We show that every tempered distribution, which is a solution of the (homogenous) Klein-Gordon equation, admits a ``tame'' restriction to the characteristic (hyper)surface $\{x^0+x^n=0\}$ in $(1+n)$-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space $\cS'_{\partial_-}(\R^n)$ which we have introduced in \cite{PullJMP}. Moreover, we show that every element of $\cS'_{\partial_-}(\R^n)$ appears as the ``tame'' restriction of a solution of the (homogeneous) Klein-Gordon equation.
In this article we introduce a new class of non-Archimedean pseudodifferential equations of Klein-Gordon type and study the corresponding Cauchy problem for these equations. A remarkable fact is that the non-Archimedean Klein-Gordon equations exhibit many similar properties to the classical Klein-Gordon equations.
We prove a completeness result for a class of polynomial solutions of the wave equation called wave polynomials and construct generalized wave polynomials, solutions of the Klein-Gordon equation with a variable coefficient. Using the transmutation (transformation) operators and their recently discovered mapping properties we prove the completeness of the generalized wave polynomials and use them for an explicit construction of the solution of the Cauchy problem for the Klein-Gordon equation. Based on this result we develop a numerical method for solving the Cauchy problem and test its performa…
We construct a class of solutions to the Cauchy problem of the Klein-Gordon equation on any standard static spacetime. Specifically, we have constructed solutions to the Cauchy problem based on any self-adjoint extension (satisfying a technical condition: "acceptability") of (some variant of) the Laplace-Beltrami operator defined on test functions in an $L^2$-space of the static hypersurface. The proof of the existence of this construction completes and extends work originally done by Wald. Further results include the uniqueness of these solutions, their support properties, the construction of…
We consider the Klein-Gordon equation on analytic spacetimes with an analytic Cauchy surface. In this setting, we prove the existence of pure analytic Hadamard states. The proof is based on considering an elliptic operator obtained by Wick rotating the Klein-Gordon operator in a neighborhood of a Cauchy hypersurface. The Cauchy data of Hadamard two-point functions are constructed as Calderón projectors (suitably generalized if the hypersurface is non-compact) for the elliptic operator.
We consider the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry, which is shown to include examples such as exterior Kerr, Kerr-de Sitter and Kerr-Kruskal spacetimes. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseu-dodifferential calculus on Riemannian manifolds of bounded geometry. We apply this result to construct all pure regular Hadamard states (and associated Feynman inverses), where regular refers to the state's two-point function having…
The scalar Klein-Gordon equation describes wave motion in a waveguide with a cut-off. For example, the displacement of an elastic cord anchored to a solid base by elastic elements can be described by the scalar Klein-Gordon equation. We analyse this equation using the concept of analytical continuation of dispersion diagram. Particularly, it is shown that the dispersion diagram is topologically equivalent to a tube analytically embedded in two-dimensional complex space. The corresponding Fourier integral is studied on this tube using the Cauchy's theorem. The basic properties of the scalar Kle…
We consider waves, which obey the semilinear Klein-Gordon equation, propagating in the Friedmann-Lemaitre-Robertson-Walker spacetimes. The equations in the de Sitter and Einstein-de Sitter spacetimes are the important particular cases. We show the global in time existence in the energy class of solutions of the Cauchy problem.
In this article we construct the fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. We use these fundamental solutions to represent solutions of the Cauchy problem and to prove $L^p-L^q$ estimates for the solutions of the equation with and without a source term.
