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We introduce the logarithmic analogue of the Laplace-Beltrami operator on Ahlfors regular metric-measure spaces. This operator is intrinsically defined with spectral properties analogous to those of elliptic pseudo-differential operators on Riemannian manifolds. Specifically, its heat semigroup consists of compact operators which are trace-class after some critical point in time. Moreover, its domain is a Banach module over the Dini continuous functions and every Hölder continuous function is a smooth vector. Finally, the operator is compatible, in the sense of noncommutative geometry, with the action of a large class of non-isometric homeomorphisms. [ABSTRACT FROM AUTHOR]

This work considers the Neumann eigenvalue problem for the weighted Laplacian on a Riemannian manifold (M , g , ∂ M) under a singular perturbation. This perturbation involves the imposition of vanishing Dirichlet boundary conditions on a small portion of the boundary. We derive an asymptotic expansion of the perturbed eigenvalues as the Dirichlet part shrinks to a point x ∗ ∈ ∂ M in terms of the spectral parameters of the unperturbed system. This asymptotic expansion demonstrates the impact of the geometric properties of the manifold at a specific point x ∗ . Furthermore, it becomes evident that the shape of the Dirichlet region holds significance as it impacts the first terms of the asymptotic expansion. A crucial part of this work is the construction of the singularity structure of the restricted Neumann Green's function which may be of independent interest. We employ a fusion of layer potential techniques and pseudo-differential operators during this work. [ABSTRACT FROM AUTHOR]

Compared with the transverse isotropic (TI) medium, the orthorhombic anisotropic medium has both horizontal and vertical symmetry axes and it can be approximated as a set of vertical fissures developed in a group of horizontal strata. Although the full-elastic orthorhombic anisotropic wave equation can accurately simulate seismic wave propagation in the underground media, a huge computational cost is required in seismic modeling, migration, and inversion. The conventional coupled pseudo-acoustic wave equations based on acoustic approximation can be used to significantly reduce the cost of calculation. However, these equations usually suffer from unwanted shear wave artifacts during wave propagation, and the presence of these artifacts can significantly degrade the imaging quality. To solve these problems, we derived a new pure P-wave equation for orthorhombic media that eliminates shear wave artifacts while compromising computational efficiency and accuracy. In addition, the derived equation involves pseudo-differential operators and it must be solved by 3D FFT algorithms. In order to reduce the number of 3D FFT, we utilized the finite difference and pseudo-spectral methods to conduct 3D forward modeling. Furthermore, we simplified the equation by using elliptic approximation and implemented 3D reverse-time migration (RTM). Forward modeling tests on several homogeneous and heterogeneous models confirm that the accuracy of the new equation is better than that of conventional methods. 3D RTM imaging tests on three-layer and SEG/EAGE 3D salt models confirm that the ORT media have better imaging quality. [ABSTRACT FROM AUTHOR]

Let L = - Δ + | x | 2 be the Hermite operator on R n , and T be a Calderon–Zygmund type operator that is modelled on certain singular integrals related to L . We establish necessary and sufficient conditions for T to be bounded on various function spaces including the Hardy spaces and the Lipschitz spaces associated to L . We then apply our results to study the boundedness of the Riesz transforms and pseudo-multipliers associated to L . [ABSTRACT FROM AUTHOR]

Hausdorff–Young’s inequality establishes the boundedness of the Fourier transform from L p to L q spaces for 1 ≤ p ≤ 2 and q = p ′ , where p ′ denotes the Lebesgue-conjugate exponent of p. This paper extends this classical result by characterizing the L p - L q boundedness of metaplectic operators, which play a significant role in harmonic analysis. We demonstrate that metaplectic operators are bounded on Lebesgue spaces if and only if their symplectic projection is either free or lower block triangular. As a byproduct, we identify metaplectic operators that serve as homeomorphisms of L p spaces. To achieve this, we leverage a parametrization of the symplectic group by Dopico and Johnson. We use our findings to provide boundedness results within L p spaces for pseudodifferential operators with symbols in Lebesgue spaces, and quantized by means of metaplectic operators. These quantizations consists of shift-invertible metaplectic Wigner distributions, which are essential to measure local phase-space concentration of signals. Using the factorization by Dopico and Johnson, we infer a decomposition law for metaplectic operators on L 2 (R 2 d) in terms of shift-invertible metaplectic operators, establish the density of shift-invertible symplectic matrices in Sp (2 d , R) , and prove that the lack of shift-invertibility prevents metaplectic Wigner distributions to define the so-called modulation spaces M p (R d) . [ABSTRACT FROM AUTHOR]

