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We consider the propagation of surface water waves described by the Boussinesq system. Following (Molinet et al. in Nonlinearity 34:744–775, 2021), we introduce a regularized Boussinesq system obtained by adding a non-local pseudo-differential operator define by g λ [ ζ ] ˆ = | k | λ ζ ˆ k with λ ∈ ] 0 , 2 ] . In this paper, we display a twofold approach: first, we study theoretically the existence of an asymptotic expansion for the solution to the Cauchy problem associated to this regularized Boussinesq system with respect to the regularizing parameter ϵ . Then, we compute numerically the function coefficients of the expansion (in ϵ ) and verify numerically the validity of this expansion up to order 2. We also check the numerical L 2 stability of the numerical algorithm. [ABSTRACT FROM AUTHOR]

In this paper, I explore a specific class of bi-parameter pseudo-differential operators characterized by symbols σ (x 1 , x 2 , ξ 1 , ξ 2) falling within the product-type Hörmander class S ρ , δ m . This classification imposes constraints on the behavior of partial derivatives of σ with respect to both spatial and frequency variables. Specifically, I demonstrate that for each multi-index α , β , the inequality | ∂ ξ α ∂ x β σ (x 1 , x 2 , ξ 1 , ξ 2) | ≤ C α , β (1 + | ξ |) m ∏ i = 1 2 (1 + | ξ i |) − ρ | α i | + δ | β i | is satisfied. My investigation culminates in a rigorous analysis of the L p -boundedness of such pseudo-differential operators, thereby extending the seminal findings of C. Fefferman from 1973 concerning pseudo-differential operators within the Hörmander class. [ABSTRACT FROM AUTHOR]

The pseudo-differential operator with symbol | k |α has been widely used in seismic modelling and imaging when involving attenuation, anisotropy and one-way wave equation, which is usually calculated using the pseudo-spectral method. For large-scale problems, applying high-dimensional Fourier transforms to solve the wave equation that includes pseudo-differential operators is much more expensive than finite-difference approaches, and it is not suitable for parallel computing with domain decomposition. To mitigate this difficulty, we present a truncated space-domain convolution method to efficiently compute the pseudo-differential operator |$\sqrt{-\nabla ^2}$|⁠ , and then apply it to viscoacoustic reverse-time migration. Although |$\sqrt{-\nabla ^2}$| is theoretically non-local in the space domain, we take the limited frequency band of seismic data into account, and constrain the approximated convolution stencil to a finite length. The convolution coefficients are computed by solving a least-squares inverse problem in the wavenumber domain. In addition, we exploit the symmetry of the resulting convolution stencil and develop a fast spatial convolution algorithm. The applications of the proposed method in Q -compensated reverse-time migration demonstrate that it is a good alternative to the pseudo-spectral method for computing the pseudo-differential operator |$\sqrt{-\nabla ^2}$|⁠ , with almost the same accuracy but much higher efficiency. [ABSTRACT FROM AUTHOR]

We characterize the matrix-weighted Triebel–Lizorkin spaces and Besov spaces by Peetre maximal function and approximation. Using these characterizations, we obtain the boundedness of pseudo-differential operators with symbol in Hörmander's class on matrix weighted Besov and Triebel–Lizorkin spaces. [ABSTRACT FROM AUTHOR]

We study two Bregman projection methods for solving variational inequality problems in real reflexive Banach spaces. Our methods have simple and elegant structures, and they require only one Bregman projection onto the feasible set and one evaluation of the cost operator at each iteration. We prove that these methods converge weakly when the cost operator is pseudomonotone on the entire space and that they converge strongly when the cost operator is strongly pseudomonotone only on the feasible set. Extensions of our proposed methods to mixed variational inequality problems are also given. Finally, we consider some examples regarding the implementation of our methods in comparisons with known methods in the literature. [ABSTRACT FROM AUTHOR]

The main aim of this article is to derive certain continuity and boundedness properties of the coupled fractional Fourier transform on Schwartz-like spaces. We extend the domain of the coupled fractional Fourier transform to the space of tempered distributions and then study the mapping properties of pseudo-differential operators associated with the coupled fractional Fourier transform on a Schwartz-like space. We conclude the article by applying some of the results to obtain an analytical solution of a generalized heat equation. [ABSTRACT FROM AUTHOR]