In this paper, we consider the Cauchy problem for Klein-Gordon equation with a cubic convolution nonlinearity in $\R^3$. By making use of Bourgain's method in conjunction with a precise Strichartz estimate of S.Klainerman and D.Tataru, we establish the $H^s (s<1)$ global well-posedness of the Cauchy problem for the cubic convolution defocusing Klein-Gordon-Hartree equation. Before arriving at the previously discussed conclusion, we obtain global solution for this non-scaling equation with small initial data in $H^{s_0}\times H^{s_0-1}$ where $s_0=\fracγ6$ but not $\frac\gamma2-1$, for this equ…
We show non-linear stability and instability results in spherical symmetry for the interior of a charged black hole -approaching a sub-extremal Reissner-Nordström background fast enough at infinity- in presence of a massive and charged scalar field, motivated by the strong cosmic censorship conjecture in that setting : 1. Stability : We prove that spherically symmetric characteristic initial data to the Einstein-Maxwell- Klein-Gordon equations approaching a Reissner-Nordström background with a sufficiently decaying polynomial decay rate on the event horizon gives rise to a space-time possessin…
We consider the massive Klein-Gordon equation on a class of asymptotically static spacetimes (in the long range sense) with Cauchy surface of bounded geometry. We prove the existence and Hadamard property of the in and out states constructed by scattering theory methods.
In this article we study the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation $\Box_g ϕ-m^2 ϕ= -|ϕ|^p $ with the small mass $m \le n/2$ in de Sitter space-time with the metric $g$. We prove that for every $p>1$ the large energy solution blows up, while for the small energy solutions we give a borderline $p=p(m,n)$ for the global in time existence. The consideration is based on the representation formulas for the solution of the Cauchy problem and on some generalizations of the Kato's lemma.
We study the Cauchy problem for the sine-Gordon equation in the semiclassical limit with pure-impulse initial data of sufficient strength to generate both high-frequency rotational motion near the peak of the impulse profile and also high-frequency librational motion in the tails. We show that for small times independent of the semiclassical scaling parameter, both types of motion are accurately described by explicit formulae involving elliptic functions. These formulae demonstrate consistency with predictions of Whitham's formal modulation theory in both the hyperbolic (modulationally stable)…
The Euler-Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. In the 3D case Guo first constructed a global smooth irrotational solution by using dispersive Klein-Gordon effect. It has been conjectured that same results should hold in the two-dimensional case. In our recent work, we proved the existence of a family of smooth solutions by constructing the wave operators for the 2D system. In this work we completely settle the 2D Cauchy problem.
We analyze the Cauchy problem for the Klein-Gordon equation on the type IIB supergravity backgrounds $AdS^5 \times Y^{p,q}$, where $Y^{p,q}$ is any of the Sasaki-Einstein 5-manifolds recently discovered by Gaunlett, Martelli, Sparks and Waldram (Adv. Theor. Math. Phys. 8 (2004) 711--734). Although these spaces are not globally hyperbolic, we prove that there exists a unique admissible propagator and derive an integral representation thereof using spectral-theoretic techniques.
We prove that the Cauchy problem for the Dirac-Klein-Gordon equations in two space dimensions is locally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor, and an associated range of spaces of positive index for the meson field. In particular, we can go below the charge norm, that is, the $L^2$ norm of the spinor. We hope that this can have implications for the global existence problem, since the charge is conserved. Our result relies on the null structure of the system, and bilinear space-time estimates for the homogeneous wave equation.
An upper bound of the relative entanglement entropy of thermal states at an inverse temperature $β$ of linear, massive Klein-Gordon and Dirac quantum field theories across two regions, separated by a nonzero distance $d$ in a Cauchy hypersurface of an ultrastatic (spin-)spacetime has been computed. This entanglement measure is bounded by a negative constant times $\ln | \tanh (πd/ 2 β) |$ which signifies power law decay for asymptotic $d$ where the exponent depends on $β< \infty$.
We construct Hadamard states for Klein-Gordon fields in a spacetime $M_{0}$ equal to the interior of the future lightcone $C$ from a base point $p$ in a globally hyperbolic spacetime $(M, g)$. Under some regularity conditions at future infinity of $C$, we identify a boundary symplectic space of functions on $C$, which allows to construct states for Klein-Gordon quantum fields in $M_{0}$ from states on the CCR algebra associated to the boundary symplectic space. We formulate the natural microlocal condition on the boundary state on $C$ ensuring that the bulk state it induces in $M_{0}$ satisfie…
We evaluate explicitly, in terms of the Cauchy data, the constant pre-factor in the large $x$ asymptotics of the Painlevé III tau-function. Our result proves the conjectural formula for this pre-factor obtained recently by O. Lisovyy, Y. Tykhyy, and the first co-author with the help of the recently discovered connection of the Painlevé tau-functions with the Virasoro conformal blocks. Our approach does not use this connection, and it is based on the Riemann-Hilbert method.