This study explores the L p {L^{p}} -boundedness and compactness characteristics of SG pseudo-differential operators within the Sobolev space framework. We analyze different attributes of minimal-maximal pseudo-differential operators using the SG symbol σ ⁢ ( x , λ ) {\sigma(x,\lambda)} within the class S m 1 , m 2 {S^{m_{1},m_{2}}} , where m 1 {m_{1}} and m 2 {m_{2}} are real numbers. The investigation delves into various aspects of these operators, and we derive a weak solution for the SG pseudo-differential equation by employing the Dunkl transform as part of the aforementioned theoretical framework. [ABSTRACT FROM AUTHOR]

The main aim of this article is to focus on the fundamental properties of the gyrator transform on the Schwartz spaces. A more generalization Hörmander symbol class is introduced and studied the boundedness properties of pseudo‐differential operators associated with the gyrator transform. Integral representation and kernel of the pseudo‐differential operators are derived. It is shown that pseudo‐differential operators satisfies certain norm inequalities on Sobolev space. As an application, we have successfully applied the gyrator transform to solve heat equation and Fredholm integral equation. [ABSTRACT FROM AUTHOR]

We study ground states of a relativistic Fermi system involved with the pseudo-differential operator - c 2 Δ + c 4 m 2 - c 2 m in the L 2 -subcritical case, where m > 0 denotes the rest mass of fermions, and c ≥ 1 represents the speed of light. By employing Green's function and the variational principle of many-fermion systems, we prove the existence of ground states for the system. The asymptotic behavior of ground states for the system is also analyzed in the non-relativistic limit where c → ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we first give characterizations of weighted Besov spaces with variable exponents via Peetre's maximal functions. Then we obtain decomposition characterizations of these spaces by atom, molecule and wavelet. As an application, we obtain the boundedness of the pseudo-differential operators on these spaces. [ABSTRACT FROM AUTHOR]

The novel fractional Stockwell transform is extended to the suitably defined Schwartz space and the generalized fractional Stockwell transform is introduced. The pseudo-differential operator associated with the fractional Stockwell transform is defined and it is proved to be continuous on the Schwartz space. Examples are illustrated for the generalized fractional Stockwell transform. The generalized convection-diffusion equation is solved using the fractional Stockwell transform. [ABSTRACT FROM AUTHOR]

The term Kirchhoff migration refers to a collection of approximate linearized inversion formulas for solving the inverse problem of seismic tomography which entails reconstructing the Earth's subsurface from reflected wave fields. A number of such formulas exists, the first dating from the 1950 s. As far as we know, these formulas have not yet been mathematically compared with respect to their imaging properties. This shortcoming is to be alleviated by the present work: we systematically discuss the advantages and disadvantages of the formulas in 2D from a microlocal point of view. To this end we consider the corresponding imaging operators in an unified framework as pseudodifferential or Fourier integral operators. Numerical examples illustrate the theoretical insights and allow a visual comparison of the different formulas. [ABSTRACT FROM AUTHOR]

In this paper, we establish the sharp Calderón–Vaillancourt theorem on L p spaces for bi-parameter and bilinear pseudo-differential operators with symbols of critical order by deriving a sufficient and necessary condition on its symbol. This sharpens the result of [G. Lu and L. Zhang, Bi-parameter and bilinear Calderón–Vaillancourt theorem with subcritical order, Forum Math. 28 2016, 6, 1087–1094] which was only proved for symbols of subcritical order. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new type of pseudo-differential equations naturally connected with non-archimedean pseudo-differential operators and whose symbols are new classes of negative definite functions in the p-adic context and in arbitrary dimension. These equations are proposed as a mathematical models to study the spreading of infectious diseases (say COVID-19) through a random walk on a complex energy landscape and taking into account social clusters in a situation of extreme social isolation. [ABSTRACT FROM AUTHOR]

