Ricardo Cordero-Soto

Propagator of a charged particle with a spin in uniform magnetic and perpendicular electric fields

We construct an explicit solution of the Cauchy initial value problem for the time-dependent Schrödinger equation for a charged particle with a spin moving in a uniform magnetic field and a perpendicular electric field varying with time. The corresponding Green function (propagator) is given in terms of elementary functions and certain integrals of the fields with a characteristic function, which should be found as an analytic or numerical solution of the equation of motion for the classical oscillator with a time-dependent frequency. We discuss a particular solution of a related nonlinear Schrödinger equation and some special and limiting cases are outlined.

Quantum integrals of motion for variable quadratic Hamiltonians

Abstract We construct integrals of motion for several models of the quantum damped oscillators in a framework of a general approach to the time-dependent Schrodinger equation with variable quadratic Hamiltonians. An extension of the Lewis–Riesenfeld dynamical invariant is given. The time-evolution of the expectation values of the energy-related positive operators is determined for the oscillators under consideration. A proof of uniqueness of the corresponding Cauchy initial value problem is discussed as an application.

Solution of the Cauchy problem for a time-dependent Schrödinger equation

We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrodinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU(1,1) symmetry of the harmonic oscillator wave functions, Bargmann’s functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner–Pollaczek polynomials, a Hankel-type integral transform, and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as by-products.

The degenerate parametric oscillator and Ince's equation

We construct Greens function for the quantum degenerate parametric oscillator in the coordinate representation in terms of standard solutions of Inces equation in a framework of a general approach to variable quadratic Hamiltonians. Exact time-dependent wavefunctions and their connections with dynamical invariants and SU(1, 1) group are also discussed. An extension to the degenerate parametric oscillator with time-dependent amplitude and phase is also mentioned.

Intrinsic spectral geometry of the Kerr-Newman event horizon

We uniquely and explicitly reconstruct the instantaneous intrinsic metric of the Kerr-Newman event horizon from the spectrum of its Laplacian. In the process we find that the angular momentum parameter, radius, area; and in the uncharged case, mass, can be written in terms of these eigenvalues. In the uncharged case this immediately leads to the unique and explicit determination of the Kerr metric in terms of the spectrum of the event horizon. Robinson’s “no hair” theorem now yields the corollary: One can “hear the shape” of noncharged stationary axially symmetric black hole space-times by listening to the vibrational frequencies of its event horizon only.