Upcoming Seminars & Events
Quantum graph models are extremely useful but they also have some drawbacks. One of them concerns the physical meaning of the vertex coupling. The self-adjointness requirement leaves a substantial freedom expressed through parameters appearing in the conditions matching the wave function at the graph vertices. It is a longstanding problem whether one can motivate their choice by approximating the graph Hamiltonian by operators on a family of networks, i.e. systems of tubular manifolds the transverse size of which tends to zero. It appears that the answer depends on the conditions imposed on tube boundaries. In this talk we present a complete solution for Neumann networks: we demonstrate that adding properly scaled potentials, both scalar and vector ones, and changing locally the graph topology, one can approximate any admissible vertex coupling. The result comes from a common work with Taksu Cheon, Olaf Post, and Ond\vrej Turek.
The reasonable effectiveness of mathematical deformation theory in physics, especially quantum mechanics and maybe elementary particle symmetries
In 1960 Wigner marveled about ``the unreasonable effectiveness of mathematics in the natural sciences," referring mainly to physics. In that spirit we shall first explain how a posteriori relativity and quantum mechanics can be obtained from previously known theories using the mathematical theory of deformations. After a tachyonic overview of how the standard model of elementary particles arose from empirically guessed symmetries we indicate how these symmetries could (very reasonably) be obtained from those of relativity using deformations (including quantization). This poses difficult and interesting mathematical problems with potentially challenging applications to physics.