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Methods Of Modern Mathematical Physics

Distance-Regular Graphs

Oxford University Press in Africa, 1927â80

在1954年，科学界庆祝了Poincare的诞辰100周年纪念日，在那个时候，Poincare在数学家们当中的名声并没有达到最高峰，而希尔伯特的精神则主导了很多数学家的思想，Poincare在物理学家们当中的名声也没有达到最高峰，因为在那时候，物理学在本质上是和量子理论相关的。

Henri Poincare，为科学服务的一生

Greybody factors in black-hole physics modify the naive Planckian spectrum that is predicted for Hawking radiation when working in the limit of geometrical optics. We consider the Schwarzschild geometry in $(3+1)$ dimensions, and analyze the Regge-Wheeler equation for arbitrary particle spin $s$ and wave-mode angular momentum $\ensuremath{\ell}$, deriving rigorous bounds on the greybody factors as a function of $s$, $\ensuremath{\ell}$, wave frequency $\ensuremath{\omega}$, and the black-hole mass $m$.

Bounding the greybody factors for Schwarzschild black holes

Second order equations of the form $\ddot{z}(t) + A_0z(t) + D\dot{z}(t) = 0$ are considered. Such equations are often used as a model for transverse motions of thin beams in the presence of damping. We derive various properties of the operator matrix $\mathcal{A} = \left[ {\begin{array}{*{20}c} 0 & I\\ { -A_0 } & { -D}\\ \end{array}} \right]$ associated with the second order problem above. We develop sufficient conditions for analyticity of the associated semigroup and for the existence of a Riesz basis consisting of eigenvectors and associated vectors of ${\mathcal{A}}$ in the phase space.

Analyticity and Riesz basis property of semigroups associated to damped vibrations

In this paper we study for a given azimuthal quantum number κ the eigenvalues of the Chandrasekhar–Page angular equation with respect to the parameters μ≔am and ν≔aω, where a is the angular momentum per unit mass of a black hole, m is the rest mass of the Dirac particle and ω is the energy of the particle (as measured at infinity). For this purpose, a self-adjoint holomorphic operator family A(κ;μ,ν) associated to this eigenvalue problem is considered. At first we prove that for fixed κ∊R\(−12,12) the spectrum of A(κ;μ,ν) is discrete and that its eigenvalues depend analytically on (μ,ν)∊C2. Moreover, it will be shown that the eigenvalues satisfy a first order partial differential equation with respect to μ and ν, whose characteristic equations can be reduced to a Painleve III equation. In addition, we derive a power series expansion for the eigenvalues in terms of ν−μ and ν+μ, and we give a recurrence relation for their coefficients. Further, it will be proved that for fixed (μ,ν)∊C2 the eigenvalues of A(...

On the Eigenvalues of the Chandrasekhar-Page Angular Equation

In this paper we give a brief outline of the applications of the generalized Heun equation (GHE) in the context of quantum field theory in curved spacetimes. In particular, we relate the separated radial part of a massive Dirac equation in the Kerr–Newman metric to the static perturbations for the non-extremal Reissner–Nordstrom solution to a GHE.

https://iopscience.iop.org/article/10.1088/0305-4470/39/40/019/pdf

The generalized Heun equation in QFT in curved spacetimes

We investigate the existence of time-periodic solutions of the Dirac equation in the Kerr-Newman background metric. To this end, the solutions are expanded in a Fourier series with respect to the time variable t, and the Chandrasekhar separation ansatz is applied so that the question of existence of a time-periodic solution is reduced to the solvability of a certain coupled system of ordinary differential equations. First, we prove the already known result that there are no time-periodic solutions in the nonextreme case. Then, it is shown that in the extreme case for fixed black hole data there is a sequence of particle masses (mN)N∊N for which a time-periodic solution of the Dirac equation does exist. The period of the solution depends only on the data of the black hole described by the Kerr-Newman metric.

Spectral analysis of radial Dirac operators in the Kerr-Newman metric and its applications to time-periodic solutions

We investigate the local energy decay of solutions of the Dirac equation in the non-extreme Kerr–Newman metric. First, we write the Dirac equation as a Cauchy problem and define the Dirac operator. It is shown that the Dirac operator is selfadjoint in a suitable Hilbert space. With the RAGE theorem, we show that for each particle its energy located in any compact region outside the event horizon of the Kerr–Newman black hole decays in the time mean.

Monika Winklmeier

Papers

Analyticity and Riesz basis property of semigroups associated to damped vibrations

On the Eigenvalues of the Chandrasekhar-Page Angular Equation

The generalized Heun equation in QFT in curved spacetimes

Spectral analysis of radial Dirac operators in the Kerr-Newman metric and its applications to time-periodic solutions

A spectral approach to the Dirac equation in the non-extreme Kerr–Newmann metric