We consider a partial data Calderón problem for elliptic boundary value problems with unbounded coefficients. In particular, we will show that partial measurement of the Neumann-Dirichlet data for the operator $ \Delta + q $ can uniquely determine $ q\in L^{n/2}(\Omega) $. This requires that we construct an explicit Green's function for the conjugated Laplacian with specified Neumann boundary conditions. The explicit construction of the Green's function allows us to deduce $ L^p $ type estimates which would otherwise be out of reach using the previous methods for constructing such Green's functions for bounded potentials. [ABSTRACT FROM AUTHOR]

We consider the case of regular singularity for a class of equations with pseudodifferential operator Dα, α > 0, on radial functions t p α D α t p = A t p u t p. Under certain conditions, we prove the existence of a solution specified in the form of a locally absolutely convergent power series. [ABSTRACT FROM AUTHOR]

The primary aim is to develop a new class of pseudo-differential operators by incorporating the Special Affine Fourier Transform (SAFT) and some of its fundamental properties in Schwartz and tempered distribution space. The secondary aim is to explore the utility of the SAFT in constructing a new generalized heat equation and derive its analytical solution. Further, we have investigated particular cases of the proposed generalized heat equation. Furthermore, we have visually illustrated solutions of this equation via MATLAB R2023b. [ABSTRACT FROM AUTHOR]

We treat the notion of principal symbol mapping on a compact smooth manifold as a *-homomorphism of C*-algebras. Principal symbol mapping is built from the ground, without referring to the pseudodifferential calculus on the manifold. Our concrete approach allows us to extend Connes trace theorem for compact Riemannian manifolds. [ABSTRACT FROM AUTHOR]

In this work we show that many operators from harmonic analysis associated with the semigroup generated by the Schrödinger operator L = -Δ+V in Rn, where n > 2 and the non-negative potential V belongs to the reverse Hölder class RHq with q > n/2- such as maximal operators, the Littlewood-Paley function, pseudo-differential operators, singular integrals, and their commutators are bounded on the weighted variable Lebesgue space LP(.)(w). We do so by applying the theory of weighted norm inequalities and extrapolation. [ABSTRACT FROM AUTHOR]

Let Q be the d-dimensional space of finite adeles over the algebraic number field K and let P = Q ∗ be its dual space. For a certain type of Vladimirov type time-dependent Hamiltonian H V (t) : Q × P → C we construct the Feynman formulas for the solution of the Cauchy problem with the Schrödinger operator where the caret operator stands for the qp- or pq-quantization. [ABSTRACT FROM AUTHOR]

In this paper, the boundedness and compactness of the Hankel wavelet multiplier associated with the unitary representation on Lp space for 1 ≤ p ≤∞, are obtained by using the Hankel transform technique. The application of Hankel wavelet multipliers in form of the Landau–Pollak–Slepian operator is given. With the help of the Hankel wavelet multiplier, the Sobolev space associated with the Hankel transform is constructed and its various properties are studied. [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]

This paper discusses the properties of the canonical quantum decomposition of the classical Pearson random variables. We show that this leads to the problem of representing the creation–annihilation–preservation (CAP) operators canonically associated to a real-valued random variable X with all moments as (normally ordered) differential operators with polynomial coefficients — a problem already studied in the literature (see references in the introduction). We deduce a formula, for the polynomial coefficients in the representation of the CAP operators of X as pseudo-differential operators, more explicit than the one existing in the literature. We give a new characterization of the Pearson distributions in terms of the Hermitianity of the associated Sturm–Liouville operators. In the second part of the paper, we introduce the notion of finite type random variable X and characterize type- 2 and type- 3 real-valued random variables. We prove that a necessary condition for X to be of finite type is the polynomial growth of the corresponding principal Jacobi sequence. This allows to single out three classes of random variables of infinite type and to prove that the Beta and the uniform distributions are of infinite type. [ABSTRACT FROM AUTHOR]

