Showing papers in "Communications in Mathematical Physics in 1983"

Thermodynamics of black holes in anti-de Sitter space

Abstract: The Einstein equations with a negative cosmological constant admit black hole solutions which are asymptotic to anti-de Sitter space. Like black holes in asymptotically flat space, these solutions have thermodynamic properties including a characteristic temperature and an intrinsic entropy equal to one quarter of the area of the event horizon in Planck units. There are however some important differences from the asymptotically flat case. A black hole in anti-de Sitter space has a minimum temperature which occurs when its size is of the order of the characteristic radius of the anti-de Sitter space. For larger black holes the red-shifted temperature measured at infinity is greater. This means that such black holes have positive specific heat and can be in stable equilibrium with thermal radiation at a fixed temperature. It also implies that the canonical ensemble exists for asymptotically anti-de Sitter space, unlike the case for asymptotically flat space. One can also consider the microcanonical ensemble. One can avoid the problem that arises in asymptotically flat space of having to put the system in a box with unphysical perfectly reflecting walls because the gravitational potential of anti-de Sitter space acts as a box of finite volume.

Nonlinear Schrödinger equations and sharp interpolation estimates

Abstract: A sharp sufficient condition for global existence is obtained for the nonlinear Schrodinger equation $$\begin{array}{*{20}c} {(NLS)} & {2i\phi _t + \Delta \phi + \left| \phi \right|^{2\sigma } \phi = 0,} & {x \in \mathbb{R}^N } & {t \in \mathbb{R}^ + } \\ \end{array} $$ in the case σ=2/N. This condition is in terms of an exact stationary solution (nonlinear ground state) of (NLS). It is derived by solving a variational problem to obtain the “best constant” for classical interpolation estimates of Nirenberg and Gagliardo.

Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids

Abstract: The equations of motion of compressible viscous and heat-conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region. The global solution in time is proved to exist uniquely and approach the stationary state ast→∞, provided the prescribed initial data and the external force are sufficiently small.

Absence of diffusion in the Anderson tight binding model for large disorder or low energy

Abstract: We prove that the Green's function of the Anderson tight binding Hamiltonian decays exponentially fast at long distances on ℤ v , with probability 1. We must assume that either the disorder is large or the energy is sufficiently low. Our proof is based on perturbation theory about an infinite sequence of block Hamiltonians and is related to KAM methods.

On the Schrodinger Equation and the Eigenvalue Problem

Abstract: If λk is thekth eigenvalue for the Dirichlet boundary problem on a bounded domain in ℝn, H. Weyl's asymptotic formula asserts that\(\lambda _k \sim C_n \left( {\frac{k}{{V(D)}}} \right)^{2/n} \), hence\(\sum\limits_{i = 1}^k {\lambda _i \sim \frac{{nC_n }}{{n + 2}}k^{\frac{{n + 2}}{n}} V(D)^{ - 2/n} } \). We prove that for any domain and for all\(\sum\limits_{i = 1}^k {\lambda _i \geqq \frac{{nC_n }}{{n + 2}}k^{\frac{{n + 2}}{n}} V(D)^{ - 2/n} } \). A simple proof for the upper bound of the number of eigenvalues less than or equal to -α for the operator Δ−V(x) defined on ℝn (n≧3) in terms of\(\int\limits_{\mathbb{R}^n } {(V + \alpha )_ - ^{n/2} } \) is also provided.

Supersymmetry and the Atiyah-Singer Index Theorem

Abstract: Using a recently introduced index for supersymmetric theories, we present a simple derivation of the Atiyah-Singer index theorem for classical complexes and itsG-index generalization using elementary properties of quantum mechanical supersymmetric systems.

Convergence of the Viscosity Method for Isentropic Gas Dynamics

Abstract: A convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established. Convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves.

Nonlinear Schrödinger equations and simple Lie algebras

Abstract: We associate a system of integrable, generalised nonlinear Schrodinger (NLS) equations with each Hermitian symmetric space. These NLS equations are considered as reductions of more general systems, this time associated with a reductive homogeneous space. The nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space. The Hamiltonian structure is investigated using “r-matrix” techniques and shown to be “canonical” for all these equations. Throughout the reduction procedure this Hamiltonian structure does not degenerate. Each of the above systems of equations is gauge equivalent to a generalised ferromagnet. Reductions of the latter are discussed in terms of the corresponding NLS type equations.

Positive mass theorems for black holes

Abstract: We extend Witten's proof of the positive mass theorem at spacelike infinity to show that the mass is positive for initial data on an asymptotically flat spatial hypersurface Σ which is regular outside an apparent horizonH. In addition, we prove that if a black hole has electromagnetic charge, then the mass is greater than the modulus of the charge. These results are also valid for the Bondi mass at null infinity. Finally, in the case of the Einstein equation with a negative cosmological constant, we show that a suitably defined mass is positive for data on an asymptotically anti-de Sitter surface Σ which is regular outside an apparent horizon.

