The time-dependent Schrödinger equation in non-integer dimensions for constrained quantum motion. (English) Zbl 1448.81311
Summary: We propose a theoretical model, based on a generalized Schrödinger 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 \(X \times Y\), 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.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81Q37 | Quantum dots, waveguides, ratchets, etc. |
Keywords:
Schrödinger equation; non-integer dimension; Green’s function; Bessel functions; fox \(H\)-functionReferences:
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