Abstract: In this paper, we define the wavelet multiplier and Landau–Pollak–Slepian (L.P.S) operators on the Hilbert space \(L^2(G)\) , where G is a locally compact abelian topological group and investigate some of their properties. In particular, we show that they are bounded linear operators, and are in Schatten p-class spaces, \(1 \le p \le \infty \) , and we determine their trace class. PubDate: 2019-03-06
Abstract: We study the Cauchy problem for the linear double dispersion equation $$\begin{aligned} u_{tt}-\Delta u_{tt}+\Delta ^2 u-\Delta u-\Delta u_t =0, \quad t\ge 0,\ x\in {\mathbb {R}}^n \end{aligned}$$ and we derive long time decay estimates for the solution in \(L^p\) spaces and in real Hardy spaces. We employ the obtained results to study the equation with nonlinearity \(\Delta f(u)\) and nonsmooth f. PubDate: 2019-03-02
Abstract: We study the Cauchy problem for an evolution equation of Schrödinger type. The Hamiltonian is the Weyl quantization of a real homogeneous quadratic form with a pseudodifferential perturbation of negative order from Shubin’s class. We prove that the propagator is a Fourier integral operator of Shubin type of order zero. Using results for such operators and corresponding Lagrangian distributions, we study the propagator and the solution, and derive phase space estimates for them. PubDate: 2019-03-02
Abstract: This paper is devoted to introduce an efficient solver using a combination of the symbol of the operator and the windowed Fourier frames (WFFs) of the coupled system of second order ordinary differential equations. The given system has a basic importance in modeling various phenomena like, Cascades and Compartment Analysis, Pond Pollution, Home Heating, Chemostats and Microorganism Culturing, Nutrient Flow in an Aquarium, Biomass Transfer and others. The proposed method reduces the system of differential equations to a system of algebraic equations in the coefficients of WFFs. The introduced method is computer oriented with highly accurate solution. To demonstrate the efficiency of the proposed method, two examples are presented and the results are displayed graphically. Finally, we convert the presented coupled systems of BVPs to a first order system of ODEs to compare the obtained numerical solution with those solutions using the fourth-order Runge–Kutta method (RK4). PubDate: 2019-03-01
Abstract: We consider the Cauchy problem for weakly hyperbolic m-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that in general one has to impose Levi conditions to get \(C^\infty \) or Gevrey well-posedness even if the coefficients are smooth. We use moduli of continuity to describe the regularity of the coefficients with respect to time, weight sequences for the characterization of their regularity with respect to space and weight functions to define the solution spaces. Furthermore, we propose a generalized Levi condition that models the influence of multiple characteristics more freely. We establish sufficient conditions for the well-posedness of the Cauchy problem, that link the Levi condition as well as the modulus of continuity and the weight sequence of the coefficients to the weight function of the solution space. Additionally, we obtain that the influences of the Levi condition and the low regularity of coefficients on the weight function of the solution space are independent of each other. PubDate: 2019-03-01
Abstract: In this article, we consider the Schrödinger semigroup related to the Dunkl–Laplacian \(\Delta _{\mu }\) (associated to finite reflection group G) on \(\mathbb {R}^n\) . We characterize the image of \(L^2(\mathbb {R}^n, e^{u^2} h_{\mu }(u) du)\) under the Schrödinger semigroup as a reproducing kernel Hilbert space. We define Dunkl–Sobolev space in \(L^2(\mathbb {R}^n, e^{u^2} h_{\mu }(u) du)\) and characterize it’s image under the Schrödinger semigroup associated to \(G=\mathbb {Z}_2^n\) as a reproducing kernel Hilbert space up to equivalence of norms. Also we provide similar results for Schrödinger semigroup associated to Dunkl–Hermite operator. PubDate: 2019-03-01
Abstract: In this short note we show results on the compactness of the commutator of pseudodifferential operator and operator of multiplication in both \(\mathrm{L}^2\) and \(\mathrm{L}^p\) setting. Our results use the boundedness results of pseudodifferential operators by Hwang and Hwang-Lee, and the Krasnoselskij type interpolation lemma. We use the obtained results to construct a variant of microlocal defect functionals via pseudodifferential operators and derive its localisation principle. PubDate: 2019-03-01
Abstract: We study the convergence of means of spectral expansions corresponding to positive self-adjoint elliptic pseudo-differential operators for distributions from the Sobolev–Liouville class. PubDate: 2019-03-01
Abstract: We present a construction of algebras of generalized functions of Colombeau-type which, instead of asymptotic estimates with respect to a regularization parameter, employs only topological estimates on certain spaces of kernels for its definition. PubDate: 2019-03-01
Abstract: The concern of this article is a semiclassical Weyl calculus on an infinite dimensional Hilbert space H. If (i, H, B) is a Wiener triplet associated to H, the quantum state space will be the space of \(L^2\) functions on B with respect to a Gaussian measure with h / 2 variance, where h is the semiclassical parameter. We prove the boundedness of our pseudodifferential operators (PDO) in the spirit of Calderón–Vaillancourt with an explicit bound, a Beals type characterization, and metaplectic covariance. An application to a model of quantum electrodynamics is added in the last section (Sect. 7), for fixed spin 1 / 2 particles interacting with the quantized electromagnetic field (photons). We prove that some observable time evolutions, the spin evolutions, the magnetic and electric evolutions when subtracting their free evolutions, are PDO in our class. PubDate: 2019-03-01