We introduce the control conditions for 0th-order pseudodifferential operators |$\textbf{P}$| whose real parts satisfy the Morse–Smale dynamical condition. We obtain microlocal control estimates under the control conditions. As a result, we show that there are no singular profiles in the solution to the evolution equation |$(i\partial _{t}-\textbf{P})u=f$| when |$\textbf{P}$| has a damping term that satisfies the control condition and |$f\in C^{\infty }$|⁠. This is motivated by the study of a microlocal model for the damped internal waves. [ABSTRACT FROM AUTHOR]

Spectral and singular value symbols are valuable tools to analyse the eigenvalue or singular value distributions of matrix-sequences in the Weyl sense. More recently, Generalized Locally Toeplitz (GLT) sequences of matrices have been introduced for the spectral/singular value study of the numerical approximations of differential operators in several contexts. As an example, such matrix-sequences stem from the large linear systems approximating Partial Differential Equations (PDEs), Fractional Differential Equations (FDEs), Integro Differential Equations (IDEs), using any discretization on reasonable grids via local methods, such as Finite Differences, Finite Elements, Finite Volumes, Isogeometric Analysis, Discontinuous Galerkin etc. Studying the asymptotic spectral behaviour of GLT sequences is useful in analysing classical techniques for the solution of the corresponding PDEs/FDEs/IDEs and in designing novel fast and efficient methods for the corresponding large linear systems or related large eigenvalue problems. The theory of GLT sequences, in combination with the results concerning the asymptotic spectral distribution of perturbed sequences of matrices, is one of the most powerful and successful tools for computing the spectral symbol f. In this regard, it would be beneficial to design an automatic procedure to compute the spectral symbols of such matrix-sequences and Ahmed Ratnani partially pursued it. Here, in the case of one-dimensional and two-dimensional differential problems, we continue in this direction by proposing an automatic procedure for computing the symbol of the underlying sequences of matrices, assuming that it is a GLT sequence satisfying mild conditions. [ABSTRACT FROM AUTHOR]

Given p ∈ (1 , ∞) and the power weight ω (x) = | x + i | β on R + = (0 , ∞) with β ∈ (− 1 / p , 1 − 1 / p) , let L N p (R + , ω) be the weighted Lebesgue space of functions f : R + → C N. The paper deals with index calculation for the Fredholm on L N p (R + , ω) operators of the form A = ∑ j = 1 m a j W (b j) , where a j are continuous on R + matrix coefficients that slowly oscillate at 0 and ∞, and W (b j) are Wiener-Hopf operators with piecewise continuous matrix symbols b j with finite sets of discontinuities on R , which are Mellin multipliers on L N p (R + , ω). To this end we apply a modification of Duduchava's decomposition scheme, results on Fourier and Mellin pseudodifferential operators, and results on convolution type operators with piecewise slowly oscillating data. Fredholm symbols constructed for the Banach algebra generated by the operators of the considered form are N × N matrix functions of specific structure. [ABSTRACT FROM AUTHOR]

We construct discrete analogs of multidimensional singular integral operators and study their invertibility. Moreover, we give a comparison between continual and discrete case. We give the theory of periodic Riemann problem also, because it is needed for studying invertibility of so‐called paired equations. For more general case of pseudo‐differential operators, we construct the solvability theory for discrete pseudo‐differential equations in discrete analogs of Sobolev–Slobodetskii spaces. Some comparison results for discrete and continuous solutions are given also in appropriate discrete normed spaces. [ABSTRACT FROM AUTHOR]

In this paper we discuss discrete pseudo‐differential operators in the context of hypercomplex analysis. We will provide the definitions and establish the necessary calculus which will be used to discuss questions like boundedness and compactness of operators. In the end, we will discuss well‐known examples from hypercomplex analysis as well as give new examples including discrete Dirac operators with non‐constant coefficients. [ABSTRACT FROM AUTHOR]

