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Papers by Victor Villalba
AIP Conference Proceedings, 2006
We apply the algebraic method of separation of variables in order to reduce the Dirac equation to... more We apply the algebraic method of separation of variables in order to reduce the Dirac equation to a set of coupled first-order ordinary differential equations. We obtain the sufficient conditions for partial or complete separability corresponding to homogeneous cosmological backgrounds.
Physical review, Mar 27, 2007
The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological... more The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological perturbations in the power-law inflationary model. The phase-integral formulas for the scalar and tensor power spectra are explicitly obtained up to ninth-order of the phase-integral approximation. We show that, the phase-integral approximation exactly reproduces the shape of the power spectra for scalar and tensor perturbations as well as the spectral indices. We compare the accuracy of the phase-integral approximation with the results for the power spectrum obtained with the slow-roll and uniform approximation methods.
Revista Mexicana De Fisica, 1996
In the present article we revisit the problem of a relativistic Dirac electron. Using a second or... more In the present article we revisit the problem of a relativistic Dirac electron. Using a second order formalism which reduces the problem of finding the energy spectrum to solving the Whittaker equation, we show that the only physical solution is obtained by truncating the hypergeometric series. Therefore the energy spectrum does not depend on any free parameters for 119 < Z < 137. RESUMEN. En el presente articulo se estudia el problema de un electron relativista de Dirac. Haciendo uso de un formalismo de segundo orden que reduce el problema de encontrar el espectro de energia a resolver la ecuacion de vVhittaker, se muestra que la unica solucion fisicamente aceptable se obtiene truncando la serie hipergeometrica. Por consiguiente, el espectro de energia no depende de ningun parametro libre para 119 < Z < 137.
Physical Review D, 2009
The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological... more The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological perturbations in the quadratic chaotic inflationary model. The phase-integral formulas for the scalar and tensor power spectra are explicitly obtained up to fifth order of the phase-integral approximation. We show that, the phase integral gives a very good approximation for the shape of the power spectra associated with scalar and tensor perturbations as well as the spectral indices. We find that the accuracy of the phase-integral approximation compares favorably with the numerical results and those obtained using the slow-roll and uniform approximation methods.
Physical Review B, 2007
We analyze the role of resonances in two-fermion entanglement production for a quasi onedimension... more We analyze the role of resonances in two-fermion entanglement production for a quasi onedimensional two channel scattering problem. We solve exactly for the problem of a two-fermion antisymmetric product state scattering off a double delta well potential. It is shown that the twoparticle concurrence of the post-selected state has an oscillatory behavior where the concurrence vanishes at the values of momenta for virtual bound states in the double well. These concurrence zeros are interpreted in terms of the uncertainty in the knowledge of the state of the one particle subspace reduced one particle density matrix. Our results suggest manipulation of fermion entanglement production through the resonance structure of quantum dots.
Journal of Physics: Condensed Matter, 2001
We study the energy spectrum of the two-electron spherical parabolic quantum dot using the exact ... more We study the energy spectrum of the two-electron spherical parabolic quantum dot using the exact Schrödinger, the Hartree-Fock, and the Kohn-Sham equations. The results obtained by applying the shifted-1/N method are compared with those obtained by using an accurate numerical technique, showing that the relative error is reasonably small, although the first method consistently underestimates the correct values. The approximate groundstate Hartree-Fock and local-density Kohn-Sham energies, estimated using the shifted-1/N method, are compared with accurate numerical self-consistent solutions. We make some perturbative analyses of the exact energy in terms of the confinement strength, and we propose some interpolation formulae. Similar analysis is made for both mean-field approximations and interpolation formulae are also proposed for these exchange-only ground-state cases.
Physica Scripta, 2007
We solve the two-component Dirac equation in the presence of a spatially one dimensional symmetri... more We solve the two-component Dirac equation in the presence of a spatially one dimensional symmetric attractive cusp potential. The components of the spinor solution are expressed in terms of Whittaker functions. We compute the bound states solutions and show that, as the potential amplitude increases, the lowest energy state sinks into the Dirac sea becoming a resonance. We characterize and compute the lifetime of the resonant state with the help of the phase shift and the Breit-Wigner relation. We discuss the limit when the cusp potential reduces to a delta point interaction.
