Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMultipole matrix elements of Green function of Laplace equation
165
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Multipole matrix elements of Green function of Laplace equation are calculated. The multipole matrix elements of Green function in electrostatics describe potential on a sphere which is produced by a charge distributed on the surface of a different (possibly overlapping) sphere of the same radius. The matrix elements are defined by double convolution of two spherical harmonics with the Green function of Laplace equation. The method we use relies on the fact that in the Fourier space the double convolution has simple form. Therefore we calculate the multipole matrix from its Fourier transform. An important part of our considerations is simplification of the three dimensional Fourier transformation of general multipole matrix by its rotational symmetry to the one-dimensional Hankel transformation.
rate research
Read More
We have constructed a sequence of solutions of the Helmholtz equation forming an orthogonal sequence on a given surface. Coefficients of these functions depend on an explicit algebraic formulae from the coefficient of the surface. Moreover, for exterior Helmholtz equation we have constructed an explicit normal derivative of the Dirichlet Green function. In the same way the Dirichlet-to-Neumann operator is constructed. We proved that normalized coefficients are uniformly bounded from zero.
The Laguerre functions constitute one of the fundamental basis sets for calculations in atomic and molecular electron-structure theory, with applications in hadronic and nuclear theory as well. While similar in form to the Coulomb bound-state eigenfunctions (from the Schroedinger eigenproblem) or the Coulomb-Sturmian functions (from a related Sturm-Liouville problem), the Laguerre functions, unlike these former functions, constitute a complete, discrete, orthonormal set for square-integrable functions in three dimensions. We construct the SU(1,1)xSO(3) dynamical algebra for the Laguerre functions and apply the ideas of factorization (or supersymmetric quantum mechanics) to derive shift operators for these functions. We use the resulting algebraic framework to derive analytic expressions for matrix elements of several basic radial operators (involving powers of the radial coordinate and radial derivative) in the Laguerre function basis. We illustrate how matrix elements for more general spherical tensor operators in three dimensional space, such as the gradient, may then be constructed from these radial matrix elements.
Quadratic matrix equations occur in a variety of applications. In this paper we introduce new permutationally invariant functions of two solvents of the n quadratic matrix equation X^2- L1X - L0 = 0, playing the role of the two elementary symmetric functions of the two roots of a quadratic scalar equation. Our results rely on the connection existing between the QME and the theory of linear second order difference equations with noncommutative coefficients. An application of our results to a simple physical problem is briefly discussed.
In this paper, we consider the discrete power function associated with the sixth Painleve equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with $widetilde{W}(3A_1^{(1)})$ symmetry. By constructing the action of $widetilde{W}(3A_1^{(1)})$ as a subgroup of $widetilde{W}(D_4^{(1)})$, i.e., the symmetry group of P$_{rm VI}$, we show how to relate $widetilde{W}(D_4^{(1)})$ to the symmetry group of the lattice. Moreover, by using translations in $widetilde{W}(3A_1^{(1)})$, we explain the odd-even structure appearing in previously known explicit formulas in terms of the $tau$ function.
We derive upper bounds for the trace of the heat kernel $Z(t)$ of the Dirichlet Laplace operator in an open set $Omega subset R^d$, $d geq 2$. In domains of finite volume the result improves an inequality of Kac. Using the same methods we give bounds on $Z(t)$ in domains of infinite volume. For domains of finite volume the bound on $Z(t)$ decays exponentially as $t$ tends to infinity and it contains the sharp first term and a correction term reflecting the properties of the short time asymptotics of $Z(t)$. To prove the result we employ refined Berezin-Li-Yau inequalities for eigenvalue means.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