This paper is devoted to prove the invariance of the microsupport under the influence of the evolution of the solution in L2(ℝn)$$ {L}^2\left({\mathrm{\mathbb{R}}}^n\right) $$ of the following Cauchy problem: 1(hDt+P)u=0,$$ \left(h{D}_t+P\right)u=0, $$ 2u(0,x)=u0(x),$$ u\left(0,x\right)={u}_0(x), $$where h$$ h $$ is a semiclassical parameter, Dt=1i∂∂t$$ {D}_t=\frac{1}{i}\frac{\partial }{\partial_t} $$ and P=Opth(p)$$ P=O{p}_t^h(p) $$ (t∈[0,1]$$ t\in \left[0,1\right] $$) is a pseudo‐differential operator of symbol p(x,ξ,h)=p0(x,ξ)+ε(h)r(x,ξ,h)∈S2nhol(1),$$ p\left(x,\xi, h\right)={p}_0\left(x,\xi \right)+\varepsilon (h)r\left(x,\xi, h\right)\in {S}_{2n}^{hol}(1), $$where ε(h)→0$$ \varepsilon (h)\to 0 $$ as h→0$$ h\to 0 $$ and p0(x,ξ)$$ {p}_0\left(x,\xi \right) $$ is real valued, that is, p0(x,ξ)∈ℝfor all(x,ξ)∈ℝ2n.$$ {p}_0\left(x,\xi \right)\in \mathrm{\mathbb{R}}\kern0.30em \mathrm{for}\ \mathrm{all}\kern0.1em \left(x,\xi \right)\in {\mathrm{\mathbb{R}}}^{2n}. $$ [ABSTRACT FROM AUTHOR]

In this note, we consider the Fourier integral operator T ϕ , a ⁢ f ⁢ (x) = ∫ R n e i ⁢ ϕ ⁢ (x , ξ) ⁢ a ⁢ (x , ξ) ⁢ f ^ ⁢ (ξ) ⁢ d ξ with 푎 in the forbidden Hörmander class S ρ , 1 m and ϕ ∈ Φ 2 satisfying the strong non-degeneracy condition. For 0 ≤ ρ ≤ 1 , set m ⁢ (n , ρ , p) = − (n − ρ) ⁢ (1 2 − 1 p) + n p ⁢ (ρ − 1) . When 2 ≤ p ≤ ∞ , 1 ≤ q ≤ ∞ and s > m − m ⁢ (n , ρ , p) , we show that T ϕ , a is bounded from the Besov space B p , q s to B p , q s − m + m ⁢ (n , ρ , p) . This result is a generalization of some theorems proved by Stein, Meyer, Runst and Bourdaud for the pseudo-differential operator T a with a ∈ S 1 , 1 m , and indices 푠 and m ⁢ (n , ρ , p) are sharp in some cases. [ABSTRACT FROM AUTHOR]

The main aim of this article is to study the Poisson type problem for anisotropic p-Laplace type equation on long cylindrical domains. The rate of convergence is shown to be exponential, thereby improving earlier known results for similar type of operators. The Poincaré inequality for a pseudo p-Laplace operator on an infinite strip-like domain is also studied and the best constant, like in many other situations in literature for other operators, is shown to be the same with the best Poincaré constant of an analogous problem set on a lower dimension. [ABSTRACT FROM AUTHOR]

In seismic modeling and reverse time migration (RTM), incorporating anisotropy is crucial for accurate wavefield modeling and high-quality images. Due to the trade-off between computational cost and simulation accuracy, the pure quasi-P-wave equation has good accuracy to describe wave propagation in tilted transverse isotropic (TTI) media. However, it involves a fractional pseudo-differential operator that depends on the anisotropy parameters, making it unsuitable for resolution using conventional solvers for fractional operators. To address this issue, we propose a novel pure quasi-P-wave equation with a generalized fractional convolution operator in TTI media. First, we decompose the conventional pure quasi-P-wave equation into an elliptical anisotropy equation and a fractional pseudo-differential correction term. Then, we use a generalized fractional convolution stencil to approximate the spatial-domain pseudo-differential term through the solution of an inverse problem. The proposed approximation method is accurate, and the wavefield modeling method based on it also accurately describes quasi-P-wave propagation in TTI media. Moreover, it only increases the computational cost for calculating mixed partial derivatives compared to those in vertical transverse isotropic (VTI) media. Finally, the proposed wavefield modeling method is utilized in RTM to correct the anisotropic effects in seismic imaging. Numerical RTM experiments demonstrate the flexibility and viability of the proposed method. [ABSTRACT FROM AUTHOR]

We consider the inverse random potential scattering problem for the two- and threedimensional biharmonic wave equation in lossy media. The potential is assumed to be a microlocally isotropic Gaussian rough field. The main contributions of the work are twofold. First, the unique continuation principle is proved for the fourth order biharmonic wave equation with rough potentials, and the well-posedness of the direct scattering problem is established in the distribution sense. Second, the correlation strength of the random potential is shown to be uniquely determined by the high frequency limit of the second moment of the backscattering data averaged over the frequency band. Moreover, we demonstrate that the expectation in the data can be removed and the data of a single realization is sufficient for the uniqueness of the inverse problem with probability one when the medium is lossless. [ABSTRACT FROM AUTHOR]