We obtain exact solutions of the Klein-Gordon and Pauli Schroedinger equations for a two-dimensio... more We obtain exact solutions of the Klein-Gordon and Pauli Schroedinger equations for a two-dimensional hydrogen-like atom in the presence of a constant magnetic field. Analytic solutions for the energy spectrum are obtained for particular values of the magnetic field strength. The results are compared to those obtained in the non-relativistic and spinless case. We obtain that the relativistic spectrum does not present s states.
The phase-integral approximation devised by Fr\"oman and Fr\"oman, is used for computin... more The phase-integral approximation devised by Fr\"oman and Fr\"oman, is used for computing cosmological perturbations in the quartic chaotic inflationary model. The phase-integral formulas for the scalar power spectrum are explicitly obtained up to fifth order of the phase-integral approximation. As in previous reports [1-3], we point out that the accuracy of the phase-integral approximation compares favorably with the numerical results and those obtained using the slow-roll and uniform approximation methods.
We solve the Klein-Gordon and Dirac equations in an open cosmological universe with a partially h... more We solve the Klein-Gordon and Dirac equations in an open cosmological universe with a partially horn topology in the presence of a time dependent magnetic field. Since the exact solution cannot be obtained explicitly for arbitrary time-dependence of the field, we discuss the asymptotic behavior of the solutions with the help of the relativistic Hamilton-Jacobi equation.
We solve the Klein-Gordon equation in the presence of a spatially one-dimensional cusp potential.... more We solve the Klein-Gordon equation in the presence of a spatially one-dimensional cusp potential. The scattering solutions are obtained in terms of Whittaker functions and the condition for the existence of transmission resonances is derived. We show the dependence of the zero-reflection condition on the shape of the potential. In the low momentum limit, transmission resonances are associated with half-bound states. We express the condition for transmission resonances in terms of the phase shifts.
We show that the energy spectrum of the one-dimensional Dirac equation in the presence of a spati... more We show that the energy spectrum of the one-dimensional Dirac equation in the presence of a spatial confining point interaction exhibits a resonant behavior when one includes a weak electric field. After solving the Dirac equation in terms of parabolic cylinder functions and showing explicitly how the resonant behavior depends on the sign and strength of the electric field, we derive an approximate expression for the value of the resonance energy in terms of the electric field and delta interaction strength.
We solve the two-component Dirac equation in the presence of a spatially one-dimensional symmetri... more We solve the two-component Dirac equation in the presence of a spatially one-dimensional symmetric cusp potential. We compute the scattering and bound state solutions and we derive the conditions for transmission resonances as well as for supercriticality.
Physics Letters A, 1989
An exact solution to the Dirac equation in the presence of an exact gravitational plane wave is p... more An exact solution to the Dirac equation in the presence of an exact gravitational plane wave is presented. by passing, an exact solution which represents neutrinos is obtained.
Physical Review A, 2003
We solve the two-component Dirac equation in the presence of a spatially one-dimensional symmetri... more We solve the two-component Dirac equation in the presence of a spatially one-dimensional symmetric cusp potential. We compute the scattering and bound state solutions and we derive the conditions for transmission resonances as well as for supercriticality.
Journal of Mathematical Physics, 1990
In this paper the separation of variables is presented in the Dirac equation in open, flat, and c... more In this paper the separation of variables is presented in the Dirac equation in open, flat, and closed expanding cosmological Robertson-Walker universes. The equations governing the radial variable and the evolution of the time-dependent factor are obtained. An exact solution to the Weyl equation is derived for an arbitrary expansion factor of the Robertson-Walker metrics. An exact solution to Dirac equation in a universe filled with radiation is also presented.
Can J Phys, 1992
In this article we obtain, by separation of variables, an exact solution to the Klein–Gordon and ... more In this article we obtain, by separation of variables, an exact solution to the Klein–Gordon and Dirac equations in a cosmological, spatially-closed, Robertson–Walker space-time with a positive cosmological constant Λ. The model is associated with a universe filled with radiation. We analyze the phenomenon of particle creation for different values of the dimensionless coupling constant ξ.
J Phys Condens Matter, 1996
In the present article we study the energy levels of a 2D hydrogenic atom when a constant magneti... more In the present article we study the energy levels of a 2D hydrogenic atom when a constant magnetic field is applied. We compute the energy spectrum with the help of a generalization of the mesh point technique recently proposed by Schwartz. We also estimate, via a variational method, the upper energy bound for small and large values of the external
AIP Conference Proceedings, 2006
We apply the algebraic method of separation of variables in order to reduce the Dirac equation to... more We apply the algebraic method of separation of variables in order to reduce the Dirac equation to a set of coupled first-order ordinary differential equations. We obtain the sufficient conditions for partial or complete separability corresponding to homogeneous cosmological backgrounds.