In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein-Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point argument. Unlike the unperturbed case, a noteworthy difficulty here arises from the possible non-unitarity of the semigroup generating the corresponding linear evolution. We then show that the equation is Hamiltonian and we establish several stability/instability results for its standing waves. Our analysis relies on a detailed study of the spectral properti…
The Cauchy problem for the Maxwell-Klein-Gordon equations in Lorenz gauge in $n$ space dimensions ($n \ge 4$) is shown to be locally well-posed for low regularity (large) data. The result relies on the null structure for the main bilinear terms which was shown to be not only present in Coulomb gauge but also in Lorenz gauge by Selberg and Tesfahun, who proved global well-posedness for finite energy data in three space dimensions. This null structure is combined with product estimates for wave-Sobolev spaces. Crucial for the improvement are the solution spaces introduced by Klainerman-Selberg. …
We study the Cauchy problem for the sine-Gordon equation in the semiclassical limit with pure-impulse initial data of sufficient strength to generate both high-frequency rotational motion near the peak of the impulse profile and also high-frequency librational motion in the tails. Subject to suitable conditions of a general nature, we analyze the fluxon condensate solution approximating the given initial data for small time near points where the initial data crosses the separatrix of the phase portrait of the simple pendulum. We show that the solution is locally constructed as a universal curv…
In this paper, we develop a Cauchy matrix reduction technique that enables us to obtain solutions for the reduced local and nonlocal complex equations from the Cauchy matrix solutions of the original before-reduction systems. Specifically, by imposing local and nonlocal complex reductions on some Ablowitz-Kaup-Newell-Segur-type equations, we study some local and nonlocal complex equations, involving the local and nonlocal complex modified Korteweg-de Vries equation, the local and nonlocal complex sine-Gordon equation, the local and nonlocal potential nonlinear Schrödinger equation and the loca…
Hawking's singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a nonnegative effective energy density (EED), thereby guaranteeing the timelike convergence property. However, there are models that do not satisfy the SEC and therefore lie outside the scope of Hawking's hypotheses, an important example being the massive Klein-Gordon field. Here we derive lower bounds on local averages of the EED for solutions to the Klein-Gordon equation, allowing nonzero mass and nonminimal coupling to the scalar curvature. The averages are take…
Strong cosmic censorship for the spherically symmetric Einstein-Maxwell-charged-Klein-Gordon system with positive $Λ$: stability of the Cauchy horizon and $H^1$ extensions
Massive holomorphicity of near-critical dimers and sine-Gordon model
University of Vienna（维也纳大学）；Courant Institute, New York University（纽约大学柯朗数学科学研究所）；PSL University-Dauphine, CNRS, UMR 7534, CEREMADE（巴黎文理研究大学-达芬奇分校，法国国家科学研究中心，联合研究单位7534，CEREMADE）
We study in this paper an abstract class of Klein-Gordon equations: \[ \p_{t}^{2}ϕ(t)- 2ık \p_{t}ϕ(t)+ h ϕ(t)=0, \] where $ϕ: \rr\to \cH$, $\cH$ is a (complex) Hilbert space, and $h$, $k$ are self-adjoint, resp. symmetric operators on $\cH$. We consider their generators $H$ (resp. $K$) in the two natural spaces of Cauchy data, the energy (resp. charge) spaces. We do not assume that the dynamics generated by $H$ or $K$ has any positive conserved quantity, in particular these operators may have complex spectrum. Assuming conditions on $h$ and $k$ which allow to use the theory of selfadjoint oper…
We present a condition on the self-interaction term that guaranties the existence of the global in time solution of the Cauchy problem for the semilinear Klein-Gordon equation in the Friedmann-Lama$\hat{i}$tre-Robertson-Walker model of the contracting universe. For the Klein-Gordon equation with the Higgs potential we give a lower estimate for the lifespan of solution.