On the Construction of Monopoles

Abstract: We show that any self-dual SU (2) monopole may be constructed either by Ward's twistor method, or Nahm's use of the ADHM construction. The common factor in both approaches is an algebraic curve whose Jacobian is used to linearize the non-linear ordinary differential equations which arise in Nahm's method. We derive the non-singularity condition for the monopole in terms of this curve and apply the result to prove the regularity of axially symmetric solutions.

Bounds for effective parameters of heterogeneous media by analytic continuation

Abstract: We give a mathematical formulation of a method for obtaining bounds on effective parameters developed by D. Bergman and G. W. Milton. This method, in contrast to others used before, does not rely on a variational principle, but exploits the properties of the effective parameter as an analytic function of the component parameters. The method is at present restricted to two-component media.

Almost periodic Schrödinger operators: III. The absolutely continuous spectrum in one dimension

Abstract: We studyH=−d2/dx2+V(x) withV(x) limit periodic, e.g.V(x)=Σancos(x/2n) with Σ∣an∣<∞. We prove that for a genericV (and for generican in the explicit example), σ(H) is a Cantor (≡ nowhere dense, perfect) set. For a dense set, the spectrum is both Cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous.

Cauchy problems for the conformal vacuum field equations in general relativity

Abstract: Cauchy problems for Einstein's conformal vacuum field equations are reduced to Cauchy problems for first order quasilinear symmetric hyperbolic systems. The “hyperboloidal initial value” problem, where Cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two-surface, is discussed and translated into the conformally related picture. It is shown that for conformal hyperboloidal initial data of classH S,s≧4, there is a unique (up to questions of extensibility) development which is a solution of the conformal vacuum field equations of classH S. It provides a solution of Einstein's vacuum field equations which has a smooth structure at past null infinity.

Stable standing waves of nonlinear Klein-Gordon equations

Abstract: In this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear Klein-Gordon equations.

Some Remarks on BRS Transformations, Anomalies and the Cohomology of the Lie Algebra of the Group of Gauge Transformations

Abstract: We show that ghosts in gauge theories can be interpreted as Maurer-Cartan forms in the infinite dimensional group ℊ of gauge transformations. We examine the cohomology of the Lie algebra of ℊ and identify the coboundary operator with the BRS operator. We describe the anomalous terms encountered in the renormalization of gauge theories (triangle anomalies) as elements of these cohomology groups.

Symmetries of cosmological Cauchy horizons

Abstract: We consider analytic vacuum and electrovacuum spacetimes which contain a compact null hypersurface ruled byclosed null generators. We prove that each such spacetime has a non-trivial Killing symmetry. We distinguish two classes of null surfaces, degenerate and non-degenerate ones, characterized by the zero or non-zero value of a constant analogous to the “surface gravity” of stationary black holes. We show that the non-degenerate null surfaces are always Cauchy horizons across which the Killing fields change from spacelike (in the globally hyperbolic regions) to timelike (in the acausal, analytic extensions).

$H$-surfaces in Lorentzian manifolds

Abstract: We consider surfaces of prescribed mean curvature in a Lorentzian manifold and show the existence of a foliation by surfaces of constant mean curvature.

Potentials for the supersymmetric nonlinear σ-model

Abstract: The most general structure for potential terms compatible withN=1,N=2, andN=4 supersymmetry in the nonlinear σ-model in two space-time dimensions is determined. The differential geometry of the internal manifold of the model plays an important role in the method used and in the results. An interesting application of nontrivial dimensional reduction is found.

Unitary representations of non-compact supergroups

Abstract: We give a general theory for the construction of oscillator-like unitary irreducible representations (UIRs) of non-compact supergroups in a super Fock space. This construction applies to all non-compact supergroupsG whose coset spaceG/K with respect to their maximal compact subsupergroupK is “Hermitean supersymmetric”. We illustrate our method with the example of SU(m, p/n+q) by giving its oscillator-like UIRs in a “particle state” basis as well as “supercoherent state basis”. The same class of UIRs can also be realized over the “super Hilbert spaces” of holomorphic functions of aZ variable labelling the coherent states.

The existence of a black hole due to condensation of matter

Abstract: When enough matter is condensed in a small region, gravitational effects will be strong enough to cause collapse and a black hole will be formed. We formulate and prove here such a statement in the language of general relativity. (This is Theorem 2 of this paper.)