Physical review, Mar 27, 2007
The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological... more The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological perturbations in the power-law inflationary model. The phase-integral formulas for the scalar and tensor power spectra are explicitly obtained up to ninth-order of the phase-integral approximation. We show that, the phase-integral approximation exactly reproduces the shape of the power spectra for scalar and tensor perturbations as well as the spectral indices. We compare the accuracy of the phase-integral approximation with the results for the power spectrum obtained with the slow-roll and uniform approximation methods.
Revista Mexicana De Fisica, 1996
In the present article we revisit the problem of a relativistic Dirac electron. Using a second or... more In the present article we revisit the problem of a relativistic Dirac electron. Using a second order formalism which reduces the problem of finding the energy spectrum to solving the Whittaker equation, we show that the only physical solution is obtained by truncating the hypergeometric series. Therefore the energy spectrum does not depend on any free parameters for 119 < Z < 137. RESUMEN. En el presente articulo se estudia el problema de un electron relativista de Dirac. Haciendo uso de un formalismo de segundo orden que reduce el problema de encontrar el espectro de energia a resolver la ecuacion de vVhittaker, se muestra que la unica solucion fisicamente aceptable se obtiene truncando la serie hipergeometrica. Por consiguiente, el espectro de energia no depende de ningun parametro libre para 119 < Z < 137.
Physical Review D, 2009
The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological... more The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological perturbations in the quadratic chaotic inflationary model. The phase-integral formulas for the scalar and tensor power spectra are explicitly obtained up to fifth order of the phase-integral approximation. We show that, the phase integral gives a very good approximation for the shape of the power spectra associated with scalar and tensor perturbations as well as the spectral indices. We find that the accuracy of the phase-integral approximation compares favorably with the numerical results and those obtained using the slow-roll and uniform approximation methods.
Physical Review B, 2007
We analyze the role of resonances in two-fermion entanglement production for a quasi onedimension... more We analyze the role of resonances in two-fermion entanglement production for a quasi onedimensional two channel scattering problem. We solve exactly for the problem of a two-fermion antisymmetric product state scattering off a double delta well potential. It is shown that the twoparticle concurrence of the post-selected state has an oscillatory behavior where the concurrence vanishes at the values of momenta for virtual bound states in the double well. These concurrence zeros are interpreted in terms of the uncertainty in the knowledge of the state of the one particle subspace reduced one particle density matrix. Our results suggest manipulation of fermion entanglement production through the resonance structure of quantum dots.
Journal of Physics: Condensed Matter, 2001
We study the energy spectrum of the two-electron spherical parabolic quantum dot using the exact ... more We study the energy spectrum of the two-electron spherical parabolic quantum dot using the exact Schrödinger, the Hartree-Fock, and the Kohn-Sham equations. The results obtained by applying the shifted-1/N method are compared with those obtained by using an accurate numerical technique, showing that the relative error is reasonably small, although the first method consistently underestimates the correct values. The approximate groundstate Hartree-Fock and local-density Kohn-Sham energies, estimated using the shifted-1/N method, are compared with accurate numerical self-consistent solutions. We make some perturbative analyses of the exact energy in terms of the confinement strength, and we propose some interpolation formulae. Similar analysis is made for both mean-field approximations and interpolation formulae are also proposed for these exchange-only ground-state cases.
Physica Scripta, 2007
We solve the two-component Dirac equation in the presence of a spatially one dimensional symmetri... more We solve the two-component Dirac equation in the presence of a spatially one dimensional symmetric attractive cusp potential. The components of the spinor solution are expressed in terms of Whittaker functions. We compute the bound states solutions and show that, as the potential amplitude increases, the lowest energy state sinks into the Dirac sea becoming a resonance. We characterize and compute the lifetime of the resonant state with the help of the phase shift and the Breit-Wigner relation. We discuss the limit when the cusp potential reduces to a delta point interaction.
