Schrödinger equation on a Dini's surface

We formulate the problem of a scalar particle constrained to move along a Dini's surface using da Costa's thin-layer quantization method. The Schrödinger equation for this problem can be separated into a straightforward azimuthal-like equation and a rather complicated angular equation. Thi...

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Bibliographic Details
Published in:Physics letters. A 2024-08, Vol.517, p.129674, Article 129674
Main Authors: M. Schmidt, Alexandre G., Pereira, Matheus E.
Format: Article
Language:English
Subjects:
Generalized Heun's equation
Schrödinger equation
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Summary:We formulate the problem of a scalar particle constrained to move along a Dini's surface using da Costa's thin-layer quantization method. The Schrödinger equation for this problem can be separated into a straightforward azimuthal-like equation and a rather complicated angular equation. This differential equation has six singularities, and we solved it exactly in terms of Heun local functions when the quantum number m vanishes. We also present applications of this problem considering hard walls, periodic boundary conditions, and interaction with an external, non-central potential. In this last case, the differential becomes Schäfke equation, which is slightly more complicated than Heun's equation. •Dynamics of a quantum scalar particle on the surface of a Dini helicoid.•The thin-layer method results in a geometric potential for the particle.•Eigenstates and eigenenergies using Heun special functions.•Physical application of Schäfke η-functions with an external potential.
ISSN:0375-9601
1873-2429
DOI:10.1016/j.physleta.2024.129674