Narrow Results

The relativistic Lippmann–Schwinger equation was earlier formulated in terms of the limit values of the corresponding resolvent. In the present paper, we write down the limit values of the resolvent in an explicit form, and so the relativistic Lippmann–Schwinger equation is presented as an integral equation. Using this integral equation, we investigate the stationary scattering problems (relativistic case, Dirac equation). We consider the dynamical scattering problems (relativistic case, Dirac equation) as well.

In this paper, we prove that if the composition symbols φ and ψ are linear fractional non-automorphisms of 𝔻 such that φ(ζ) and ψ(ζ) belong to ∂𝔻 for some ζ ∈ ∂𝔻 and u, v ∈ H^∞ are continuous on ∂𝔻 with u(ζ)v(ζ) ≠ 0, then is compact on H² if and only if ζ is the common boundary fixed point of φ and ψ and one of the following statements holds: (i) both φ and ψ are parabolic; (ii) both φ and ψ are hyperbolic and another fixed point of φ is where w is the fixed point of ψ other than ζ. We also study the commutant of a weighted composition operator on H². We verify that if φ is an analytic self-map of 𝔻 with Denjoy–Wolff point b ∈ 𝔻 and u ∈ H^∞\{0}, then every weighted composition operator in the commutant {W_{u, φ}}′ has {f ∈ H² : f(b) = 0} as its nontrivial invariant subspace.

In this paper, the boundedness and compactness of the Hankel wavelet multiplier associated with the unitary representation on $L^p$ space for $1\le p\le \infty$, 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.

In this paper, we introduce Fibonacci backward difference operator $F{\mathrm{\Delta }}^{Bl}$ of fractional order l by the composition of Fibonacci band matrix $F$ and difference operator ${\mathrm{\Delta }}^{Bl}$ of fractional order l, defined by ${({\mathrm{\Delta }}^{Bl}x)}_s={\sum }_{i=0}^{\infty }{(-1)}^i\frac{\mathrm{\Gamma }(l+1)}{i!\mathrm{\Gamma }(l-i+1)}{x}_{s-i}$ and introduce sequence spaces ${\ell }_k(F{\mathrm{\Delta }}^{Bl})$ and ${\ell }_{\infty }(F{\mathrm{\Delta }}^{Bl}).$ We present some topological properties, obtain Schauder basis and determine $\alpha$-, $\beta$- and $\gamma$-duals of the spaces ${\ell }_k(F{\mathrm{\Delta }}^{Bl})$ and ${\ell }_{\infty }(F{\mathrm{\Delta }}^{Bl}).$ We characterize certain classes of matrix mappings from the spaces ${\ell }_k(F{\mathrm{\Delta }}^{Bl})$ and ${\ell }_{\infty }(F{\mathrm{\Delta }}^{Bl})$ to any of the space ${\ell }_{\infty },$$c,$$c_0$ or ${\ell }_1.$ Finally we compute necessary and sufficient conditions for a matrix operator to be compact on the spaces ${\ell }_k(F{\mathrm{\Delta }}^{Bl})$ and ${\ell }_{\infty }(F{\mathrm{\Delta }}^{Bl}).$

By using nonuniform (geometric) grid network, a new high-order finite-difference compact scheme has been obtained for the numerical solution of three-space dimensions partial differential equations of elliptic type. Single cell discretization to the elliptic equation makes it easier to compute and exhibit stability of the numerical solutions. The monotone and irreducible property of the Jacobian matrix to the system of difference equations analyses the converging behavior of the numerical solution values. As an experiment, applications of the compact scheme to Schrödinger equations, sine-Gordon equations, elliptic Allen–Cahn equation and Poisson’s equation have been presented with root mean squared errors of exact and approximate solution values. The results corroborate the reliability and efficiency of the scheme.

An existence theory for solutions of a parabolic problem u_t – div(A(x)∇u) + q(x)u = f(x, t) for x ∈ D and t ∈ (0,T] under a dynamical boundary condition σ(x)u_t + ∇u^T A(x)n = g(x, t) for x ∈ ∂D and t ∈ (0, T] is developed and a spectral representation formula is derived. It extends the results of [2] to problems with variable coefficients. We are interested in the case where the dynamical coefficient σ is a sign-changing or negative function. The one-dimensional parabolic problem is well-posed in the space C([0,T], H¹(D)). This is not true in higher dimensions. Our approach is based on the spectral theory of an associated elliptic problem with the eigenvalue parameter both in the equation and the boundary condition. By means of the theory of compact operators the spectrum is analyzed. Qualitative properties of the eigenfunctions are derived, e.g. strict positivity of the principal two eigenfunctions follows from a Harnack-type inequality. An interesting phenomenon is the "parameter-resonance", where for a specific value of the mean of the dynamical coefficient σ(x), two eigenvalues of the elliptic problem cross.

We introduce our three types of numerical inversion formulas of the Laplace transform based on a Fredholm integral equation of the second kind, the Sinc functions (the Sinc method) and the numerical singular value decomposition of the Laplace transform in a certain reproducing kernel Hilbert space. However, we will need the power of the high-accuracy numerical method with multiple-precision arithmetic for difficult situations. We shall give computational experiments for some very difficult situations for the real inversion.