We describe the Friedrichs extension of elliptic symmetric pseudodifferential operators on a closed smooth manifold with the domain consisting of functions vanishing on a given submanifold. In summary, the Friedrichs extension is an elliptic Sobolev problem defined in terms of boundary and coboundary operators, and the number of boundary and coboundary conditions in the problem depends on the order of the operator and the codimension of the submanifold. In this paper, the discreteness of the spectrum is proved, and singularities of eigenfunctions are described. [ABSTRACT FROM AUTHOR]

The probabilistic symbol is the right-hand side derivative of the characteristic functions corresponding to the one-dimensional marginals of a stochastic process. This object, as long as the derivative exists, provides crucial information concerning the stochastic process. For a Lévy process, one obtains the characteristic exponent while the symbol of a (rich) Feller process coincides with the classical symbol which is well known from the theory of pseudodifferential operators. Leaving these classes behind, the most general class of processes for which the symbol still exists are Lévy-type processes. It has been an open question, whether further generalizations are possible within the framework of Markov processes. We answer this question in the present article: within the class of Hunt semimartingales, Lévy-type processes are exactly those for which the probabilistic symbol exists. Leaving Hunt behind, one can construct processes admitting a symbol. However, we show, that the applicability of the symbol might be lost for these processes. Surprisingly, in our proofs the upper and lower Dini derivatives corresponding to certain singular functions play an important rôle. [ABSTRACT FROM AUTHOR]

The critical CR GJMS operator on a strictly pseudoconvex CR manifold is a nonhypoelliptic CR invariant differential operator. We prove that, under the embeddability assumption, it is essentially self-adjoint and has closed range. Moreover, its spectrum is discrete, and the eigenspace corresponding to each non-zero eigenvalue is a finite-dimensional subspace of the space of smooth functions. As an application, we obtain a necessary and sufficient condition for the existence of a contact form with zero CR Q-curvature. [ABSTRACT FROM AUTHOR]

We prove the reducibility of a class of quasi-periodically time dependent linear operators, which are derived from linearizing the dispersive third order Benjamin–Ono (BO) equation on the circle at a small amplitude quasi-periodic function, with a diophantine frequency vector ω ∈ O 0 ⊂ R ν . It is shown that there exists a set O ∞ ⊂ O 0 of asymptotically full Lebesgue measure such that for any ω ∈ O ∞ , the operators can be reduced to the ones with constant coefficients by some linear transformations depending on time quasi-periodically. These transformations include a change of variable induced by a diffeomorphism of the torus, the flow of some partial differential equations and a pseudo-differential operator of order zero. We first reduce the linearized operator of order three to the one with constant coefficients plus a remainder of order zero, and then a perturbative reducibility scheme is performed. The major difficulties encountered are brought by the non-smooth character of the dispersive relation in view of the presence of the Hilbert operator H. We look for several appropriate transformations which are real, reversibility-preserving and satisfy the sharp tame bounds which are used for the reducibility. This work will be the first fundamental step in proving the existence of time quasi-periodic solutions for the dispersive third order BO equation. [ABSTRACT FROM AUTHOR]

Let X , Y be real, infinite-dimensional Banach spaces. Let ℒ ⁢ (X , Y) be the space of bounded operators. An important aspect of understanding differentiability of the operator norm at A ∈ ℒ ⁢ (X , Y) is to estimate the limit (which always exists) lim t → 0 + ⁡ ∥ A + t ⁢ B ∥ - ∥ A ∥ t for ⁢ B ∈ ℒ ⁢ (X , Y) , using the values of B on the state space S A = { τ ∈ ℒ ⁢ (X , Y) ∗ : τ ⁢ (A) = ∥ A ∥ , ∥ τ ∥ = 1 }. In this paper, we give several examples of Banach spaces, including the ℓ p spaces (for 1 < p < ∞ ) where a more tangible estimate is possible, under additional hypotheses on A. We also use the notion of norm-weak upper-semi-continuity (usc, for short) of the preduality map to achieve this. Our results also show that the operator subdifferential limit is related to the corresponding subdifferential limit of the vectors in the range space, when A ∗ ∗ attains its norm. [ABSTRACT FROM AUTHOR]