We obtain exact solutions of the Klein-Gordon and Pauli Schroedinger equations for a two-dimensio... more We obtain exact solutions of the Klein-Gordon and Pauli Schroedinger equations for a two-dimensional hydrogen-like atom in the presence of a constant magnetic field. Analytic solutions for the energy spectrum are obtained for particular values of the magnetic field strength. The results are compared to those obtained in the non-relativistic and spinless case. We obtain that the relativistic spectrum does not present s states.
The phase-integral approximation devised by Fr\"oman and Fr\"oman, is used for computin... more The phase-integral approximation devised by Fr\"oman and Fr\"oman, is used for computing cosmological perturbations in the quartic chaotic inflationary model. The phase-integral formulas for the scalar power spectrum are explicitly obtained up to fifth order of the phase-integral approximation. As in previous reports [1-3], we point out that the accuracy of the phase-integral approximation compares favorably with the numerical results and those obtained using the slow-roll and uniform approximation methods.
We solve the Klein-Gordon and Dirac equations in an open cosmological universe with a partially h... more We solve the Klein-Gordon and Dirac equations in an open cosmological universe with a partially horn topology in the presence of a time dependent magnetic field. Since the exact solution cannot be obtained explicitly for arbitrary time-dependence of the field, we discuss the asymptotic behavior of the solutions with the help of the relativistic Hamilton-Jacobi equation.
We solve the Klein-Gordon equation in the presence of a spatially one-dimensional cusp potential.... more We solve the Klein-Gordon equation in the presence of a spatially one-dimensional cusp potential. The scattering solutions are obtained in terms of Whittaker functions and the condition for the existence of transmission resonances is derived. We show the dependence of the zero-reflection condition on the shape of the potential. In the low momentum limit, transmission resonances are associated with half-bound states. We express the condition for transmission resonances in terms of the phase shifts.
We show that the energy spectrum of the one-dimensional Dirac equation in the presence of a spati... more We show that the energy spectrum of the one-dimensional Dirac equation in the presence of a spatial confining point interaction exhibits a resonant behavior when one includes a weak electric field. After solving the Dirac equation in terms of parabolic cylinder functions and showing explicitly how the resonant behavior depends on the sign and strength of the electric field, we derive an approximate expression for the value of the resonance energy in terms of the electric field and delta interaction strength.
We solve the two-component Dirac equation in the presence of a spatially one-dimensional symmetri... more We solve the two-component Dirac equation in the presence of a spatially one-dimensional symmetric cusp potential. We compute the scattering and bound state solutions and we derive the conditions for transmission resonances as well as for supercriticality.
Physics Letters A, 1989
An exact solution to the Dirac equation in the presence of an exact gravitational plane wave is p... more An exact solution to the Dirac equation in the presence of an exact gravitational plane wave is presented. by passing, an exact solution which represents neutrinos is obtained.
Physical Review A, 2003
We solve the two-component Dirac equation in the presence of a spatially one-dimensional symmetri... more We solve the two-component Dirac equation in the presence of a spatially one-dimensional symmetric cusp potential. We compute the scattering and bound state solutions and we derive the conditions for transmission resonances as well as for supercriticality.
Journal of Mathematical Physics, 1990
In this paper the separation of variables is presented in the Dirac equation in open, flat, and c... more In this paper the separation of variables is presented in the Dirac equation in open, flat, and closed expanding cosmological Robertson-Walker universes. The equations governing the radial variable and the evolution of the time-dependent factor are obtained. An exact solution to the Weyl equation is derived for an arbitrary expansion factor of the Robertson-Walker metrics. An exact solution to Dirac equation in a universe filled with radiation is also presented.
Can J Phys, 1992
In this article we obtain, by separation of variables, an exact solution to the Klein–Gordon and ... more In this article we obtain, by separation of variables, an exact solution to the Klein–Gordon and Dirac equations in a cosmological, spatially-closed, Robertson–Walker space-time with a positive cosmological constant Λ. The model is associated with a universe filled with radiation. We analyze the phenomenon of particle creation for different values of the dimensionless coupling constant ξ.
J Phys Condens Matter, 1996
In the present article we study the energy levels of a 2D hydrogenic atom when a constant magneti... more In the present article we study the energy levels of a 2D hydrogenic atom when a constant magnetic field is applied. We compute the energy spectrum with the help of a generalization of the mesh point technique recently proposed by Schwartz. We also estimate, via a variational method, the upper energy bound for small and large values of the external