6533b7defe1ef96bd1275f2e
RESEARCH PRODUCT
Lévy flights in an infinite potential well as a hypersingular Fredholm problem.
Vladimir A. StephanovichPiotr GarbaczewskiMariusz ŻAbaE. V. Kirichenkosubject
Quantum PhysicsMathematical analysisSpectrum (functional analysis)Orthogonal functionsFredholm integral equationEigenfunctionParticle in a boxMathematics::Spectral Theory01 natural sciences010305 fluids & plasmasSchrödinger equationMathematics - Spectral Theorysymbols.namesakeSpectrum of a matrix0103 physical sciencessymbols010306 general physicsEigenvalues and eigenvectorsCondensed Matter - Statistical MechanicsMathematical PhysicsMathematics - ProbabilityMathematicsdescription
We study L\'evy flights {{with arbitrary index $0< \mu \leq 2$}} inside a potential well of infinite depth. Such problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems. The major technical tool is a transformation of the eigenvalue problem for initial fractional Schr\"odinger equation into that for Fredholm integral equation with hypersingular kernel. The latter equation is then solved by means of expansion over the complete set of orthogonal functions in the domain $D$, reducing the problem to the spectrum of a matrix of infinite dimensions. The eigenvalues and eigenfunctions are then obtained numerically with some analytical results regarding the structure of the spectrum.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2016-04-09 | Physical review. E |