TY  - JOUR
A1  - Petreska, Irina
A1  - de Castro, Antonio S. M.
A1  - Sandev, Trifce
A1  - Lenzi, Ervin K.
T1  - The time-dependent Schrödinger equation in non-integer dimensions for constrained quantum motion
JF  - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics
N2  - We propose a theoretical model, based on a generalized Schroedinger equation, to study the behavior of a constrained quantum system in non-integer, lower than two-dimensional space. The non-integer dimensional space is formed as a product space X x Y, comprising x-coordinate with a Hausdorff measure of dimension alpha(1) = D -1 (1 < D < 2) and y-coordinate with the Lebesgue measure of dimension of length (alpha(2) = 1). Geometric constraints are set at y = 0. Two different approaches to find the Green's function are employed, both giving the same form in terms of the Fox H-function. For D = 2, the solution for two-dimensional quantum motion on a comb is recovered. (C) 2020 Elsevier B.V. All rights reserved.
KW  - Schrödinger equation
KW  - non-integer dimension
KW  - Green's function
KW  - Bessel functions
KW  - Fox H-function
Y1  - 2020
U6  - https://doi.org/10.1016/j.physleta.2020.126866
SN  - 0375-9601
SN  - 1873-2429
VL  - 384
IS  - 34
PB  - Elsevier
CY  - Amsterdam
ER  - 
TY  - JOUR
A1  - Petreska, Irina
A1  - Pejov, Ljupco
A1  - Sandev, Trifce
A1  - Kocarev, Ljupčo
A1  - Metzler, Ralf
T1  - Tuning of the dielectric relaxation and complex susceptibility in a system of polar molecules: a generalised model based on rotational diffusion with resetting
JF  - Fractal and fractional
N2  - The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately. 
The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation. 
In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process. 
The autocorrelation function and complex susceptibility are analysed in detail. 
We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate. 
The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules.
KW  - rotational diffusion
KW  - memory kernel
KW  - Fokker-Planck equation
KW  - non-exponential relaxation
KW  - autocorrelation function
KW  - complex
KW  - susceptibility
Y1  - 2022
U6  - https://doi.org/10.3390/fractalfract6020088
SN  - 2504-3110
VL  - 6
IS  - 2
PB  - MDPI AG, Fractal Fract Editorial Office
CY  - Basel
ER  -