We study the solvability of discrete elliptic pseudodifferential equations in a sector of the plane. Using special factorization of the symbol, the problem is reduced to a boundary value problem for analytic functions of two variables. A periodic analog of one integral transformation is obtained that was used to construct solutions of elliptic pseudodifferential equations in conical domains. The formula for the general solution of the discrete equation under consideration and some boundary value problems are described in terms of this transformation. [ABSTRACT FROM AUTHOR]

We consider a special boundary value problem in the Sobolev–Slobodetskii space for a model elliptic pseudodifferential equation in a multidimensional cone. Taking into account the special factorization of the elliptic symbol, we write the general solution of the pseudodifferential equation that contains an arbitrary function. To determine it unambiguously, some integral condition is added to the equation, which makes it possible to write the solution in Fourier transforms. [ABSTRACT FROM AUTHOR]

Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time–frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (Grossmann in Commun Math Phys 48(3):191–194, 1976. https://doi.org/10.1007/BF01617867), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (Cohen in J Math Phys 7(5):781–786, 1966. https://doi.org/10.1063/1.1931206) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born–Jordan distribution which originates from the Born–Jordan quantization (Born and Jordan in Z Phys 34(1):858–888, 1925. https://doi.org/10.1007/BF01328531). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born–Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born–Jordan distribution. [ABSTRACT FROM AUTHOR]

We consider a Dirichlet elliptic equation driven by the (p , q) -Laplacian and a reaction which includes a singular term, a parametric convection and a Carathéodory perturbation. Under minimal conditions on the perturbation, using the theory of nonlinear operators of pseudomonotone type, we show that for all small values of the parameter, the problem has a positive solution and determines the properties of the solution set. [ABSTRACT FROM AUTHOR]

We study an evolutionary equation with an operator (i∂/∂x), where φ is a smooth function satisfying certain conditions. As special cases of this equation, we get a partial differential equation of parabolic type with derivatives of finite and infinite orders and an equation with certain operators of fractional differentiation. It is shown that the restriction of the operator (i∂/∂x) to some spaces of type S coincides with a pseudodifferential operator constructed according to the function φ regarded as a symbol. We establish the correct solvability of a nonlocal multipoint (in time) problem for an equation of this kind with an initial function, which is an element of the space of generalized functions of ultradistribution type. The properties of the fundamental solution of this problem are analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a class of pseudo-differential operators on the lattice ℤ n . More preciously, we consider a weighted symbol class M ρ , Λ m (ℤ n × n) , m ∈ ℝ associated to a suitable weight function Λ on ℤ n . We study elements of the symbolic calculus for pseudo-differential operators associated with M ρ , Λ m (ℤ n × n) by deriving formulae for the composition, adjoint and transpose. We define the notion of M -ellipticity for symbols belonging to M ρ , Λ m (ℤ n × n) and construct the parametrix of M -elliptic pseudo-differential operators. Further, we investigate the minimal and maximal extensions for M -elliptic pseudo-differential operators and show that they coincide on ℓ 2 (ℤ n) subject to the M -ellipticity of symbols. We also determine the domains of the minimal and maximal operators. Finally, we discuss Fredholmness and compute the index of M -elliptic pseudo-differential operators on ℤ n . [ABSTRACT FROM AUTHOR]

We consider non-local Schrödinger operators H = - L - V in L 2 (R d) , d ⩾ 1 , where the kinetic terms L are pseudo-differential operators which are perturbations of the fractional Laplacian by bounded non-local operators and V is the fractional Hardy potential. We prove pointwise estimates of eigenfunctions corresponding to negative eigenvalues and upper finite-time horizon estimates for heat kernels. We also analyze the relation between the matching lower estimates of the heat kernel and the ground state near the origin. Our results cover the relativistic Schrödinger operator with Coulomb potential. [ABSTRACT FROM AUTHOR]

We are concerned with the existence as well as the exponential stability in H1-level for the damped defocusing Schrödinger equation posed in a two-dimensional exterior domain Ω with smooth boundary ∂Ω. The proofs of the existence are based on the properties of pseudo-differential operators introduced in Dehman et al. [Math. Z. 254, 729–749 (2006)] and a Strichartz estimate proved by Anton [Bull. Soc. Math. Fr. 136, 27–65 (2008)], while the exponential stability is achieved by combining arguments firstly considered by Zuazua [J. Math. Pures Appl. 9, 513–529 (1991)] for the wave equation adapted to the present context and a global uniqueness theorem. In addition, we proved propagation results for the linear Schrödinger equation for any dimensional and for any (reasonable) boundary conditions employing microlocal analysis. [ABSTRACT FROM AUTHOR]

