299 research outputs found

    Distributional Borel Summability for Vacuum Polarization by an External Electric Field

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    It is proved that the divergent perturbation expansion for the vacuum polarization by an external constant electric field in the pair production sector is Borel summable in the distributional sense.Comment: 14 page

    Distributional Borel Summability of Odd Anharmonic Oscillators

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    It is proved that the divergent Rayleigh-Schrodinger perturbation expansions for the eigenvalues of any odd anharmonic oscillator are Borel summable in the distributional sense to the resonances naturally associated with the system

    Perturbation theory of PT-symmetric Hamiltonians

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    In the framework of perturbation theory the reality of the perturbed eigenvalues of a class of \PTsymmetric Hamiltonians is proved using stability techniques. We apply this method to \PTsymmetric unperturbed Hamiltonians perturbed by \PTsymmetric additional interactions

    Canonical Expansion of PT-Symmetric Operators and Perturbation Theory

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    Let HH be any \PT symmetric Schr\"odinger operator of the type 2Δ+(x12+...+xd2)+igW(x1,...,xd) -\hbar^2\Delta+(x_1^2+...+x_d^2)+igW(x_1,...,x_d) on L2(Rd)L^2(\R^d), where WW is any odd homogeneous polynomial and gRg\in\R. It is proved that H\P H is self-adjoint and that its eigenvalues coincide (up to a sign) with the singular values of HH, i.e. the eigenvalues of HH\sqrt{H^\ast H}. Moreover we explicitly construct the canonical expansion of HH and determine the singular values μj\mu_j of HH through the Borel summability of their divergent perturbation theory. The singular values yield estimates of the location of the eigenvalues \l_j of HH by Weyl's inequalities.Comment: 20 page

    Some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant

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    Comparison between the exact value of the spectral zeta function, ZH(1)=56/5[32cos(π/5)]Γ2(1/5)/Γ(3/5)Z_{H}(1)=5^{-6/5}[3-2\cos(\pi/5)]\Gamma^2(1/5)/\Gamma(3/5), and the results of numeric and WKB calculations supports the conjecture by Bessis that all the eigenvalues of this PT-invariant hamiltonian are real. For one-dimensional Schr\"odinger operators with complex potentials having a monotonic imaginary part, the eigenfunctions (and the imaginary parts of their logarithmic derivatives) have no real zeros.Comment: 6 pages, submitted to J. Phys.

    Scalar Quantum Field Theory with Cubic Interaction

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    In this paper it is shown that an i phi^3 field theory is a physically acceptable field theory model (the spectrum is positive and the theory is unitary). The demonstration rests on the perturbative construction of a linear operator C, which is needed to define the Hilbert space inner product. The C operator is a new, time-independent observable in PT-symmetric quantum field theory.Comment: Corrected expressions in equations (20) and (21

    PTPT symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum

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    Consider in L2(Rd)L^2(R^d), d1d\geq 1, the operator family H(g):=H0+igWH(g):=H_0+igW. \ds H_0= a^\ast_1a_1+... +a^\ast_da_d+d/2 is the quantum harmonic oscillator with rational frequencies, WW a PP symmetric bounded potential, and gg a real coupling constant. We show that if g<ρ|g|<\rho, ρ\rho being an explicitly determined constant, the spectrum of H(g)H(g) is real and discrete. Moreover we show that the operator \ds H(g)=a^\ast_1 a_1+a^\ast_2a_2+ig a^\ast_2a_1 has real discrete spectrum but is not diagonalizable.Comment: 20 page

    On the eigenproblems of PT-symmetric oscillators

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    We consider the non-Hermitian Hamiltonian H= -\frac{d^2}{dx^2}+P(x^2)-(ix)^{2n+1} on the real line, where P(x) is a polynomial of degree at most n \geq 1 with all nonnegative real coefficients (possibly P\equiv 0). It is proved that the eigenvalues \lambda must be in the sector | arg \lambda | \leq \frac{\pi}{2n+3}. Also for the case H=-\frac{d^2}{dx^2}-(ix)^3, we establish a zero-free region of the eigenfunction u and its derivative u^\prime and we find some other interesting properties of eigenfunctions.Comment: 21pages, 9 figure

    CPT-conserving Hamiltonians and their nonlinear supersymmetrization using differential charge-operators C

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    A brief overview is given of recent developments and fresh ideas at the intersection of PT and/or CPT-symmetric quantum mechanics with supersymmetric quantum mechanics (SUSY QM). We study the consequences of the assumption that the "charge" operator C is represented in a differential-operator form. Besides the freedom allowed by the Hermiticity constraint for the operator CP, encouraging results are obtained in the second-order case. The integrability of intertwining relations proves to match the closure of nonlinear SUSY algebra. In an illustration, our CPT-symmetric SUSY QM leads to non-Hermitian polynomial oscillators with real spectrum which turn out to be PT-asymmetric.Comment: 25 page
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