A Simple Diffractive Boundary Value Problem on an Asymptotically Anti-de Sitter Space

In this paper we describe the propagation of C°° and Sobolev singularities for the wave equation on C°° manifolds with corners M equipped with a Riemannian metric g. That is, for X = M x R t , P = D 2 t - Δ M , and u ∈ H 1 loc (X) solving Pu = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WF b (u) is a union of maximally extended generalized broken bicharacteristics. This result is a C°° counterpart of Lebeau's results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary, [11]. Our methods rely on b-microlocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if M has a smooth boundary (and no corners).

This paper develops a rigorous semiclassical and microlocal framework to analyze energy dissipation and singularity propagation in systems governed by dissipative pseudodifferential operators. We consider semiclassical operators \( P_h \) with symbols \( a(x, \xi) \in S^m(\mathbb{R}^n) \), where \( m < 0 \), and study families of solutions \( u_h \) to \( P_h u_h = 0 \). By constructing microlocal partitions of unity, we derive asymptotic formulas for the local and total dissipation rate \( \varepsilon_h \), proving its convergence to an explicit phase-space integral as \( h \to 0^+ \). Our analysis reveals that the semiclassical wavefront set \( \mathrm{WF}_h(u_h) \) propagates along the generalized bicharacteristics of the real part of the principal symbol \( p_0(x, \xi) \), while the imaginary part \( \mathrm{Im}\, p_0(x, \xi) \leq 0 \) governs the amplitude decay along these flows. This establishes a precise, quantitative relationship between geometric propagation and physical dissipation, unifying the description of singularity transport and energy loss in the semiclassical regime. The framework bridges deterministic and statistical perspectives, offering a robust tool for investigating multiscale energy dynamics in turbulent systems and beyond.

The notion of a second microlocalization along an involutive submanifold WCIS introduced by M. Kashiwara [6] .Y. Laurent developed this notion and established a theory of the 2-microdifferential operators [8]. In [Ill, 3 . Sjostrand defined the second analytic wave front set (and even a kth wave front set) along the lagrangian submanifolds.So he was able to get a result of M. Kashiwara and to generalize it. This result provides very strict relations between the support and the wave front set, and, as a generalization, between the (k-l)th wave front set and the kth one. J.M.Bony gave a definition adapted to the frame of the C∞o-singularities. Finally G. Lebeau spread the notion to the isotropic manifolds and applied it to a degenerated diffraction problem.

We study the propagation of singularities for semilinear Schrodinger equations with quadratic Hamiltonians, in particular for the semilinear harmonic oscillator. We show that the propagation still occurs along the flow the Hamiltonian flow, but for Sobolev regularities in a certain range and provided the notion of Sobolev-wave front set is conveniently modified. The proof makes use of a weighted version of the paradifferential calculus, adapted to our situation. The results can be regarded as the Schrodinger counterpart of those known for semilinear hyperbolic equations, which hold with the usual wave front set.

We survey the correct definition of a generalized Dirac operator on a Space-Time and the classicla result about propagation of singularities. This says that light travels along light-like geodesics. Finally we show this is also true for generalized Dirac operators.

The concept of wave front set was introduced in 1969–1970 by Sato in the hyperfunctions context [1, 34] and by Hörmander [23] in the $\mathcal C^{\infty }$ context. Howe in [25] used the theory of wave front sets in the study of Lie groups representations. Heifetz in [22] defined a notion of wave front set for distributions in the $p$-adic setting and used it to study some representations of $p$-adic Lie groups. In this article, we work in the $k\mathopen{(\!(} t \mathopen{)\!)}$-setting with $k$ a Characteristic 0 field. In that setting, balls are no longer compact but working in a definable context provides good substitutes for finiteness and compactness properties. We develop a notion of definable distributions in the framework of [13] and [14] for which we define notions of singular support and $\Lambda$-wave front sets (relative to some multiplicative subgroups $\Lambda$ of the valued field) and we investigate their behavior under natural operations like pullback, tensor product, and products of distributions.

We consider the spaces of ultradifferentiable functions $$\mathcal {S}_\omega $$ as introduced by Bjorck (and its dual $$\mathcal {S}'_\omega $$ ) and we use time-frequency analysis to define a suitable wave front set in this setting and obtain several applications: global regularity properties of pseudodifferential operators of infinite order and the micro-pseudolocal behaviour of partial differential operators with polynomial coefficients and of localization operators with symbols of exponential growth. Moreover, we prove that the new wave front set, defined in terms of the Gabor transform, can be described using only Gabor frames. Finally, some examples show the convenience of the use of weight functions to describe more precisely the global regularity of (ultra)distributions.