We extend the stability and spectral invariance of convolution-dominated matrices to the case of quasi-Banach algebras p < 1 . As an application, we construct a spectrally invariant quasi-Banach algebra of pseudodifferential operators with non-smooth symbols that generalize Sjöstrand's results. [ABSTRACT FROM AUTHOR]

In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely 퐒퐆 -class. [ABSTRACT FROM AUTHOR]

In this paper, the weighted estimates for multilinear pseudo-differential operators were systematically studied in rearrangement invariant Banach and quasi-Banach spaces. These spaces contain the Lebesgue space, the classical Lorentz space and Marcinkiewicz space as typical examples. More precisely, the weighted boundedness and weighted modular estimates, including the weak endpoint case, were established for multilinear pseudo-differential operators and their commutators. As applications, we show that the above results also hold for the multilinear Fourier multipliers, multilinear square functions, and a class of multilinear Calderón–Zygmund operators. [ABSTRACT FROM AUTHOR]

The article "Lefschetz Formula for Nonlocal Elliptic Problems Associated with a Bundle" published in Mathematical Notes discusses the generalization of the classical Lefschetz formula to endomorphisms of elliptic complexes of pseudodifferential operators. The paper presents a Lefschetz formula for nonlocal problems associated with a bundle, focusing on operator complexes, ellipticity conditions, and the Lefschetz number of endomorphisms. The main result of the paper is Theorem 2, which provides a formula for the Lefschetz number under certain conditions. The article was authored by N. R. Orlova and was supported by ongoing institutional funding. [Extracted from the article]

Given an action of a discrete group on a closed smooth manifold, we consider the class of nonlocal elliptic operators generated by pseudodifferential operators on the manifold and shift operators. The operators in this class are Fredholm under suitable ellipticity conditions. In the present paper, we give an index formula for these operators for the case of the group , where is an arbitrary finite group. To solve this index problem, we explicitly write the contribution to the index by the fixed points of finite-order elements of the group. The index formula is stated in terms of characteristic classes in periodic cyclic cohomology. [ABSTRACT FROM AUTHOR]

In this paper we establish the well-posedness of the Cauchy problem for a class of pseudo-differential hyperbolic equations on the torus. The class considered here includes a space-like fractional order Laplacians. By applying the toroidal pseudo-differential calculus we establish regularity estimates, existence and uniqueness in the scale of the standard Sobolev spaces on the torus. [ABSTRACT FROM AUTHOR]

In this paper, some properties of resonances for multi-dimensional quantum walks are studied. Resonances for quantum walks are defined as eigenvalues of complex translated time evolution operators in the pseudo momentum space. For some typical cases, we show some results of existence or nonexistence of resonances. One is a perturbation of an elastic scattering of a quantum walk which is an analogue of classical mechanics. Another one is a shape resonance model which is a perturbation of a quantum walk with a non-penetrable barrier. [ABSTRACT FROM AUTHOR]

We consider the linear elasticity system for a body coated with a thin elastic layer. The effect of the thin layer on that body can be described through an impedance operator linking the displacement and the traction on the interface. This operator cannot be in general explicitly computed for an arbitrary shape. In this paper, we consider the case of a linear elastic slab with Hooke's law. We compute by the Fourier transform the exact symbol of the impedance operator and prove that it is a pseudodifferential operator of order one. Then, we give a Padé-like approximation at order three (with respect to the thickness of the layer) for the impedance operator. The pattern of this approximation will be a helpful guide for approximating the impedance operator for a body whose boundary has nonzero curvature. [ABSTRACT FROM AUTHOR]

In this paper we consider 0-th order pseudodifferential operators on the circle. We show that inside any interval disjoint from the critical values of the principal symbol, the spectrum is absolutely continuous with possibly finitely many embedded eigenvalues. We also give an example of embedded eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper we show the invertibility of the geodesic X-ray transform on one forms and 2-tensors on asymptotically conic manifolds, up to the natural obstruction, allowing existence of certain kinds of conjugate points. We use the 1-cusp pseudodifferential operator algebra and its semiclassical foliation version introduced and used by Vasy and Zachos, who showed the same type invertibility on functions.The complication of the invertibility of the tensorial X-ray transform, compared with X-ray transform on functions, is caused by the natural kernel of the transform consisting of 'potential tensors'. We overcome this by arranging a modified solenoidal gauge condition, under which we have the invertibility of the X-ray transform. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel approach to the fractional power of the Laplace-Bessel operator Δv, defined as... The fractional power of this operator is introduced as a pseudo-differential operator through the multidimensional Bessel transform. Our primary contributions encompass a normalized singular integral representation, Bochner subordination, and intertwining relations. [ABSTRACT FROM AUTHOR]

