XXXIII Workshop on Geometric Methods in Physics 29.06-5.07.2014

Andreas Ruffing

Quantum Difference-Differential Equations

Differential equations which contain the parameter of a scaling process are referred to by the name Quantum Difference--Differential Equations. Some of their applications to discrete models of the Schr\"odinger equation are presented and some of their rich, filigrane und sometimes unexpected analytic structures are revealed.

A Lie-algebraic concept for obtaining basic adaptive discretizations is explored, generalizing the concept of deformed Heisenberg algebras by Julius Wess.

Some of the moment problems of the underlying basic difference equations are investigated.
Applications to discrete Schr\"odinger theory are worked out and some spectral properties of the arising operators are presented, also in the case of Schr\"odinger operators with basic shift--potentials and in the case of ground state difference--differential operators.

For the arising orthogonal function systems, the concept of inherited orthogonality is explained.
The results in this talk are mainly related to a recent joint work with Sophia Ro{\ss}kopf and Lucia Birk.

An analogous situation on an equidistant lattice has been
worked out.

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