Wave front set for solutions to Schrödinger equations

In this paper we introduce and study global wave front sets in terms of the tau -Wigner transform in global ultradifferentiable classes of Beurling type modulated with weight functions in the sense of Braun, Meise, and Taylor, and we compare it with other wave front sets existing in the literature defined by different time-frequency analysis tools, such as the short-time Fourier transform or Gabor frames. Conditions for the equality of these wave front sets are provided and some examples are given.

A unipotent support for irreducible representations

We discuss the algebraic quantization of a real, massive scalar field in the Poincaré patch of the [Formula: see text]-dimensional anti-de Sitter spacetime, with arbitrary boundary conditions. By using the functional formalism, we show that it is always possible to associate to such system an algebra of observables enjoying the standard properties of causality, time-slice axiom and F-locality. In addition, we characterize the wavefront set of the ground state associated to the system under investigation. As a consequence, we are able to generalize the definition of Hadamard states and construct a global algebra of Wick polynomials.

We consider a model case for a strictly convex domain of dimension $d\geq 2$ with smooth boundary and we describe dispersion for the wave equation with Dirichlet boundary conditions. More specifically, we obtain the optimal fixed time decay rate for the smoothed out Green function: a $t^{1/4}$ loss occurs with respect to the boundary less case, due to repeated occurrences of swallowtail type singularities in the wave front set.

An analysis for a direct calculation of the embeddings in flat spacetimes of static spherically symmetric manifolds with Lorentz metric is worked out. For each manifold with non-constant curvature we arrive at a parametrical embedding which represents an infinite geometrical multiplicity of the embedded surface. The embeddings of manifolds with constant curvature are not parametrical and can be determined univocally. Examples concerning Schwarzschild, Reissner-Weyl, de Sitter and anti-de Sitter spacetimes are considered.

Filtration problems are actual for the design of underground structures and foundations, strengthening of loose soil and construction of watertight walls in the porous rock. A liquid grout pumped under pressure penetrates deep into the porous rock. Solid particles of the suspension retained in the pores, strengthen the loose soil and create watertight partitions. The aim of the study is to construct an explicit analytical solution of the filtration problem. A one-dimensional model of deep bed filtration of a monodisperse suspension in a homogeneous porous medium with size-exclusion mechanism of particles retention is considered. Solid particles are freely transferred by the carrier fluid through large pores and get stuck in the throats of small pores. The mathematical model of deep bed filtration includes the mass balance equation for suspended and retained particles and the kinetic equation for the deposit growth. The model describes the movement of concentrations front of suspended and retained particles in an empty porous medium. Behind the concentrations front, solid particles are transported by a carrier fluid, accompanied by the formation of a deposit. The complex model has no explicit exact solution. To construct the asymptotic solution in explicit form, methods of nonlinear asymptotic analysis are used. The new coordinate transformation allows to obtain a parameter that is small at all points of the porous sample at any time. In this paper, a global asymptotic solution of the filtration problem is constructed using a new small parameter. Numerical calculations are performed for a nonlinear filtration coefficient found experimentally. Calculations confirm the closeness of the asymptotics to the solution in the entire filtration domain. For a nonlinear filtration coefficient, the asymptotics is closer to the numerical solution than the exact solution of the problem with a linear coefficient. The analytical solution obtained in the paper can be used to analyze solutions of problems of underground fluid mechanics and fine-tune laboratory experiments.

The discovery of the radiation properties of black holes prompted the search for a natural candidate quantum ground state for a massless scalar field theory on the Schwarzschild spacetime. Among the several available proposals in the literature, an important physical role is played by the so-called Unruh state which is supposed to be appropriate to capture the physics of a real spherically symmetric black hole formed by collapsing matter. One of the aims of this paper, referring to a massless Klein-Gordon field, is to rigorously construct that state globally, i.e. on the algebra of Weyl observables localized in the union of the static external region, the future event horizon and the non-static black hole region. The Unruh state is constructed following the traditional recipe that it is the vacuum state with respect to the affine parameter U of the geodesic forming the whole past horizon whereas it is the vacum state with respect to the Schwarzschild Killing time t on the past light infinity, interpreting these data within our algebraic formalism. Eventually, making use of the microlocal-analysis approach, we prove that the Unruh state built up following our procedure fulfills the so-called Hadamard condition everywhere it is defined and, hence, it is perturbatively stable, realizing the natural candidate with which one could study purely quantum phenomena such as the role of the back reaction of Hawking's radiation. The achieved results are obtained by means of a bulk-to-boundary reconstruction technique which exploits the Killing (horizon) structure and the conformal asymptotic structure of the underlying background, employing Hormander's theorem on propagation of singularities, some recent results about passive state extended to our case, and a careful analysis of the remaining part of the wavefront set of the state. A crucial technical role is played by the recent results due to Dafermos and Rodnianski on the peeling behaviour of the solutions of Klein-Gordon equations in Schwarzschild spacetime.