We prove well-posedness of the Cauchy problem for a class of third order quasilinear evolution equations with variable coefficients in projective Gevrey spaces. The class considered is connected with several equations in Mathematical Physics as the KdV and KdVB equation and some of their many generalizations. [ABSTRACT FROM AUTHOR]

In this paper we study a finite‐depth layer of viscous incompressible fluid in dimension n≥2, modeled by the Navier‐Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity‐capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e., time‐independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension n≥2 and without surface tension in dimension n=2, for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well‐known, to the best of our knowledge this is the first construction of viscous traveling wave solutions. Our proof involves a number of novel analytic ingredients, including: the study of an overdetermined Stokes problem and its underdetermined adjoint problem, a delicate asymptotic development of the symbol for a normal‐stress to normal‐Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

In this paper, bilinear pseudo‐differential operators with symbols in the bilinear Hörmander class BS0,0m$BS^m_{0,0}$ are considered. In particular, the boundedness of these operators on Sobolev spaces is established. Our main result is proved by using symbolic calculus and the boundedness of those operators with certain S0, 0‐type symbols on Lebesgue spaces. [ABSTRACT FROM AUTHOR]

This work concerns the asymptotic analysis of high-frequency wave propagation in randomly layered media with fast variations and long-range correlations. The analysis takes place in the three-dimensional physical space and weak-coupling regime. The role played by the slow decay of the correlations on a propagating pulse is twofold. First we observe a random travel time characterized by a fractional Brownian motion that appears to have a standard deviation larger than the pulse width, which is in contrast with the standard O'Doherty-Anstey theory for random propagation media with mixing properties. Second, a deterministic pulse deformation is described as the solution of a paraxial wave equation involving a pseudodifferential operator. This operator is characterized by the autocorrelation function of the medium fluctuations. In case of fluctuations with long-range correlations this operator is close to a fractional Weyl derivative whose order, between 2 and 3, depends on the power decay of the autocorrelation function. In the frequency domain, the pseudodifferential operator exhibits a frequency-dependent power-law attenuation with exponent corresponding to the order of the fractional derivative, and a frequency-dependent phase modulation, both ensuring the causality of the limiting paraxial wave equation as well as the Kramers-Kronig relations. The mathematical analysis is based on an approximation-diffusion theorem for random ordinary differential equations with long-range correlations. [ABSTRACT FROM AUTHOR]

In this paper, we first study two nonlinear dynamic equations on time scales. By means of the theory of dynamic equations on time scales and the properties of the shift operators δ± as well as the Schauder's fixed point theorem, some existence theorems of almost periodic solution in shifts δ± and pseudo (v-pseudo) almost periodic solution in shifts δ± of the equations are established. Secondly, based on the obtained results, we bring two ecosystems under investigation on some specific time scales to obtain more general results. [ABSTRACT FROM AUTHOR]

In this paper, we study small solutions of the non-linear Klein–Gordon equation on R d with quadratic potential. The goal is to understand for which masses, we can apply a normal form procedure. We prove two mains theorems. Our first contribution gives explicit for which the procedure of partial normal form works due to algebraic assumptions. The second one shows that this strategy works for almost all m in the sense of Lebesgue measure. [ABSTRACT FROM AUTHOR]

In this paper, we study some operator theoretical properties of pseudo-differential operators with operator-valued symbols on the Heisenberg motion group. Specifically, we investigate L 2 – L p boundedness of pseudo-differential operators on the Heisenberg motion group for the range 2 ≤ p ≤ ∞. We also provide a necessary and sufficient condition on the operator-valued symbols in terms of λ-Weyl transforms such that the corresponding pseudo-differential operators on the Heisenberg motion group are in the class of Hilbert–Schmidt operators. As a consequence, we obtain a characterization of the trace class pseudo-differential operators on the Heisenberg motion group and provide a trace formula for these trace class operators. [ABSTRACT FROM AUTHOR]

Ground states are a well-known class of Hadamard states in smooth spacetimes. In this paper, we show that the ground state of the Klein–Gordon field in a non-smooth ultrastatic spacetime is an adiabatic state. The order of the state depends linearly on the regularity of the metric. We obtain the result by combining a propagation of singularities result for non-smooth pseudodifferential operators, properties of the causal propagator, and eigenvalue asymptotics for elliptic operators of low regularity. [ABSTRACT FROM AUTHOR]

