable from the macro-level. However, the situation can change if a given algebra is noncommutative. An analogous situation occurs when as stages, instead of Weil algebras, we take many other C^∞-algebras, for example those that describe spaces in which with ordinary points coexist “parametrised points”, for example closed curves (loops). We also discuss some other consequences of putting Einstein algebras into the conceptual environment of category theory.

It has been shown that a positive semi-definite Hamiltonian H, that has a tridiagonal matrix representation in a given basis$\left\{|{\varphi }_n〉\right\}$, can be represented in the form H = A^†A, where A is a forward shift operator satisfying$A|{\varphi }_n〉=c_n|{\varphi }_n〉+d_n|{\varphi }_n-1〉$ playing the role of an annihilation operator. Here, the coherent states |z) are defined as eigenstates of A. We also exhibit a complete set of coherent states {|c_n)}, labeled by the discrete points c_n, which we may also use as a basis. We find a solution of the coherent state |z), indexed by the continuous parameter z as a series expansion in terms of the {|c_n)}. We further show how to compute the time development of the coherent state |z) and we illustrate this with examples. As a major result, we find the explicit closed form solution |z) for the Morse oscillator.