This paper investigates wave-equations on spacetimes with a metric which is locally analytic in the time. We use recent results in the theory of the non-characteristic Cauchy problem to show that a solution to a wave-equation vanishing in an open set vanishes in the ``envelope'' of this set, which may be considerably larger and in the case of timelike tubes may even coincide with the spacetime itself. We apply this result to the real scalar field on a globally hyperbolic spacetime and show that the field algebra of an open set and its envelope coincide. As an example there holds an analog of B…
We construct states on the algebra of the Klein-Gordon field that minimize the energy density in homogeneous and in inhomogeneous spacetimes, both with compact Cauchy hypersurfaces. The energy density is measured by geodesic observers and smeared over a spacelike slab of spacetime, entirely containing a Cauchy hypersurface and extended in time. We further show that these states are Hadamard states. The present construction generalizes the construction of States of Low Energy in Robertson-Walker spacetimes presented by Olbermann (2007).
We prove that semiclassical gravity in conformally static, globally hyperbolic spacetimes with a massless, conformally coupled Klein-Gordon field is well posed, when viewed as a coupled theory for the dynamical conformal factor of the metric and the Klein-Gordon theory. Namely, it admits unique and stable solutions whenever constrained fourth-order initial data for the conformal factor and suitably defined Hadamard initial data for the Klein-Gordon state are provided on a spacelike Cauchy surface. As no spacetime symmetries are imposed on the conformal factor, the present result implies that, …
We consider the pseudodifferential operators $H_{m,Ω}$ associated by the prescriptions of quantum mechanics to the Klein-Gordon Hamiltonian $\sqrt{|{\bf P}|^2+m^2}$ when restricted to a compact domain $Ω$ in ${\mathbb R}^d$. When the mass $m$ is 0 the operator $H_{0,Ω}$ coincides with the generator of the Cauchy stochastic process with a killing condition on $\partial Ω$. (The operator $H_{0,Ω}$ is sometimes called the {\it fractional Laplacian} with power 1/2, cf. \cite{Gie}.) We prove several universal inequalities for the eigenvalues $0 < β_1 < β_2 \le >...$ of $H_{m,Ω}$ and their means $\o…
We construct Hadamard states for the Yang-Mills equation linearized around a smooth, space-compact background solution. We assume the spacetime is globally hyperbolic and its Cauchy surface is compact or equal $\rr^d$. We first consider the case when the spacetime is ultra-static, but the background solution depends on time. By methods of pseudodifferential calculus we construct a parametrix for the associated vectorial Klein-Gordon equation. We then obtain Hadamard two-point functions in the gauge theory, acting on Cauchy data. A key role is played by classes of pseudodifferential operators t…
This paper deals with the Klein-Gordon equation on the Poincaré chart of the 5-dimensional Anti-de Sitter universe. When the mass $μ$ is larger than $-{1}{4}$, the Cauchy problem is well posed despite the loss of global hyperbolicity due to the time-like horizon. We express the finite energy solutions in the form of a continuous Kaluza-Klein tower and we deduce a uniform decay as $\mid t\mid^{-{3}{2}}$. We investigate the case $μ={ν^2-1}{2}$, $ν\in\NN^*$, which encompasses the gravitational fluctuations, $ν=4$, and the electromagnetic waves, $ν=2$. The propagation of the wave front set shows t…
This is a survey of the author's recent work rather than a broad survey of the literature. The survey is concerned with the global in time solutions of the Cauchy problem for matter waves propagating in the curved spacetimes, which can be, in particular, modeled by cosmological models. We examine the global in time solutions of some class of semililear hyperbolic equations, such as the Klein-Gordon equation, which includes the Higgs boson equation in the Minkowski spacetime, de Sitter spacetime, and Einstein & de Sitter spacetime. The crucial tool for the obtaining those results is a new appro…
The Semiclassical Einstein-Klein-Gordon System: Asymptotic Analysis of Minkowski Spacetime
