Tobias Kuna, Ludwig Streit, Werner Westerkamp
We construct the Feynman integrands for a class of exponentially growing time-dependent potentials as white noise functionals. We show that they solve the Schroedinger equation. The Morse potential is considered as a special case.
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The Feynman integral for the Schroedinger propagator is constructed as a generalized function of white noise, for a linear space of potentials spanned by measures and Laplace transforms of measures, i.e., locally singular as well as rapidly growing at infinity. Remarkably, all these propagators admit a perturbation expansion.
The Feynman path integrals for the magnetic Schroedinger equations are defined mathematically, in particular, with polynomially growing potentials in the spatial direction. For example, we can handle electromagnetic potentials $(V,A_{1},A_{2},...,A_{d})$ such that $V(t,x) = |x|^{2(l+1)} + $`` a polynomial of degree $(2l + 1)$ in $x$ " ($l = 0,1,2,...$) and $A_{j}(t,x)$ are polynomials of degree $l$ in $x$. The Feynman path integrals are defined as $L^2$-valued continuous functions with respect to the time variable.
We show the existence of exponentially growing finite energy solutions for the charged Klein-Gordon equation on the De Sitter-Kerr-Newman metric for small charge and mass of the field and small angular momentum of the black hole. The mechanism behind is that the zero resonance that exists for zero charge, mass and angular momentum moves into the upper half plane.
For any sub-extremal Kerr spacetime with non-zero angular momentum, we find an open family of non-zero masses for which there exist smooth, finite energy, and exponentially growing solutions to the corresponding Klein-Gordon equation. If desired, for any non-zero integer m, an exponentially growing solution can be found with mass arbitrarily close to |am|/2Mr_+. In addition to its direct relevance for the stability of Kerr as a solution to the Einstein-Klein-Gordon system, our result provides the first rigorous construction of a superradiant instability. Finally, we note that this linear insta…
We define the class of multivariate group entropies as a novel set of information - theoretical measures, which extends significantly the family of group entropies. We propose new examples related to the "super-exponential" universality class of complex systems; in particular, we introduce a general entropy, representing a suitable information measure for this class. We also show that the group-theoretical structure associated with our multivariate entropies can be used to define a large family of exactly solvable discrete dynamical models. The natural mathematical framework allowing us to for…
Exponentially-growing Mode Instability on Reissner-Nordström--Anti-de-Sitter black holes
研究Reissner-Nordström--Anti-de-Sitter黑洞上的Klein-Gordon方程增长模式解，发现其在某些参数下存在不稳定性，与Schwarzschild-AdS不同，且不依赖超辐射现象。
The Cauchy problem is studied for the self-adjoint and non-self-adjoint Schroedinger equations. We first prove the existence and uniqueness of solutions in the weighted Sobolev spaces. Secondly we prove that if potentials are depending continuously and differentiably on a parameter, so are the solutions, respectively. The non-self-adjoint Schroedinger equations that we study are those used in the theory of continuous quantum measurements. The results on the existence and uniqueness of solutions in the weighted Sobolev spaces will play a crucial role in the proof for the convergence of the Feyn…
A nonzero non-Hermitian winding number indicates that a gapped system is in a nontrivial topological class due to the non-Hermiticity of its Hamiltonian. While for Hermitian systems nontrivial topological quantum numbers are reflected by the existence of edge states, a nonzero non-Hermitian winding number impacts a system's bulk response. To establish this relation, we introduce the bulk Green function, which describes the response of a gapped system to an external perturbation on timescales where the induced excitations have not propagated to the boundary yet, and show that it will grow in sp…
Many stochastic complex systems are characterized by the fact that their configuration space doesn't grow exponentially as a function of the degrees of freedom. The use of scaling expansions is a natural way to measure the asymptotic growth of the configuration space volume in terms of the scaling exponents of the system. These scaling exponents can, in turn, be used to define universality classes that uniquely determine the statistics of a system. Every system belongs to one of these classes. Here we derive the information geometry of scaling expansions of sample spaces. In particular, we pre…
A Feynman-Kac type formula of relativistic Schrödinger operators with unbounded vector potential and spin 1/2 is given in terms of a three-component process consisting of Brownian motion, a Poisson process and a subordinator. This formula is obtained for unbounded magnetic fields and magnetic field with zeros. From this formula an energy comparison inequality is derived. Spatial decay of bound states is established separately for growing and decaying potentials by using martingale methods.
The wave equation $\square_{g_{M,a}}ψ=0$ on subextremal Kerr spacetimes $(\mathcal{M}_{M,a},g_{M,a})$, $0<|a|<M$, does not admit real mode solutions, as was established by Shlapentokh-Rothman. In this paper, we show that the absence of real modes does not persist under the addition of an arbitrary short-range non-negative potential $V$ to the wave equation or under changes of the metric $g_{M,a}$ in the far away region of $\mathcal{M}_{M,a}$ (retaining the causality of $T$ there). In particular, we first establish, for any $0<|a|<M$, the existence of real mode solutions $ψ$ to equation $\squar…
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