We consider evolutionary equations with fractional differentiation operators whose restrictions to certain spaces of type S coincide with pseudodifferential operators constructed on the basis of smooth symbols, which are multipliers in these spaces. We establish the well-posedness of the time-nonlocal multipoint problem for these equations with initial function, which is an element of the space of generalized functions of the ultradistribution type. It is shown that the solutions of these problems stabilize to zero in the spaces of generalized functions of the type S′ (weak stabilization) and also stabilize to zero uniformly on R in the case where the initial generalized function has a bounded support. [ABSTRACT FROM AUTHOR]

In this paper, the authors prove that bilinear pseudo-differential operator... [ABSTRACT FROM AUTHOR]

To strike a balance between modeling accuracy and computational efficiency for simulations of ultrasound waves in soft tissues, we derive a pseudodifferential factorization of the wave operator with fractional attenuation. This factorization allows us to approximately solve the Helmholtz equation via one-way (transmission) or two-way (transmission and reflection) sweeping schemes tailored to high-frequency wave fields. We provide explicitly the three highest order terms of the pseudodifferential expansion to incorporate the well-known square-root first order symbol for wave propagation, the zeroth order symbol for amplitude modulation due to changes in wave speed and damping, and the next symbol to model fractional attenuation. We also propose wide-angle Padé approximations for the pseudodifferential operators corresponding to these three highest order symbols. Our analysis provides insights regarding the role played by the frequency and the Padé approximations in the estimation of error bounds. We also provide a proof-of-concept numerical implementation of the proposed method and test the error estimates numerically. [ABSTRACT FROM AUTHOR]

This paper concerns the asymmetric transport observed along interfaces separating two-dimensional bulk topological insulators modeled by (continuous) differential Hamiltonians and how such asymmetry persists after numerical discretization. We first demonstrate that a relevant edge current observable is quantized and robust to perturbations for a large class of elliptic Hamiltonians. We then establish a bulk edge correspondence stating that the observable equals an integer-valued bulk difference invariant depending solely on the bulk phases. We next show how to extend such results to periodized Hamiltonians amenable to standard numerical discretizations. A form of no-go theorem implies that the asymmetric transport of periodized Hamiltonians necessarily vanishes. We introduce a filtered version of the edge current observable and show that it is approximately stable against perturbations and converges to its quantized limit as the size of the computational domain increases. To illustrate the theoretical results, we finally present numerical simulations that approximate the infinite domain edge current with high accuracy and show that it is approximately quantized even in the presence of perturbations. [ABSTRACT FROM AUTHOR]

We derive rigorously from the water waves equations new irrotational shallow‐water models for the propagation of surface waves in the case of uneven topography in horizontal dimensions one and two. The systems are made to capture the possible change in the waves' propagation, which can occur in the case of large amplitude topography. The main contribution of this work is the construction of new multiscale shallow‐water approximations of the Dirichlet–Neumann operator. We prove that the precision of these approximations is given at the order O(με)$O(\mu {\varepsilon })$, O(με+μ2β2)$O(\mu \varepsilon +\mu ^2\beta ^2)$, and O(μ2ε+μεβ+μ2β2)$O(\mu ^2\varepsilon +\mu {\varepsilon }\beta + \mu ^2\beta ^2)$. Here, μ$\mu$, ε$\varepsilon$, and β$\beta$ denote, respectively, the shallow‐water parameter, the nonlinear parameter, and the bathymetry parameter. From these approximations, we derive models with the same precision as the ones above. The model with precision O(με)$O(\mu {\varepsilon })$ is coupled with an elliptic problem, while the other models do not present this inconvenience. [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]

