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  - Fischer, Florian
A1  - Keller, Matthias
T1  - Riesz decompositions for Schrödinger operators on graphs
JF  - Journal of mathematical analysis and applications
N2  - We study superharmonic functions for Schrodinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem.
KW  - Potential theory
KW  - Green's function
KW  - Schrödinger operator
KW  - Weighted
KW  - graph
KW  - Subcritical
KW  - Greatest harmonic minorant
Y1  - 2021
U6  - https://doi.org/10.1016/j.jmaa.2020.124674
SN  - 0022-247X
SN  - 1096-0813
VL  - 495
IS  - 1
PB  - Elsevier
CY  - Amsterdam
ER  -