This article is devoted to the spectral analysis of the electromagnetic Schrödinger operator on the Euclidean plane. In the semiclassical limit, we derive a pseudo-differential effective operator that allows us to describe the spectrum in various situations and appropriate regions of the complex plane. Not only results of the self-adjoint case are proved (or recovered) in the proposed unifying framework, but also new results are established when the electric potential is complex-valued. In such situations, when the non-self-adjointness comes with its specific issues (lack of a "spectral theorem", resolvent estimates), the analogue of the "low-lying eigenvalues" of the self-adjoint case are still accurately described and the spectral gaps estimated. [ABSTRACT FROM AUTHOR]

We study properties of pseudodifferential operators which arise in their use in boundary value problems. Smooth domains as well as intersections of smooth domains are considered. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of pseudodifferential operators known as time-frequency localization operators, which depend on a symbol ς and two windows functions g 1 and g 2 . We first present some basic properties of the windowed Opdam–Cherednik transform. Then, we use modulation spaces associated with the Opdam–Cherednik transform as appropriate classes for symbols and windows and study the boundedness and compactness of the localization operators associated with the windowed Opdam–Cherednik transform on modulation spaces. Finally, we show that these operators are in the Schatten–von Neumann class. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of solutions to various Dirichlet sublinear-type problems involving the fractional Laplacian when the fractional parameter s tends to zero. Depending on the type on nonlinearity, positive solutions may converge to a characteristic function or to a positive solution of a limit nonlinear problem in terms of the logarithmic Laplacian, that is, the pseudodifferential operator with Fourier symbol ln (| ξ | 2) . In the case of a logistic-type nonlinearity, our results have the following biological interpretation: in the presence of a toxic boundary, species with reduced mobility have a lower saturation threshold, higher survival rate, and are more homogeneously distributed. As a result of independent interest, we show that sublinear logarithmic problems have a unique least-energy solution, which is bounded and Dini continuous with a log-Hölder modulus of continuity. [ABSTRACT FROM AUTHOR]

A model elliptic pseudodifferential equation in a polyhedral cone is considered, and the situation when some of the parameters of the cone tend to their limiting values is investigated. In Sobolev–Slobodetskii spaces, a solution of the equation in the cone is constructed in the case of a special wave factorization of the elliptic symbol. It is shown that a limit solution of the boundary value problem with an additional integral condition can exist only under additional constraints on the boundary function. [ABSTRACT FROM AUTHOR]

We discuss the solvability of two Cauchy problems in matrix pseudodifferential operators. The first is associated with a set of matrix pseudodifferential operators of negative order, a prominent example being the set of strict integral operator parts of products of a solution of the -hierarchy, where is a maximal commutative subalgebra of . We show that it can be solved in the case of compatibility completeness of the adopted setting. The second Cauchy problem is slightly more general and relates to a set of matrix pseudodifferential operators of order zero or less. The key example here is the collection of integral operator parts of the different products of a solution of the strict -hierarchy. This system is solvable if two properties hold: the Cauchy solvability in dimension and the compatibility completeness. Both conditions are shown to hold in the formal power series setting. [ABSTRACT FROM AUTHOR]

In this paper we derive generalizations of comparison results for semimartingales. Our results are based on Markov projections and on known comparison results for Markov processes. The first part of the paper is concerned with an alternative method for the construction of Markov projections of semimartingales. In comparison to the construction in Bentata and Cont (2009) which is based on the solution of a well-posed martingale problem, we make essential use of pseudo-differential operators as investigated in Böttcher (2008) and of fundamental solutions of related evolution problems. This approach allows to dismiss with some boundedness assumptions on the differential characteristics in the martingale approach. As consequence of the construction of Markov projections, comparison results for path-independent functions (European options) of semimartingales can be reduced to the well investigated problem of comparison of Markovian semimartingales. The Markov projection approach to comparison results does not require one of the semimartingales to be Markovian, which is a common assumption in literature. An idea of Brunick and Shreve (2013) to mimick updated processes leads to a related reduction result to the Markovian case and thus to the comparison of related generators. As consequence, a general comparison result is also obtained for path-dependent functions of semimartingales. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the L2 boundedness of the Fourier integral operator Tø,a with smooth and rough symbols and phase functions which satisfy certain non-degeneracy conditions. In particular, if the symbol a ∈ L ∞ S ρ m , the phase function ø satisfies some measure conditions and ∥ ∇ ξ k ϕ (⋅ , ξ) ∥ L ∞ ≤ C ∣ ξ ∣ ϵ − k for all k ≥ 2,ξ ≠ 0, and some ϵ > 0, we obtain that Tø,a is bounded on L2 if m < n 2 min { ρ − 1 , − ϵ 2 } . This result is a generalization of a result of Kenig and Staubach on pseudo-differential operators and it improves a result of Dos Santos Ferreira and Staubach on Fourier integral operators. Moreover, the Fourier integral operator with rough symbols and inhomogeneous phase functions we study in this paper can be used to obtain the almost everywhere convergence of the fractional Schrödinger operator. [ABSTRACT FROM AUTHOR]

