Abstract
In Classical Mechanics constraints describe forces restricting the motion of two systems when they are in In Quantum Mechanics in the Schrodinger representation the state of the system is described by a (probability) wave and the Schrodinger equation is dispersive (it does not preserve locality of the wave function): at any time the wave function is spread out over all space. It is difficult to define contact. To avoid this difficulty it is convenient to use the Heisenberg representation and describe the system by means of self-adjoint operators on some function space. Each self-adjoint operator operator has a domain of definition. We consider first in some detail the dynamics in R^3 and later consider the case of dimension two and dimension one. In three dimension contact interactions gives the Efimov spectrum of trimers and quadrimers in low energy physics and the Bose-Einstein condensate, both in low density and in high density. The case of dimension one is particularly interesting because the system may represent three particles on a Y-shaped graph with contact interaction at the vertex. We consider both the case of Bose particles which satisfy the Schrodinger equation and the case of spin 1\2 fermions which satisfy the Pauli equation. They form respectively Bose crystals and Fermi crystals (not necessary periodic). In both cases there is only one bound state at each vertex. In an extended crystal, due to Fermi-Dirac statistics, all bound states may be occupied to form the Fermi sea. The particles on the surface have a Dirac spectrum. We prove that in the semiclassical limit the dynamics of particles on the surface of the Fermi sea is the (classical) motion studied in detail by Novikov and Maltsev. In this semiclassical description the Fermi surface can be deformed by a magnetic field and one may have topological resonances.
Highlights
In Classical Mechanics constraints we describe forces restricting the motion of two systems when they are in contact
Point interactions [3] can be seen as a weak contact interaction between two particles one of which is infinitely massive
Theorem 1: The hamiltonian of a system describing the strong contact interactions of a particle with two identical bosons is limit, in the strong resolvent sense, of hamiltonians with two body negative potentials of class C1 that have support that shrinks to a point with law Vǫ(|x|)
Summary
In Classical Mechanics constraints we describe forces restricting the motion of two systems when they are in contact. Vi,j(|xi − xj|) and Ui,j(|xi − xj|) hat are distributions supported by the boundary This can be verified by taking the scalar product with a function in the domain of H 0 (the free hamiltonian restricted to functions that vanish in a neighborhood of Ŵ) and integrating by parts. This condition implies a very singular modification of the free dynamics at coincidence hyper-planes. The solutions are not in the domain of the free hamiltonian; solution of the Schrödinger equation is only meant in a weak sense, after averaging with a smooth function and integrating by parts.
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