Haydar Uncu

Bose-Einstein condensate in a harmonic trap with an eccentric dimple potential

We investigate Bose-Einstein condensation of noninteracting gases in a harmonic trap with an offcenter dimple potential. We specifically consider the case of a tight and deep dimple potential, which is modeled by a point interaction. This point interaction is represented by a Dirac delta function. The atomic density, chemical potential, critical temperature and condensate fraction, and the role of the relative depth and the position of the dimple potential are analyzed by performing numerical calculations.

One-dimensional semirelativistic Hamiltonian with multiple Dirac delta potentials

In this paper, we consider the one-dimensional semirelativistic Schrödinger equation for a particle interacting with N Dirac delta potentials. Using the heat kernel techniques, we establish a resolvent formula in terms of anN × N matrix, called the principal matrix. This matrix essentially includes all the information about the spectrum of the problem. We study the bound state spectrum by working out the eigenvalues of the principal matrix. With the help of the Feynman-Hellmann theorem, we analyze how the bound state energies change with respect to the parameters in the model. We also prove that there are at most N bound states and explicitly derive the bound state wave function. The bound state problem for the two-center case is particularly investigated. We show that the ground state energy is bounded below, and there exists a selfadjoint Hamiltonian associated with the resolvent formula. Moreover, we prove that the ground state is nondegenerate. The scattering problem for N centers is analyzed by exactly solving the semirelativistic Lippmann-Schwinger equation. The reflection and the transmission coefficients are numerically and asymptotically computed for the two-center case. We observe the so-called threshold anomaly for two symmetrically located centers. The semirelativistic version of the Kronig-Penney model is shortly discussed, and the band gap structure of the spectrum is illustrated. The bound state and scattering problems in the massless case are also discussed. Furthermore, the reflection and the transmission coefficients for the two delta potentials in this particular case are analytically found. Finally, we solve the renormalization group equations and compute the beta function nonperturbatively.

Bose-Einstein Condensate in a Linear Trap With a Dimple Potential

We study Bose-Einstein condensation in a linear trap with a dimple potential where we model dimple potentials by Dirac \del function. Attractive and repulsive dimple potentials are taken into account. This model allows simple, explicit numerical and analytical investigations of noninteracting gases. Thus, the \Sch is used instead of the Gross-Pitaevski equation. We calculate the atomic density, the chemical potential, the critical temperature and the condensate fraction. The role of the relative depth of the dimple potential with respect to the linear trap in large condensate formation at enhanced temperatures is clearly revealed. Moreover, we also present a semi-classical method for calculating various quantities such as entropy analytically. Moreover, we compare the results of this paper with the results of a previous paper in which the harmonic trap with a dimple potential in 1D was investigated.

A parabolic model for dimple potentials

We study the truncated parabolic function and demonstrate that it is a representation of the Dirac function. We also show that the truncated parabolic function, used as a potential in the Schr¨ odinger equation, has the same bound state spectrum, tunneling and reflection amplitudes as the Dirac potential, as the width of the parabola approximates to zero. Dirac potential is used to model dimple potentials which are utilized to increase the phase-space density of a Bose‐Einstein condensate in a harmonic trap. We show that a harmonic trap with a function at the origin is a limiting case of the harmonic trap with a symmetric truncated parabolic potential around the origin. Hence, the truncated parabolic is a better candidate for modeling the dimple potentials.

A new method for derivation of statistical weight of the Gentile Statistics

We present a new method for obtaining the statistical weight of the Gentile Statistics. In a recent paper, Perez and Tun presented an approximate combinatoric and an exact recursive formula for the statistical weight of Gentile Statistics, beginning from bosonic and fermionic cases, respectively Hernandez-Perez and Tun (2007). In this paper, we obtain two exact, one combinatoric and one recursive, formulae for the statistical weight of Gentile Statistics, by another approach. The combinatoric formula is valid only for special cases, whereas recursive formula is valid for all possible cases. Moreover, for a given q-maximum number of particles that can occupy a level for Gentile statistics-the recursive formula we have derived gives the result much faster than the recursive formula presented in Hernandez-Perez and Tun (2007), when one uses a computer program. Moreover we obtained the statistical weight for the distribution proposed by Dai and Xie (2009).