Exact Methods for Self Interacting Neutrinos

The effective many-body Hamiltonian which describes vacuum oscillations and self interactions of neutrinos in a two flavor mixing scheme under the single angle approximation has the same dynamical symmetries as the well known BCS pairing Hamiltonian. These dynamical symmetries manifest themselves in terms of a set of constants of motion and can be useful in formulating the collective oscillation modes in an intuitive way. In particular, we show that a neutrino spectral split can be simply viewed as an avoided level crossing between the eigenstates of a mean field Hamiltonian which includes a Lagrange multiplier in order to fix the value of an exact many-body constant of motion. We show that the same dynamical symmetries also exist in the three neutrino mixing scheme by explicitly writing down the corresponding constants of motion.Comment: To appear in the proceedings of CETUP* 201

An Exactly Solvable Model of Interacting Bosons

We introduce a class of exactly solvable boson models. We give explicit analytic expressions for energy eigenvalues and eigenvectors for an sd-boson Hamiltonian, which is related to the SO(6) chain of the Interacting Boson Model Hamiltonian.Comment: 8 pages of LATE

Supersymmetry and Nuclear Pairing

We show that nuclear pairing Hamiltonian exhibits supersymmetry in the strong-coupling limit. The underlying supersymmetric quantum mechanical structure explains the degeneracies between the energies of the N and Nmax-N+1 pair eigenstates. The supersymmetry transformations connecting these states are given.Comment: 4 pages of REVTEX, one figur

Symmetry and Supersymmetry in Nuclear Pairing: Exact Solutions

Pairing plays a crucial role in nuclear spectra and attempts to describe it has a long history in nuclear physics. The limiting case in which all single particle states are degenerate, but with different s-wave pairing strengths was only recently solved. In this strong coupling limit the nuclear pairing Hamiltonian also exhibits a supersymmetry. Another solution away from those limits, namely two non-degenerate single particle states with different pairing strengths, was also given. In this contribution these developments are summarized and difficulties with possible generalizations to more single particle states and d-wave pairing are discussed.Comment: 6 pages of LATEX, to be published in the Proceedings of the "10th Int. Spring Seminar on Nuclear Physics: New Quests in Nuclear Structure", Vietri Sul Mare, May 21-25, 201

An Exactly Solvable Supersymmetric Model of Semimagic Nuclei

A simple model of nucleons coupled to angular momentum zero (s-pairs) occupying the valance shell of a semi-magic nuclei is considered. The model has a separable, orbit dependent pairing interaction which dominates over the kinetic term. It is shown that such an interaction leads to an exactly solvable model whose (0+) eigenstates and energies can be computed very easily with the help of the algebraic Bethe ansatz method. It is also shown that the model has a supersymmetry which connects the spectra of some semimagic nuclei. The results obtained from this model for the semimagic Ni isotopes from 58Ni to 68Ni are given. In addition, a new and easier technique for calculating the energy eigenvalues from the Bethe ansatz equations is also presented.Comment: Talk given at the International Conference on Nuclear Physics and Astrophysics: From Stable Beams to Exotic Nuclei, Cappadocia, June 200

Lacunary statistical cluster points of sequences

In this note we introduce the concept of a lacunary statistical cluster (l.s.c.) point and prove some of its properties in finite dimensional Banach spaces. We develop the method suggested by S. Pehlivan and M.A. Mamedov [20] where it was proved that under some conditions optimal paths have the same unique stationary limit point and stationary cluster point. We also show that the set $Gamma_x^{theta}$ of l.s.c. points is nonempty and compact