In mathematical research, it is well-known that many interesting linear
operators, such as the derivative, are unbounded, i.e. they do not have any real
Lipschitz constant. In this project, we suggest to instead consider such linear
maps as bounded operators, but with a bound given by an infinite number in the
ring of Colombeau generalized numbers. Linear maps with an (infinite) Lipschitz
constant are called infinite bounded operators. The work packages we propose to
develop are:
1. Theory of infinite bounded operators;
2. Applications to mathematical analysis and quantum mechanics;
3. Universal properties of generalized functions and of basic infinite bounded
operators.
The project aims at showing the flexibility of a non -Archimedean framework (i.e.
of numbers which includes also infinitesimal and infinite constants), such as
the Colombeau ring of generalized numbers, in strongly extending classical
results of mathematical analysis using a simpler setting, and in showing
important applications in solving singular differential equations and in quantum
mechanics. The theory of infinite bounded operators represents a clear example
of strong simplification due to a fundamental properties of infinitesimal and
infinite numbers. It will surely stimulate further research of functional
analysis in this framework. The possibility to use infinitesimal and infinite
numbers in modeling physical systems is another important further feature of our
approach.
The primary researchers involved are: The applicant P. Giordano, Prof. M.
Kunzinger and Prof. H. Vernaeve are the senior researchers working on this
project. We also plan to employ two PhD candidates for the development of this
project.