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New method for computation of discrete spectrum of radical Schrödinger operator

Ivan Úlehla, Miloslav Havlíček (1980)

Aplikace matematiky

A new method for computation of eigenvalues of the radial Schrödinger operator - d 2 / d x 2 + v ( x ) , x 0 is presented. The potential v ( x ) is assumed to behave as x - 2 + ϵ if x 0 + and as x - 2 - ϵ if x + , ϵ 0 . The Schrödinger equation is transformed to a non-linear differential equation of the first order for a function z ( x , ) . It is shown that the eigenvalues are the discontinuity points of the function z ( , ) . Moreover, it is shown how to obtain an arbitrarily accurate approximation of eigenvalues. The method seems to be much more economical in comparison...

On m -sectorial Schrödinger-type operators with singular potentials on manifolds of bounded geometry

Ognjen Milatovic (2004)

Commentationes Mathematicae Universitatis Carolinae

We consider a Schrödinger-type differential expression H V = * + V , where is a C -bounded Hermitian connection on a Hermitian vector bundle E of bounded geometry over a manifold of bounded geometry ( M , g ) with metric g and positive C -bounded measure d μ , and V is a locally integrable section of the bundle of endomorphisms of E . We give a sufficient condition for m -sectoriality of a realization of H V in L 2 ( E ) . In the proof we use generalized Kato’s inequality as well as a result on the positivity of u L 2 ( M ) satisfying the...

On phase segregation in nonlocal two-particle Hartree systems

Walter Aschbacher, Marco Squassina (2009)

Open Mathematics

We prove the phase segregation phenomenon to occur in the ground state solutions of an interacting system of two self-coupled repulsive Hartree equations for large nonlinear and nonlocal interactions. A self-consistent numerical investigation visualizes the approach to this segregated regime.

On the convergence of SCF algorithms for the Hartree-Fock equations

Eric Cancès, Claude Le Bris (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The present work is a mathematical analysis of two algorithms, namely the Roothaan and the level-shifting algorithms, commonly used in practice to solve the Hartree-Fock equations. The level-shifting algorithm is proved to be well-posed and to converge provided the shift parameter is large enough. On the contrary, cases when the Roothaan algorithm is not well defined or fails in converging are exhibited. These mathematical results are confronted to numerical experiments performed by chemists.

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