137 research outputs found

    Bogoliubov Coefficients of 2D Charged Black Holes

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    We exactly calculate the thermal distribution and temperature of Hawking radiation for a two-dimensional charged dilatonic black hole after it has settled down to an "equilibrium" state. The calculation is carried out using the Bogoliubov coefficients. The background of the process is furnished by a preexisting black hole and not by collapsing matter as considered by Giddings and Nelson for the case of a Schwarzschild black hole. Furthermore, the vanishing of the temperature and/or the Hawking radiation in the extremal case is obtained as a regular limit of the general case.Comment: 9 pages, 1 eps figur

    Bagchi's Theorem for families of automorphic forms

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    We prove a version of Bagchi's Theorem and of Voronin's Universality Theorem for family of primitive cusp forms of weight 22 and prime level, and discuss under which conditions the argument will apply to general reasonable family of automorphic LL-functions.Comment: 15 page

    Completeness of the Coulomb scattering wave functions

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    Completeness of the eigenfunctions of a self-adjoint Hamiltonian, which is the basic ingredient of quantum mechanics, plays an important role in nuclear reaction and nuclear structure theory. However, until now, there was no a formal proof of the completeness of the eigenfunctions of the two-body Hamiltonian with the Coulomb interaction. Here we present the first formal proof of the completeness of the two-body Coulomb scattering wave functions for repulsive unscreened Coulomb potential. To prove the completeness we use the Newton's method [R. Newton, J. Math Phys., 1, 319 (1960)]. The proof allows us to claim that the eigenfunctions of the two-body Hamiltonian with the potential given by the sum of the repulsive Coulomb plus short-range (nuclear) potentials also form a complete set. It also allows one to extend the Berggren's approach of modification of the complete set of the eigenfunctions by including the resonances for charged particles. We also demonstrate that the resonant Gamow functions with the Coulomb tail can be regularized using Zel'dovich's regularization method.Comment: 12 pages and 1 figur

    Eigenvalues of Laplacian with constant magnetic field on non-compact hyperbolic surfaces with finite area

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    We consider a magnetic Laplacian ΔA=(id+A)(id+A)-\Delta_A=(id+A)^\star (id+A) on a noncompact hyperbolic surface \mM with finite area. AA is a real one-form and the magnetic field dAdA is constant in each cusp. When the harmonic component of AA satifies some quantified condition, the spectrum of ΔA-\Delta_A is discrete. In this case we prove that the counting function of the eigenvalues of ΔA-\Delta_{A} satisfies the classical Weyl formula, even when $dA=0.

    Properties of the series solution for Painlevé I

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    We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painlevé equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented

    On some problems involving Hardy's function

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    Some problems involving the classical Hardy function Z(t):=ζ(1/2+it)(χ(1/2+it))1/2,ζ(s)=χ(s)ζ(1s) Z(t) := \zeta(1/2+it)\bigl(\chi(1/2+it)\bigr)^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s) are discussed. In particular we discuss the odd moments of Z(t)Z(t), the distribution of its positive and negative values and the primitive of Z(t)Z(t). Some analogous problems for the mean square of ζ(1/2+it)|\zeta(1/2+it)| are also discussed.Comment: 15 page

    Arithmetical properties of Multiple Ramanujan sums

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    In the present paper, we introduce a multiple Ramanujan sum for arithmetic functions, which gives a multivariable extension of the generalized Ramanujan sum studied by D. R. Anderson and T. M. Apostol. We then find fundamental arithmetic properties of the multiple Ramanujan sum and study several types of Dirichlet series involving the multiple Ramanujan sum. As an application, we evaluate higher-dimensional determinants of higher-dimensional matrices, the entries of which are given by values of the multiple Ramanujan sum.Comment: 19 page

    A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators

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    Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with finite singular critical points are considered.Comment: 38 pages, Proposition 2.2 and its proof corrected, Remarks 2.5, 3.4, and 3.12 extended, details added in subsections 2.3 and 4.2, section 6 rearranged, typos corrected, references adde

    On parton distributions in a photon gas

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    In some cases it may be useful to know parton distributions in a photon gas. This may be relevant, e.g., for the analysis of interactions of high energy cosmic ray particles with the cosmic microwave background radiation. The latter can be considered as a gas of photons with an almost perfect blackbody spectrum. An approach to finding such parton distributions is described. The survival probability of ultra-high energy neutrinos traveling through this radiation is calculated.Comment: 5 pages, 4 figures, EPJ style files. Some changes in the text. Two new sections discussing ultra-high energy neutrino damping in the cosmic microwave background radiation are include
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