Resonance expansions of massless Dirac fields propagating in the exterior of a de Sitter-Reissner-Nordstr\"om black hole

We give an expansion of the solution of the evolution equation for the massless Dirac fields in the outer region of de Sitter-Reissner-Nordstr\"om black hole in terms of resonances. By means of this method we describe the decay of local energy for compactly supported data. The proof uses the cut-off resolvent estimates for the semi-classical Schr\"odinger operators from Bony and H\"afner, 2008. The method extends to the Dirac operators on spherically symmetric asymptotically hyperbolic manifolds.Comment: The paper is continuation of arXiv:1407.3654 and generalizes arXiv:0706.0350 to the Dirac field

An Inverse Problem for Trapping Point Resonances

We consider semi-classical Schr{\"o}dinger operator $P(h)=-h^2\Delta +V(x)$ in ${\mathbb R}^n$ such that the analytic potential $V$ has a non-degenerate critical point $x_0=0$ with critical value $E_0$ and we can define resonances in some fixed neighborhood of $E_0$ when $h>0$ is small enough. If the eigenvalues of the Hessian are \zz-independent the resonances in $h^\delta$-neighborhood of $E_0$ ($\delta >0$) can be calculated explicitly as the eigenvalues of the semi-classical Birkhoff normal form. Assuming that potential is symmetric with respect to reflections about the coordinate axes we show that the classical Birkhoff normal form determines the Taylor series of the potential at $x_0.$ As a consequence, the resonances in a $h^\delta$-neighborhood of $E_0$ determine the first $N$ terms in the Taylor series of $V$ at $x_0.$ The proof uses the recent inverse spectral results of V. Guillemin and A. Uribe

Periodic Jacobi operator with finitely supported perturbations: the inverse resonance problem

We consider a periodic Jacobi operator $H$ with finitely supported perturbations on ${\Bbb Z}.$ We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the scattering data: the inverse of the transmission coefficient and the Jost function on the right half-axis, is one-to-one and onto. We consider the problem of reconstruction of the scattering data from all eigenvalues, resonances and the set of zeros of $R_-(\lambda)+1,$ where $R_-$ is the reflection coefficient

Resonances for Dirac operators on the half-line

We consider the 1D Dirac operator on the half-line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: 1) asymptotics of counting function, 2) estimates on the resonances and the forbidden domain.Comment: arXiv admin note: text overlap with arXiv:1302.463

Resonances for 1D massless Dirac operators

We consider the 1D massless Dirac operator on the real line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: 1) asymptotics of counting function, 2) estimates on the resonances and the forbidden domain, 3) the trace formula in terms of resonances.Comment: 23 page

Resonances for the radial Dirac operators

We consider the radial Dirac operator with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: 1) asymptotics of counting function, 2) in the massless case we get the trace formula in terms of resonances.Comment: 36 pages. arXiv admin note: substantial text overlap with arXiv:1307.247