Spectral Analysis of the Dirac Polaron

A system of a Dirac particle interacting with the radiation field is considered. The Hamiltonian of the system is defined by H = \alpha\cdot(\hat\mathbf{p}-q\mathbf{A}(\hat\mathbf{x}))+m\beta + H_f where $q\in\mathbb{R}$ is a coupling constant, \mathbf{A}(\hat\mathbf{x}) denotes the quantized vector potential and $H_f$ denotes the free photon Hamiltonian. Since the total momentum is conserved, $H$ is decomposed with respect to the total momentum with fiber Hamiltonian $H(\mathbf{p}), (\mathbf{p}\in\mathbb{R}^3)$. Since the self-adjoint operator $H(\mathbf{p})$ is bounded from below, one can define the lowest energy $E(\mathbf{p},m):=\inf\sigma(H(\mathbf{p}))$. We prove that $E(\mathbf{p},m)$ is an eigenvalue of $H(\mathbf{p})$ under the following conditions: (i) infrared regularization and (ii) $E(\mathbf{p},m)<E(\mathbf{p},0)$. We also discuss the polarization vectors and the angular momenta

Asymptotic Expansion of the One-Loop Approximation of the Chern-Simons Integral in an Abstract Wiener Space Setting

In an abstract Wiener space setting, we constract a rigorous mathematical model of the one-loop approximation of the perturbative Chern-Simons integral, and derive its explicit asymptotic expansion for stochastic Wilson lines.Comment: 39 page

A two-dimensional pictorial presentation of Berele's insertion algorithm for symplectic tableaux

We give the first two-dimensional pictorial presentation of Berele's correspondence \cite{Berele}, an analogue of the Robinson-Schensted (R-S) correspondence \cite{Robinson, Schensted} for the symplectic group Sp(2n, \Cpx ). From the standpoint of representation theory, the R-S correspondence combinatorially describes the irreducible decomposition of the tensor powers of the natural representation of GL(n,\Cpx). Berele's insertion algorithm gives the bijection that describes the irreducible decomposition of the tensor powers of the natural representation of Sp(2n, \Cpx). Two-dimensional pictorial presentations of the R-S correspondence via local rules (first given by S. Fomin \cite{Fomin,FominGen}) and its many variants have proven very useful in understanding their properties and creating new generalizations. We hope our new presentation will be similarly useful.Comment: 42 page

Embedded Eigenvalues and Neumann-Wigner Potentials for Relativistic Schrodinger Operators

The existence of potentials for relativistic Schrodinger operators allowing eigenvalues embedded in the essential spectrum is a long-standing open problem. We construct Neumann-Wigner type potentials for the massive relativistic Schrodinger operator in one and three dimensions for which an embedded eigenvalue exists. We show that in the non-relativistic limit these potentials converge to the classical Neumann-Wigner and Moses-Tuan potentials, respectively. For the massless operator in one dimension we construct two families of potentials, different by the parities of the (generalized) eigenfunctions, for which an eigenvalue equal to zero or a zero-resonance exists, dependent on the rate of decay of the corresponding eigenfunctions. We obtain explicit formulae and observe unusual decay behaviours due to the non-locality of the operator

Spectral analysis of non-commutative harmonic oscillators: the lowest eigenvalue and no crossing

The lowest eigenvalue of non-commutative harmonic oscillators $Q$ is studied. It is shown that $Q$ can be decomposed into four self-adjoint operators, and all the eigenvalues of each operator are simple. We show that the lowest eigenvalue $E$ of $Q$ is simple. Furthermore a Jacobi matrix representation of $Q$ is given and spectrum of $Q$ is considered numerically.Comment: 4figures. We revised section