Distributed Order Derivatives and Relaxation Patterns

We consider equations of the form $(D_{(\rho)}u)(t)=-\lambda u(t)$, $t>0$, where $\lambda >0$, $D_{(\rho)}$ is a distributed order derivative, that is the Caputo-Dzhrbashyan fractional derivative of order $\alpha$, integrated in $\alpha\in (0,1)$ with respect to a positive measure $\rho$. Such equations are used for modeling anomalous, non-exponential relaxation processes. In this work we study asymptotic behavior of solutions of the above equation, depending on properties of the measure $\rho$

Levy stable distributions via associated integral transform

We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Levy stable probability distributions g_{\alpha}(x), 0 \leq x < \infty, 0 < \alpha < 1. We demonstrate that the knowledge of one such a distribution g_{\alpha}(x) suffices to obtain exactly g_{\alpha^{p}}(x), p=2, 3,... Similarly, from known g_{\alpha}(x) and g_{\beta}(x), 0 < \alpha, \beta < 1, we obtain g_{\alpha \beta}(x). The method is based on the construction of the integral operator, called Levy transform, which implements the above operations. For \alpha rational, \alpha = l/k with l < k, we reproduce in this manner many of the recently obtained exact results for g_{l/k}(x). This approach can be also recast as an application of the Efros theorem for generalized Laplace convolutions. It relies solely on efficient definite integration.Comment: 12 pages, typos removed, references adde

Self-adjoint extensions and spectral analysis in Calogero problem

In this paper, we present a mathematically rigorous quantum-mechanical treatment of a one-dimensional motion of a particle in the Calogero potential $\alpha x^{-2}$. Although the problem is quite old and well-studied, we believe that our consideration, based on a uniform approach to constructing a correct quantum-mechanical description for systems with singular potentials and/or boundaries, proposed in our previous works, adds some new points to its solution. To demonstrate that a consideration of the Calogero problem requires mathematical accuracy, we discuss some "paradoxes" inherent in the "naive" quantum-mechanical treatment. We study all possible self-adjoint operators (self-adjoint Hamiltonians) associated with a formal differential expression for the Calogero Hamiltonian. In addition, we discuss a spontaneous scale-symmetry breaking associated with self-adjoint extensions. A complete spectral analysis of all self-adjoint Hamiltonians is presented.Comment: 39 page

Multi-dimensional trio coherent states

We introduce a novel class of higher-order, three-mode states called K-dimensional trio coherent states. We study their mathematical properties and prove that they form a complete set in a truncated Fock space. We also study their physical content by explicitly showing that they exhibit nonclassical features such as oscillatory number distribution, sub-poissonian statistics, Cauchy-Schwarz inequality violation and phase-space quantum interferences. Finally, we propose an experimental scheme to realize the state with K=2 in the quantized vibronic motion of a trapped ion.Comment: 17 pages, 12 figures, accepted for publication in J. Phys. A: Math. Ge