On the logarithmic asymptotics of orthogonal measure for special polynomials in the problem of radiation scattering

We consider a class of polynomial quantum Hamiltonians with complete numbers of conservation laws, linear with respect to particle numbers operators. The system of special non-classical polynomials is introduced for the Hamiltonian diagonalization problem. The exact asymptotics for orthogonal measure is obtained for this system of polynomials by logarithmic potentials methods in the case of large particle numbers. The exact case of quadratic Hamiltonian is considered as an explicit example.