We study the WKB analysis of the solutions to the linear problem for a modified affine Toda field equation, which is equivalent to the higher-order ordinary differential equation (ODE) studied in the ODE/IM correspondence. After gauge transformation, we diagonalize the flat connection of the linear problem to reduce the latter to a set of independent first-order linear differential equations. We explicitly perform this procedure for classical affine Lie algebras with lower ranks. In particular, we study the WKB solutions of the D r 1 - and D r + 1 2 -type linear problems, which correspond to the higher-order ODEs with the pseudo-differential operator. The diagonalized connection is obtained from the Riccati equations of the adjoint linear problem and related to the conserved currents of the integrable hierarchy constructed by Drinfeld and Sokolov up to total derivatives. [ABSTRACT FROM AUTHOR]

Using a Hilbert space framework inspired by the methods of orthogonal projections and Hodge decompositions, we study a general class of problems (called Z-problems) that arise in effective media theory, especially within the theory of composites, for defining the effective operator. A new and unified approach is developed, based on block operator methods, for obtaining solutions of the Z-problem, formulas for the effective operator in terms of the Schur complement, and associated variational principles (e.g., the Dirichlet and Thomson minimization principles) that lead to upper and lower bounds on the effective operator. In the case of finite-dimensional Hilbert spaces, this allows for a relaxation of the standard hypotheses on positivity and invertibility for the classes of operators usually considered in such problems by replacing inverses with the Moore–Penrose pseudoinverse. As we develop the theory, we show how it applies to the classical example from the theory of composites on the effective conductivity in the periodic conductivity problem in the continuum (2d and 3d) under the standard hypotheses. After that, we consider the following three important and diverse examples (increasing in complexity) of discrete electrical network problems in which our theory applies under the relaxed hypotheses. First, an operator-theoretic reformulation of the discrete Dirichlet-to-Neumann (DtN) map for an electrical network on a finite linear graph is given and used to relate the DtN map to the effective operator of an associated Z-problem. Second, we show how the classical effective conductivity of an electrical network on a finite linear graph is essentially the effective operator of an associated Z-problem. Finally, we consider electrical networks on periodic linear graphs and develop a discrete analog to the classical example of the periodic conductivity equation and effective conductivity in the continuum. [ABSTRACT FROM AUTHOR]

In this paper, we develop a Direct Discontinuous Galerkin (DDG) method for solving a time dependent partial differential equation with convection-diffusion terms and a nonlocal term which is a pseudo-differential operator of order α ∈ (1 , 2). This kind of equation was first introduced to describe morphodynamics of dunes and was then used for signal processing methods. We consider the DDG method which is based on the direct weak formulation of the PDE into the DG function space for both numerical solution and test functions. Suitable numerical fluxes for all operators are then introduced. We prove nonlinear stability estimates along with convergence results. Finally numerical experiments are given to illustrate qualitative behaviors of solutions and to confirm convergence results. [ABSTRACT FROM AUTHOR]

In this note we study the properties of a sequence of approximate propagators for the Schrödinger equation, in the spirit of Feynman's path integrals. Precisely, we consider Hamiltonian operators arising as the Weyl quantization of a quadratic form in phase space, plus a bounded potential perturbation in the form of a pseudodifferential operator with a rough symbol. The corresponding Schrödinger propagator belongs to the class of generalized metaplectic operators, a fact that naturally motivates the introduction of a manageable time-slicing approximation scheme consisting of operators of the same type. By leveraging on this design and techniques of wave packet analysis we are able to prove several convergence results with precise rates in terms of the mesh size of the time slicing subdivision, even stronger then those that can be achieved under the same assumptions using the standard Trotter approximation scheme. In particular, we prove convergence in the norm operator topology in |$L^2$|⁠ , as well as pointwise convergence of the corresponding integral kernels for non-exceptional times. [ABSTRACT FROM AUTHOR]