University of York（约克大学）；Università di Trento（特伦托大学）；GNFM-INdAM（意大利数学国家研究委员会几何与函数论全国小组）；TIFPA-INFN（意大利国家核物理研究所特伦托前沿物理与分析中心）；Università di Genova（热那亚大学）；INFN（意大利国家核物理研究所）；Universität Regensburg（雷根斯堡大学）
The paper presents a generalization and further development of our recent publications where solutions of the Klein-Fock-Gordon equation defined on a few particular $D=(2+1)$-dim static space-time manifolds were considered. The latter involve toy models of 2-dim spaces with axial symmetry, including dimension reduction to the 1-dim space as a singular limiting case. Here the non-static models of space geometry with axial symmetry are under consideration. To make these models closer to physical reality, we define a set of "admissible" shape functions $ρ(t,z)$ as the $(2+1)$-dim Einstein equatio…
In this paper we study some classes of second order non-homogeneous nonlinear differential equations allowing a specific representation for nonlinear Green's function. In particular, we show that if the nonlinear term possesses a special multiplicativity property, then its Green's function is represented as the product of the Heaviside function and the general solution of the corresponding homogeneous equations subject to non-homogeneous Cauchy conditions. Hierarchies of specific non-linearities admitting this representation are derived. The nonlinear Green's function solution is numerically j…
We develop a scattering theory for the linear wave equation $\Box_g ψ= 0$ on the interior of Reissner-Nordström black holes, connecting the fixed frequency picture to the physical space picture. Our main result gives the existence, uniqueness and asymptotic completeness of finite energy scattering states. The past and future scattering states are represented as suitable traces of the solution $ψ$ on the bifurcate event and Cauchy horizons. The heart of the proof is to show that after separation of variables one has uniform boundedness of the reflection and transmission coefficients of the resu…
We consider the question of scattering for the boson star equation in three space dimensions. This is a semi-relativistic Klein-Gordon equation with a cubic nonlinearity of Hartree type. We combine weighted estimates, obtained by exploiting a special null structure present in the equation, and a refined asymptotic analysis performed in Fourier space, to obtain global solutions evolving from small and localized Cauchy data. We describe the behavior at infinity of such solutions by identifying a suitable nonlinear asymptotic correction to scattering. As a byproduct of the weighted energy estimat…
We study Hilbert space aspects of the Klein-Gordon equation in two-dimensional spacetime. We associate to its restriction to a spacelike wedge a scattering from the past light cone to the future light cone, which is then shown to be (essentially) the Hankel transform of order zero. We apply this to give a novel proof, solely based on the causality of this spatio-temporal wave propagation, of the theorem of de Branges and V. Rovnyak concerning Hankel pairs with a support property. We recover their isometric expansion as an application of Riemann's general method for solving Cauchy-Goursat probl…
We study the spectrum $\{λ_j(m)\}_{j=1}^{\infty}$ of a timelike Killing vector field $Z$ acting as a differential operator $D_Z$ on the Hilbert space of solutions of the massive Klein-Gordon equation $(\Box_g + m^2) u = 0$ on a globally hyperbolic stationary spacetime $(M, g)$ with compact Cauchy hypersurface. The inverse mass $m^{-1}$ is formally like the Planck constant in a Schrödinger equation, and we give Weyl asymptotics as $m \to \infty$ for the number $$N_{ν, C}(m)= \# \{j \mid \frac{λ_j(m)}{m} \in [ν- \frac{C}{m}, ν+ \frac{C}{m} ]\}$$ for a given $C > 0$. The semi-classical mass asymp…
We study solutions to the Yang-Mills-Higgs equations on the maximal Cauchy development of the data given on a ball of radius $R$ in $\mathbb{R}^3$. The energy of the data could be infinite and the solution grows at most inverse polynomially in $R-t$ as $t\rightarrow R$. As applications, we derive pointwise decay estimates for Yang-Mills-Higgs fields in the future of a hyperboloid or in the Minkowski space $\mathbb{R}^{1+3}$ for data bounded in the weighted energy space with weights $|x|^{1+ε}$. Moreover, for the abelian case of Maxwell-Klein-Gordon system, we extend the small data result of Li…
