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Spectral singularities have been extensively studied over the last one and half decades for different non-Hermitian potentials in non-Hermitian quantum mechanics. The nature of spectral singularities has not been studied for the case of quaternionic potential. In the present work, we perform an analytical study on scattering from a quaternionic point interaction represented by a delta function. New features of spectral singularities are observed, which are different than the case of a complex (non-quaternionic) point interaction. The most notable difference is the occurrence of spectral singularity from the lossy point interaction, which is forbidden in the case of standard non-Hermitian quantum mechanics. [ABSTRACT FROM AUTHOR]

We study the tunneling problem from the general Smith–Volterra–Cantor (SVC) potential of finite length L characterized by the scaling parameter ρ and stage G. We show that the SVC(ρ) potential of stage G is a special case of the super periodic potential (SPP) of order G. By using the SPP formalism developed by us earlier, we provide the closed form expression of the tunneling probability TG(k) with the help of the q-Pochhammer symbol. The profile of TG(k) with wave vector k is found to saturate with increasing stage G. Very sharp transmission resonances are found to occur in this system, which may find applications in the design of sharp transmission filters. [ABSTRACT FROM AUTHOR]

The dynamics of collisionless plasma described by the Poisson-Vlasov equations is connected to the Hamiltonian motions of particles and their symmetries. The group of canonical diffeomorphisms of particle phase space is described and adopted as the configuration space. The dual space of its Lie algebra of Hamiltonian vector fields is identified with the space of nonclosed one-forms. The Poisson equation is obtained as a constraint arising from the gauge symmetries of particle dynamics. Variational derivative constrained by the Poisson equation is used to obtain reduced dynamical equations. Usual Lie-Poisson reduction for the group of canonical diffeomorphisms gives the momentum-Vlasov equations. Plasma density is defined as the divergence of symplectic dual of momentum variables. This definition is also given a momentum map description associated with the action of additive group of functions of particle phase space. Equivalence of Hamiltonian functionals in momentum and density formulations is shown. As an alternative formulation in momentum variables, a canonical Hamiltonian system with a quadratic Hamiltonian functional is described. In the case that the groups of Hamiltonian and volume preserving diffeomorphisms coincide, a comparison of one-dimensional plasma and two-dimensional incompressible fluid is presented. [ABSTRACT FROM AUTHOR]

Karabulut and Sibert [J. Math. Phys. 38, 4815 (1997)] have constructed an orthogonal set of functions from linear combinations of equally spaced Gaussians. In this paper we show that they are actually eigenfunctions of a q-oscillator in coordinate representation. We also reinterpret the coordinate representation example of q-oscillator given by Macfarlane as the functions orthogonal with respect to an unusual inner product definition. It is shown that the eigenfunctions in both q-oscillator examples are infinitely degenerate. [ABSTRACT FROM AUTHOR]

A new Wannier function made from distributed Gaussians is reported. Convolution integrals of the Wannier functions are studied and analytical expressions for the convolution integral are given. A new expression for a set of coefficients introduced in an earlier publication [J. Math. Phys. 38, 4815 (1997)] is derived. Using this new expression, the peculiar behavior of another set of coefficients introduced in the same publication in a particular limit is explained. [ABSTRACT FROM AUTHOR]

It is shown that the Poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one-form in three dimensions. Advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability. Techniques for finding Poisson structures are presented and applied to various examples such as the Halphen system which has been studied as the two-monopole problem by Atiyah and Hitchin. It is shown that the Halphen system can be formulated in terms of a flat SL(2,R)-valued connection and belongs to a nontrivial Godbillon–Vey class. On the other hand, for the Euler top and a special case of three-species Lotka–Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable bi-Hamiltonian structures. The globally integrable bi-Hamiltonian case is a linear and the SL(2,R) structure is a quadratic unfolding of an integrable one-form in 3+1 dimensions. It is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of Poisson structure and some new techniques for incorporating arbitrary constants into the Poisson one-form are presented herein. This leads to some extensions, analogous to q extensions, of Poisson structure. The Kermack–McKendrick model and some of its generalizations describing the spread of epidemics, as well as the integrable cases of the Lorenz, Lotka–Volterra, May–Leonard, and Maxwell–Bloch systems admit globally integrable bi-Hamiltonian structure. [ABSTRACT FROM AUTHOR]

Examines an orthogonal function set called distributed Gaussian orthogonal polynomials constructed from equally placed Gaussians. Infinite chain limit of the polynomials and the corresponding quadrature; Analytical expressions for polynomials, quadrature points, weights and Lagrange functions; Construction of Wannier functions from Gaussians.

Second order degenerate Clément and Sarıoğlu–Tekin Lagrangians are casted into forms of various first order Lagrangians. The structures of the iterated tangent bundle and acceleration bundle are presented as a suitable geometric framework. Hamiltonian analyses of these equivalent formalisms are performed by means of the Dirac–Bergmann constraint algorithm. All formulations are shown to possess only second class constraints. [ABSTRACT FROM AUTHOR]

We present Skinner-Rusk and Hamiltonian formalisms of second order degenerate Clément and Sarıoğlu-Tekin Lagrangians. The Dirac-Bergmann constraint algorithm is employed to obtain Hamiltonian realizations of Lagrangian theories. The Gotay-Nester-Hinds algorithm is used to investigate Skinner-Rusk formalisms of these systems. [ABSTRACT FROM AUTHOR]