Abstract:
We describe the solution of algebraic equations for the coefficients of the normal factorization
Ut=eiˆHt=este−12(a†,Rta†)−(gt,a†)e(a†,Cta)e12(a,¯ρta)+(¯ft,a)
of the unitary group Ut with Hamiltonian
ˆH=i2((a†,Aa†)−(a,¯Aa))+(a†,Ba)+i(a†,h)−i(a,¯h)
in terms of the matrices Φt, Ψt which define the canonical transformation of the creation-annihilation operators. Such a decomposition defines the normal symbol of squeezing and the inner products of squeezed states which are necessary for constructing a basis in a linear hull generated by a finite set of squeezed states. A new class of solvable quantum problems is related to Hamiltonians with A and B such that [A¯A,B]=0 and rankA¯A⩾rankB. In this case, the solution is expressed in terms of the eigenvalues of the Hermitian matrix A¯A−B2.