Abstract: In this paper, we deal with a class of nonlinear fractional non-autonomous evolution equations with delay by using Hilfer fractional derivative, which generalized the famous Riemann–Liouville fractional derivative. Combining techniques of fractional calculus, measure of noncompactness and some fixed point theorem, we obtain new existence result of mild solutions when the associated semigroup is not compact. Furthermore, the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition. The results obtained improve and extend some related conclusions. Finally, two examples will be presented to illustrate the main results. PubDate: 2019-03-01
Abstract: By using the Hille–Yosida theorem, Phillips theorem and Fattorini theorem we prove that the M/G/1 queueing model with vacations and multiple phases of operation, which is described by infinitely many partial differential equations with integral boundary conditions, has a unique positive time-dependent solution that satisfies the probability condition. Next, by studying the spectrum of the operator, which corresponds to the model, on the imaginary axis we prove that the time-dependent solution of the model strongly converges to its steady-state solution. PubDate: 2019-03-01
Abstract: In this paper a nonlocal problem for the elliptic equation in a cylindrical domain is considered. It is shown that this problem is ill-posed as well as the Cauchy problem for the Laplace equation. The method of spectral expansion in eigenfunctions of the nonlocal problem for equations with involution establishes a criterion of the strong solvability of the considered nonlocal problem. It is shown that the ill-posedness of the nonlocal problem is equivalent to the existence of an isolated point of the continuous spectrum for a nonself-adjoint operator with involution. PubDate: 2019-03-01
Abstract: In this paper we study the perturbation \(L=H+V\) , where \(H=-\frac{{{d}^{2m}}}{d{{x}^{2m}}}+{{x}^{2m}}\) on \(\mathbb {R}\) , \(m\in {{\mathbb {N}}^{*}}\) and V is a decreasing scalar potential. Let \({{\lambda }_{k}}\) be the \(k^{th}\) eigenvalue of H. We suppose that the eigenvalues of L around \({{\lambda }_{k}}\) can be written in the form \({{\lambda }_{k}}+{{\mu }_{k}}\) . The main result of the paper is an asymptotic formula for fluctuation \(\{ {{\mu }_{k}} \}\) which is given by a transformation of V. In the case \(m=1\) we recover a result on the harmonic oscillator. PubDate: 2019-02-28
Abstract: In this paper, we study the quaternion windowed Fourier transform and prove the Beckner’s uncertainty principle in term of entropy, Lieb uncertainty principle and the Heisenberg uncertainty principle for the quaternion windowed Fourier transform, the radar quaternion ambiguity function and the quaternion-Wigner transform. PubDate: 2019-02-26
Abstract: We study the long-time behavior of solutions for a general class of nonlinear fractional differential equations. These equations involve Hadamard fractional derivatives of different orders. We determine sufficient conditions on the nonlinear terms which guarantee that solutions exist globally and decay to zero as a logarithmic function. For this purpose, we combine and generalize some versions of Gronwall–Bellman inequality, appropriate regularization techniques and several properties of the Hadamard fractional derivative. Our findings are illustrated by examples. PubDate: 2019-02-25
Abstract: The two-sided quaternion Fourier transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization of Beurling’s theorem, Hardy, Cowling–Price and Gelfand–Shilov theorems, is obtained for the two-sided quaternion Fourier transform. PubDate: 2019-02-20
Abstract: Let \({\mathbb {S}}\) be a locally compact abstract shearlet group and \(\sigma :{\mathbb {S}}\rightarrow {\mathcal {U}}({\mathcal {H}})\) be a representation of \({\mathbb {S}}\) on a separable Hilbert space \({\mathcal {H}}\) . We give a necessary and sufficient condition for the family of representation coefficients to be a continuous shearlet frame. Furthermore relations between continuous shearlet frames associated to a given representation and its irreducible sub-representations are investigated. PubDate: 2019-02-18
Abstract: We study from a pseudo-differential point of view the frame operator associated with a Gabor system. In particular we show how an application of the classical boundedness theorem of Calderón–Vaillancourt yields sufficient conditions for a Gabor system to form a frame in \(L^2\left( {\mathbb {R}}^d\right) \) . PubDate: 2019-02-15
Abstract: We define pseudo-differential operators on a locally compact, Hausdorff and abelian group G as natural extensions of pseudo-differential operators on \({\mathbb {R}}^n\) . In particular, for pseudo-differential operators with symbols in \(L^2(G\times \widehat{G})\) , where \(\widehat{G}\) is the dual group of G, we give explicit formulas for the products and adjoints, characterize them as Hilbert–Schmidt operators on \(L^2(G)\) and prove that they form a \(C^*\) -algebra, which is also a \(H^*\) -algebra. We give a characterization of trace class pseudo-differential operators in terms of symbols lying in a subspace of \(L^1(G\times \widehat{G})\cap L^2(G\times \widehat{G})\) . PubDate: 2019-02-13
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