Two Quantum Mechanical Problems on the Non-Commutative Plane

VEGA FEDERICO GASPAR; PISANI PABLO ; FALOMIR HORACIO

La Plata

Conferencia; HK2010; 2011

Humboldt Kolleg

We study the non-relativistic quantum mechanical problem of a particle in the non-commutative plane in the presence of two central potentials. The first one is given by $V(R)=V_0\Theta{(R^2-A^2)}$, where $\Theta$ is the Heaviside function, and the second one is $V(R)=V_0\delta{(R^2-A^2)}$, where $\delta$ is Dirac´s delta; $V_0$ is a constant and $R^2$ is the distance to the origin in the non-commutative plane. Using the spectral decomposition of the differential operator $R^2$ one obtains a recurrence relation that can be explicitly solved. (In this way, an implicit equation for the spectrum is obtained). This recursion determine the spectrum of the Hamiltonian in terms of trascendental functions. We also analyse the particular limit in which $V_0$ tends to infinite -for both cases- and the consequent simplification of the problem. This limit could be considered as an extension to the non-commutative plane of the problem of a particle in a disc subject to Dirichlet boundary conditions, an approach to the a non-commutative disc. Finally, we discuss the pole structure of the zeta function associated to these Hamiltonians.