In this paper, we investigate the Besov spaces on compact Lie groups in a subelliptic setting, that is, associated with a family of vector fields, satisfying the Hörmander condition, and their corresponding sub-Laplacian. Embedding properties between subelliptic Besov spaces and Besov spaces associated to the Laplacian on the group are proved. We link the description of subelliptic Sobolev spaces with the matrix-valued quantisation procedure of pseudo-differential operators to provide sharp subelliptic Sobolev and Besov estimates for operators in the $ (\rho,\delta) $ (ρ , δ) -Hörmander classes. In contrast with the available results in the literature in the setting of compact Lie groups, we allow Fefferman-type estimates in the critical case $ \rho =\delta. $ ρ = δ. Interpolation properties between Besov spaces and Triebel–Lizorkin spaces are also investigated. [ABSTRACT FROM AUTHOR]

We construct the uniform asymptotics of solutions to a third order equation whose holomorphic coefficients have an arbitrary irregular singularity in the space of functions of exponential growth. In general, the problem of constructing the asymptotics of solutions of differential equations in a neighborhood of an irregular singular point was formulated by Poincaré in his articles devoted to the analytical theory of differential equations. This problem for the equations with degeneracies of arbitrary order, if there are multiple roots, was solved only in some special cases, for example, when the equation has second order. The main method in the case of degenerations of higher order is the requantization based on the Laplace–Borel transform. This method was developed of constructing the asymptotics of solutions of differential equations in a neighborhood of irregular singular points in the case that the main symbol of the differential operator has multiple roots. The problem of constructing the asymptotics of solutions to higher order equations is much more complicated. To solve it, we use the requantization method that was not required when solving a similar problem for second order equations. We will solve some model problem that is an important step towards solving the Poincaré general problem of constructing the asymptotics of solutions in a neighborhood of an irregular singular point for an equation of arbitrary order. The problem of further research is to generalize the method of the present article to equations of arbitrary order. [ABSTRACT FROM AUTHOR]

We continue the study of the perturbation problem discussed in H. D. Cornean and R. Purice (2023) and get rid of the "slow variation" assumption by considering symbols of the form a(x+δ F(x), ξ) with α a real Hörmander symbol of class S 0,00(ℝd x ℝd) and F a smooth function with all its derivatives globally bounded, with |δ| ≤ 1. We prove that while the Hausdorff distance between the spectra of the Weyl quantization of the above symbols in a neighbourhood of δ = 0 is still of the order √|δ|, the distance between their spectral edges behaves like |δ|ν with ν ϵ [1/2,1) depending on the rate of decay of the second derivatives of F at infinity. [ABSTRACT FROM AUTHOR]

We review recent progress regarding the index theory of operators defined on non-compact manifolds that can be modeled by Lie groupoids. The structure of a particular type of almost regular foliation is recalled and the construction of the corresponding accompanying holonomy Lie groupoid. Using deformation groupoids, K-theoretical invariants can be defined and compared. We summarize how questions in index theory are addressed via the geometrization made possible by the use of deformation groupoids. The discussion is motivated by examples and applications to degenerate PDE's, diffusion processes, evolution equations and geometry. [ABSTRACT FROM AUTHOR]

In this paper, we consider the semiclassical 2D magnetic Schrödinger operator in the case where the magnetic field vanishes along a smooth closed curve. Assuming that this curve has an axis of symmetry, we prove that semiclassical tunneling occurs. The main result is an expression of the splitting of the first two eigenvalues and an explicit tunneling formula. [ABSTRACT FROM AUTHOR]

We consider the linear relaxation Boltzmann equation in a semiclassical framework. We construct a family of sharp quasimodes for the associated operator which yields sharp spectral asymptotics for its small spectrum in the low temperature regime. We deduce some information on the long time behavior of the solutions with a sharp estimate on the return to equilibrium as well as a quantitative metastability result. The main novelty is that the collision operator is a pseudo-differential operator in the critical class S 1/2 and that its action on the Gaussian quasimodes yields a superposition of exponentials. [ABSTRACT FROM AUTHOR]

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]

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]