Volume 192, Nº 1 (2017)

We investigate Hom–Lie H-pseudobialgebras. We present some examples and a theorem that allows constructing these new algebraic structures. We consider coboundary Hom–Lie H-pseudobialgebras and the corresponding classical Hom–Yang–Baxter equations.

We propose a procedure for multiplying solutions of linear and nonlinear one-dimensional wave equations, where the speed of sound can be an arbitrary function of one variable. We obtain exact solutions. We show that the functional series comprising these solutions can be used to solve initial boundary value problems. For this, we introduce a special scalar product.

We construct Hilbert spaces of particle states such that all neutrinos and also charged leptons and up and down quarks are united into multiplets and their components can be treated as different quantum states of a single particle. The phenomenon of neutrino oscillations arises in a theory based on the Lagrangian of the fermionic sector of the Standard Model modified according to the proposed approach.

It was recently shown that there are some difficulties in the solution method proposed by Laskin for obtaining the eigenvalues and eigenfunctions of the one-dimensional time-independent fractional Schrödinger equation with an infinite potential well encountered in quantum mechanics. In fact, this problem is still open. We propose a new fractional approach that allows overcoming the limitations of some previously introduced strategies. In deriving the solution, we use a method based on the eigenfunction of the Weyl fractional derivative. We obtain a solution suitable for computations in a closed form in terms of Mittag–Leffler functions and fractional trigonometric functions. It is a simple extension of the results previously obtained by Laskin et al.

We consider scale transformations (q, p) → (λq, λp) in phase space. They induce transformations of the Husimi functions H(q, p) defined in this space. We consider the Husimi functions for states that are arbitrary superpositions of n-particle states of a harmonic oscillator. We develop a method that allows finding so-called stretched states to which these superpositions transform under such a scale transformation. We study the properties of the stretched states and calculate their density matrices in explicit form. We establish that the density matrix structure can be described using negative binomial distributions. We find expressions for the energy and entropy of stretched states and calculate the means of the number-ofstates operator. We give the form of the Heisenberg and Robertson–Schrödinger uncertainty relations for stretched states.