Kotani theory for one dimensional stochastic Jacobi matrices

Abstract: We consider families of operators,Hω, on l2 given by (Hωu)(n)=u(n+1)+u(n−1)+Vω(n)u(n), whereVω is a stationary bounded ergodic sequence We prove analogs of Kotani's results, including that for ae ω,σac(Hω) is the essential closure of the set ofE where γ(E) the Lyaponov index, vanishes and the result that ifVω is non-deterministic, then σac is empty

On a sharp transition from area law to perimeter law in a system of random surfaces

Abstract: We introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a Bernoulli system of random plaquettes. The transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface, spanning a prescribed loop. As such, this transition offers a generalization of the bond percolation phenomenon. At low plaquette densities, the probability for large loops is shown to decay exponentially with the loops' area, whereas for high densities the decay is by a perimeter law. Furthermore, we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds. Thus, if a natural conjecture about the phase structure of the bond percolation model is true, then there is a sharp transition in the asymptotic behavior of the surface events. Our analysis incorporates block variables, in terms of which a non-critical system is transformed into one which is close to a trivial, high or low density, fixed point. Stochastic geometric effects like those discussed here play an important role in lattice gauge theories.

Gaussian ensembles of random hermitian matrices intermediate between orthogonal and unitary ones

Abstract: A Gaussian ensemble of Hermitian matrices depending on a parameter α is considered. When α=0, the ensemble is Gaussian Orthogonal, and when α=1, it is Gaussian Unitary. An analytic expression for then-level correlation and cluster functions is given for anyn and 0≦α≦1. This ensemble is of relevance in the study of time reversal symmetry breaking of nuclear interactions.

Pseudodifferential Operators on Supermanifolds and the Atiyah-singer Index Theorem

Abstract: Fermionic quantization, or Clifford algebra, is combined with pseudodifferential operators to simplify the proof of the Atiyah-Singer index theorem for the Dirac operator on a spin manifold.

On the iteration of a rational function: computer experiments with Newton's method

Abstract: Using Newton's method to look for roots of a polynomial in the complex plane amounts to iterating a certain rational function. This article describes the behavior of Newton iteration for cubic polynomials. After a change of variables, these polynomials can be parametrized by a single complex parameter, and the Newton transformation has a single critical point other than its fixed points at the roots of the polynomial. We describe the behavior of the orbit of the free critical point as the parameter is varied. The Julia set, points where Newton's method fail to converge, is also pictured. These sets exhibit an unexpected stability of their gross structure while the changes in small scale structure are intricate and subtle.

An uncertainty principle for fermions with generalized kinetic energy

Abstract: We derive semiclassical upper bounds for the number of bound states and the sum of negative eigenvalues of the one-particle Hamiltoniansh=f(−i∇)+V(x) acting onL2(ℝn). These bounds are then used to derive a lower bound on the kinetic energy\(\sum\limits_{j = 1}^N {\left\langle {\psi ,f( - i abla _j )\psi } \right\rangle }\) for anN-fermion wavefunction ψ. We discuss two examples in more detail:f(p)=|p| andf(p)=(p2+m2)1/2−m, both in three dimensions.

Log hölder continuity of the integrated density of states for stochastic Jacobi matrices

Abstract: We consider the integrated density of states,k(E), of a general operator on l2(ℤv) of the formh=h0+v, where\((h_0 u)(n) = \sum\limits_{\left| i \right| = 1} {u(n + i)} \) and (vu)(n)=v(n)u(n), wherev is a general bounded ergodic stationary process on ℤv. We show that |k(E)−k(E′)|≦C[−log(|E−E′|]−1 when |E−E′|≦1/2, The key is a “Thouless formula for the strip.”

Large solutions for harmonic maps in two dimensions

Abstract: We seek critical points of the functionalE(u)=\(\mathop \smallint \limits_\Omega\)|βu|2, where Ω is the unit disk in ℝ2 andu:Ω→S2 satisfies the boundary conditionu=γ on ∂Ω. We prove that if γ is not a constant, thenE has a local minimum which is different from the absolute minimum. We discuss in more details the case where γ(x, y)=(Rx,Ry,\(\sqrt {1 - R^2 }\)) andR<1.

Bifurcation to infinitely many sinks

Abstract: This paper considers one parameter families of diffeomorphisms {Ft} in two dimensions which have a curve of dissipative saddle periodic pointsPt, i.e.Ftn(Pt)=Pt and |detDFtn(Pt)|<1. The family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofPt as the parameter varies throught0. Gavirlov and Silnikov proved that if the new homoclinic intersections are created nondegenerately att0, then there is an infinite cascade of periodic sinks, i.e. there are parameter valuestn accumulating att0 for which there is a sink of periodn [GS2, Sect. 4]. We show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately. We give computer evidence to show that this latter result is probably applicable to the Henon map forA near 1.392 andB equal −0.3.

Existence of the exponentially localised Wannier functions

Abstract: A partial answer (Theorem 1 below) to a problem concerning analytic and periodic families of projections in Hilbert spaces is given. As a consequence the existence of exponentially localised Wannier functions corresponding to nondegenerated bands of arbitrary three-dimensional crystals is proved.