162 research outputs found

    Spectral Analysis of the Dirac Polaron

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    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 qRq\in\mathbb{R} is a coupling constant, \mathbf{A}(\hat\mathbf{x}) denotes the quantized vector potential and HfH_f denotes the free photon Hamiltonian. Since the total momentum is conserved, HH is decomposed with respect to the total momentum with fiber Hamiltonian H(p),(pR3)H(\mathbf{p}), (\mathbf{p}\in\mathbb{R}^3). Since the self-adjoint operator H(p)H(\mathbf{p}) is bounded from below, one can define the lowest energy E(p,m):=infσ(H(p))E(\mathbf{p},m):=\inf\sigma(H(\mathbf{p})). We prove that E(p,m)E(\mathbf{p},m) is an eigenvalue of H(p)H(\mathbf{p}) under the following conditions: (i) infrared regularization and (ii) E(p,m)<E(p,0)E(\mathbf{p},m)<E(\mathbf{p},0). We also discuss the polarization vectors and the angular momenta

    Embedded Eigenvalues and Neumann-Wigner Potentials for Relativistic Schrodinger Operators

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    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

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

    Fermionic renormalization group method based on the smooth Feshbach map

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    For a fermion system, an operator theoretic renormalization group method based on the smooth Feshbach map is constructed. By using the fermionic renormalization group method, the closed operator of the form: Hg(θ) = HS ⊗ 1 + eθν1 ⊗ Hf + Wg(θ) is analyzed, where HS is a selfadjoint operator on a separable Hilbert space and bounded from below, Hf denotes the fermionic quantization of the one fermion kinetic energy c|k|ν, k ∈ Rd (c, ν > 0), Wg(θ) is a small perturbation with respect to HS ⊗ 1 + eθν1 ⊗ Hf and θ ∈ C is a complex scaling parameter. The constant g ∈ R denotes a coupling constant such that Wg(θ) → 0(g → 0) in some sense. It is assumed that HS has a discrete simple eigenvalue E ∈ σd(HS), and proved that Hg(θ) has an eigenvalue Eg(θ) close to E for a small coupling constant g. Moreover, the eigenvalue Eg(θ) and the corresponding eigenvector Ψ(θ) is constructed by the process of the operator theoretic renormalization group method

    Stability of Discrete Ground State

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    We present new criteria for a self-adjoint operator to have a ground state. As an application, we consider models of ``quantum particles'' coupled to a massive Bose field and prove the existence of a ground state of them, where the particle Hamiltonian does not necessarily have compact resolvent
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