Department of Physics,

Moscow State University,

Vorobievy Gory,

119992 Moscow,

RUSSIA

E-mail: olegshv@mail.ru

**Quantum systems with linear constraints and quadratic
Hamiltonians**

**Abstract:**

Quantum systems with constraints are often considered in modern theoretical
physcics. All realistic field models based on the idea of gauge symmetry
are of this type. A partial case of constraints being linear in coordinate
and momenta operators is very important. Namely, when one applies semiclassical
methods to an arbitrary constraint system, the constraints in "general
position case" become linear. In the talk, different mathematicals constructions
for the Hilbert space space for the constraint system are discussed. Properties
of Gaussian and quasi-Gaussian wave functions for these systems are investigated.
An analog of the notion of Maslov complex germ is suggested. Properties
of Hamiltonians being quadratic with respect to the coordinate and momenta
operators are discussed, the Green function for the evolution equation
is constructed. The Maslov theorem (it says that there exists a Gaussian
eigenfunction of the quantum Hamiltonian iff the classical Hamiltonian
system is stable) is generalized to the constrained systems. The case of
infinite number degrees of freedom (constrained Fock space) is also discussed.