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Zawartość zarchiwizowana w dniu 2022-12-23

Constructive complex analysis and density functionals

Cel



The aim of this project is the development of methods, based on potential theory, analysis on Riemann surfaces and constructive approximation, to attack long-standing quantum-mechanical problems such as those posed by the use of the density-functional theory in physical systems. New statements of problems will be studied to develop the mathematical theory and to design new strategies for the exact solution of some specific physical problems. In particular, the determination of fundamental and/or experimentally measurable quantities (information entropies, kinetic and exchange energies, etc.) of fermionic systems (atoms, nuclei, neutron stars) can be attacked by modern methods of potential theory with external fields of harmonic functions together with the recently discovered techniques of special functions (e.g. orthogonal polynomials with varying measures and hyperelliptic functions).

At the end of the project an accurate determination of the information entropy of the two simplest physical systems will have been obtained (harmonic oscillator and hydrogen atom), at least for moderate n but also for n tending to infinity (which corresponds to the Rydberg states). From these results a conjecture improving the Bialynick-Birula Mycielski inequality will be formulated.

The meaning of entropy for orthogonal polynomials will be understood and, in particular, the ability to formulate and prove the asymptotic results for orthogonal polynomials on the interval [-1,1] only in terms of the behaviour of the norm of the monic orthogonal polynomials and the mutual energy of the zero distribution measure and the measure p2 (X) dp(x) describing the size of the polynomials.

Regarding rational approximation, the most relevant results for Pade approximation to Hermite-Pade approximation for vector functions and also for matrix functions will be extended. In particular, for the approximation of Markov functions (the general Markov system) the relevant convergence results for at least some important Markov systems will be obtained.

Finally, for Sobolev orthogonal polynomials the proper method of treating continuous Sobolev orthogonality will be found. For discrete Sobolev orthogonality one can still work with operator theory involving band matrices, but a new technique is needed for continuous Sobolev orthogonality. Presumably a differential-difference operator is needed, and it is expected that such an operator can be identified.

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Koordynator

Universidad de Granada
Wkład UE
Brak danych
Adres
Avenida Fuentenueva
18071 Granada
Hiszpania

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