Reports on Mathematical Physics最新文献

The present paper focuses on the study of fundamental group of Minkowski space with the fine topology which has been found to contain uncountably many subgroups isomorphic to the additive group of integers.

We treat the quantum dynamics of a harmonic oscillator as well as its inverted counterpart in the Schrödinger picture. Generally in the most papers in the literature, the inverted harmonic oscillator is formally obtained from the harmonic oscillator by the replacement of ωby iω, this leads to unbounded eigenvectors. This explicitly demonstrates that there are some unclear points involved in redefining the variables in the harmonic oscillator inversion. To remedy this situation, we introduce a scaling operator (Dyson transformation) by connecting the inverted harmonic oscillator to an anti-$PT$-symmetric harmonic oscillator, and we obtain the standard quasi-Hermiticity relation which would ensure the time invariance of the eigenfunction's norm. We give a complete description for the eigenproblem. We show that the wave functions for this system are normalized in the sense of the pseudo-scalar product. A Gaussian wave packet of the inverted oscillator is investigated by using the ladder operators method. This wave packet is found to be associated with the generalized coherent state that can be crucially utilized for investigating the mean values of the space and momentum operators. We find that these mean values reproduce the classical motion.

In this manuscript, we present a generalized deformed sum inspired by the nonadditive property of entropies such as those investigated by Tsallis and Shannon in the context of information theory. From this deformed sum, we define a generalization of the Vitali set and prove its nonmeasurability. Moreover, the standard sum is recovered as the deforming parameter tends to zero, and Vitali's theorem is retrieved. In particular, the present work is a generalization of the results derived in [2] for arbitrary functions satisfying general conditions, and not only nonextensive statistics.

A shape invariant nonseparable and nondiagonalizable two-dimensional model with anharmonic complex interaction, first studied by Cannata, Ioffe, and Nishnianidze, is re-examined with the purpose of providing an algebraic construction of the associated functions to the excited-state wavefunctions, needed to complete the basis. The two operators A⁺ and A^-, coming from the shape invariant supersymmetric approach, where A⁺ acts as a raising operator while A^-annihilates all wavefunctions, are completed by introducing a novel pair of operators B⁺ and B^-, where B^- acts as the missing lowering operator. It is then shown that building the associated functions as polynomials in A⁺ and B⁺ acting on the ground state provides a much more efficient approach than that used in the original paper. In particular, we have been able to extend the previous results obtained for the first two excited states of the quartic anharmonic oscillator either by considering the next three excited states or by adding a cubic or a sextic term to the Hamiltonian.

In this paper we present the example the details of the transition of the Hubbard model to the Heisenberg model in the limit of on-site repulsion constant u → ∞. We explore the models with respect to the nearest and the next-nearest-neighbour hopping. We construct the next-nearest-neighbour hopping free subspaces for the considered example and find the procedure applicable to any number and any configuration of electrons in the chain. We found that some eigenvalues permute themselves for a specific value of the ratio of on-site repulsion constant to hopping constant and the effect is more visible the greater the next-nearest-neighbour hopping is. A similar situation occurs for eigenvectors. We also confirm SU(2)× SU(2) symmetry breaking when the next-nearest-neighbour hopping are considered.

In this paper, we consider the Cauchy problem for the relativistic Boltzmann equation with near vacuum initial data where the distribution function depends on time, position and momenta. The collision kernel considered here corresponds to short range interactions and the background space-time is fixed and is of Bianchi type I. The existence of a unique global (in time) mild solution is obtained in a suitable weighted space.

We give a rigorous definition of moments of an unbounded observable with respect to a quantum state in terms of Yosida's approximations of unbounded generators of contractions semigroups. We use this notion to characterize Gaussian states in terms of the moments of the field operator, which we call Weyl moments. As a by-product, rigorous formulae for the mean value vector and the covariance matrix of a Gaussian state are obtained.

Motivated by our recent work published in [23], we achieve, in this paper, a matrix formulation of the density operator to construct a two-component vector coherent state representation for a supersymmetric harmonic oscillator. We investigate and discuss the main relevant statistical properties. We use the completeness relation to perform the thermodynamic analysis in the diagonal P-representation of the density operator.

We describe bivector fields and Poisson structures on local Calabi-Yau threefolds which are total spaces of vector bundles on a contractible rational curve. In particular, we calculate all possible holomorphic Poisson structures on the conifold.