131 research outputs found
Honoring A.V. Bitsadze's service to science (on his 100th birthday
The paper dedicated to Prof. Andrei Vasilievich Bitsadze (5.9.1916-4.6.1994) on the occasion of his 100th birthda
ITERATED DIRICHLET PROBLEM FOR THE HIGHER ORDER POISSON EQUATION
Convoluting the harmonic Green function with itself consecutively leads to a polyharmonic Green function suitable to solve an iterated Dirichlet problem for the higher order Poisson equation. The procedure works in any regular domain and is not restricted to two dimensions. In order to get explicit expressions however the situation is studied in the complex plane and sometimes in particular the unit disk is considered.Convoluting the harmonic Green function with itself consecutivelyleads to a polyharmonic Green function suitable to solve an iterated Dirichlet problem for the higher order Poisson equation. The procedure works in any regular domain and is not restricted to two dimensions. In order to get explicit expressions however the situation is studied in the complex plane and sometimes in particular the unit disk is considered
On the reduction of the multidimensional Schroedinger equation to a first order equation and its relation to the pseudoanalytic function theory
Given a particular solution of a one-dimensional stationary Schroedinger
equation (SE) this equation of second order can be reduced to a first order
linear differential equation. This is done with the aid of an auxiliary Riccati
equation. We show that a similar fact is true in a multidimensional situation
also. We consider the case of two or three independent variables. One
particular solution of (SE) allows us to reduce this second order equation to a
linear first order quaternionic differential equation. As in one-dimensional
case this is done with the aid of an auxiliary Riccati equation. The resulting
first order quaternionic equation is equivalent to the static Maxwell system.
In the case of two independent variables it is the Vekua equation from theory
of generalized analytic functions. We show that even in this case it is
necessary to consider not complex valued functions only, solutions of the Vekua
equation but complete quaternionic functions. Then the first order quaternionic
equation represents two separate Vekua equations, one of which gives us
solutions of (SE) and the other can be considered as an auxiliary equation of a
simpler structure. For the auxiliary equation we always have the corresponding
Bers generating pair, the base of the Bers theory of pseudoanalytic functions,
and what is very important, the Bers derivatives of solutions of the auxiliary
equation give us solutions of the main Vekua equation and as a consequence of
(SE). We obtain an analogue of the Cauchy integral theorem for solutions of
(SE). For an ample class of potentials (which includes for instance all radial
potentials), this new approach gives us a simple procedure allowing to obtain
an infinite sequence of solutions of (SE) from one known particular solution
Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I
Classical Schur analysis is intimately connected to the theory of orthogonal
polynomials on the circle [Simon, 2005]. We investigate here the connection
between multipoint Schur analysis and orthogonal rational functions.
Specifically, we study the convergence of the Wall rational functions via the
development of a rational analogue to the Szeg\H o theory, in the case where
the interpolation points may accumulate on the unit circle. This leads us to
generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields
asymptotics of a novel type.Comment: a preliminary version, 39 pages; some changes in the Introduction,
Section 5 (Szeg\H o type asymptotics) is extende
Transmutations for Darboux transformed operators with applications
We solve the following problem. Given a continuous complex-valued potential
q_1 defined on a segment [-a,a] and let q_2 be the potential of a Darboux
transformed Schr\"odinger operator. Suppose a transmutation operator T_1 for
the potential q_1 is known such that the corresponding Schr\"odinger operator
is transmuted into the operator of second derivative. Find an analogous
transmutation operator T_2 for the potential q_2.
It is well known that the transmutation operators can be realized in the form
of Volterra integral operators with continuously differentiable kernels. Given
a kernel K_1 of the transmutation operator T_1 we find the kernel K_2 of T_2 in
a closed form in terms of K_1. As a corollary interesting commutation relations
between T_1 and T_2 are obtained which then are used in order to construct the
transmutation operator for the one-dimensional Dirac system with a scalar
potential
Topics in Mathematical Analysis
This volume contains most of the lectures of the "Minicorsi of Mathematical Analysis" held at the University of Padova in the years 2000-2003
Riemann–Hilbert Problems for Monogenic Functions on Upper Half Ball of R^4
In this paper we are interested in finding solutions to Riemann– Hilbert boundary value problems, for short Riemann–Hilbert problems, with variable coefficients in the case of axially monogenic functions defined over the upper half unit ball centred at the origin in four-dimensional Euclidean space. Our main idea is to transfer Riemann– Hilbert problems for axially monogenic functions defined over the up- per half unit ball centred at the origin of four-dimensional Euclidean spaces into Riemann–Hilbert problems for analytic functions defined over the upper half unit disk of the complex plane. Furthermore, we extend our results to axially symmetric null-solutions of perturbed generalized Cauchy–Riemann equations
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