A class of models is considered for a quantum particle constrained on degenerate Riemannian manifolds known as Grushin cylinders, and moving freely subject only to the underlying geometry: The corresponding spectral and scattering analysis is developed in detail in viewof the phenomenon of transmission across the singularity that separates the two half-cylinders. Whereas the classical counterpart always consists of a particle falling in finite time along the geodesics onto the metric's singularity locus, the quantum models may display geometric confinement, or on the opposite partial transmission and reflection. All the local realisations of the free (Laplace-Beltrami) quantum Hamiltonian are examined as non-equivalent protocols of transmission/reflection and the structure of their spectrum is characterised, including when applicable their ground state and positivity. Besides, the stationary scattering analysis is developed and transmission and reflection coefficients are calculated. This allows to comprehend the distinguished status of the so-called 'bridging' transmission protocol previously identified in the literature, which we recover and study within our systematic analysis.

Quantum particle across Grushin singularity / M. Gallone, A. Michelangeli. - In: JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL. - ISSN 1751-8113. - 54:21(2021 May 28), pp. 215201 abeb81.1-215201 abeb81.43. [10.1088/1751-8121/abeb81]

Quantum particle across Grushin singularity

M. Gallone
Primo
;
2021-05-28

Abstract

A class of models is considered for a quantum particle constrained on degenerate Riemannian manifolds known as Grushin cylinders, and moving freely subject only to the underlying geometry: The corresponding spectral and scattering analysis is developed in detail in viewof the phenomenon of transmission across the singularity that separates the two half-cylinders. Whereas the classical counterpart always consists of a particle falling in finite time along the geodesics onto the metric's singularity locus, the quantum models may display geometric confinement, or on the opposite partial transmission and reflection. All the local realisations of the free (Laplace-Beltrami) quantum Hamiltonian are examined as non-equivalent protocols of transmission/reflection and the structure of their spectrum is characterised, including when applicable their ground state and positivity. Besides, the stationary scattering analysis is developed and transmission and reflection coefficients are calculated. This allows to comprehend the distinguished status of the so-called 'bridging' transmission protocol previously identified in the literature, which we recover and study within our systematic analysis.
Constant-fibre direct sum; Differential self-adjoint operators; Geometric quantum confinement; Grushin manifold; Krein Vishik Birman; Laplace-Beltrami operator; Quantum transmission across singularity
Settore MAT/07 - Fisica Matematica
Settore FIS/02 - Fisica Teorica, Modelli e Metodi Matematici
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/863205
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