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

For quantum optics models, polynomial Hamiltonians with respect to creation and annihilation operators are used:
$$ \hat{H}=\sum_{k=1}^{p}\omega_{k}a_{k}^{+}a_{k}^{-}+ \sum_{(r, s)\in J \subset\mathbb{Z}_{+}^{p} }{b_{rs}a^{+r}a^{-s}}+ h.c., $$
where $a^{r}=a_{1}^{r_{1}}...a_{p}^{r_{p}}$, $\omega_{i}$ and $b_{rs}=b_{sr}^{*}$ are some constants and $h.c. $ are Hermitian conjugated terms of $H$. The number of different types of particles is $p$. Standard basis consists of eigenvectors of $\hat{H}_{0}=\sum_{k=1}^{p}\omega_{k}a_{k}^{+}a_{k}^{-}$, it is $\{|n\rangle=|n_{1}\rangle_{1}...|n_{p}\rangle_{p}\}$, where $\hat{n}_{k}=a_{k}^{+}a_{k}^{-}$ is $k$-type particle number operator.

Our goal is to describe the asymptotics of the eigenvalues of the Hamiltonian for large eigenvalues of particle number operator. The system of special non-classical polynomials is introduced for the Hamiltonian diagonalization problem. An asymptotics for orthogonal measure is obtained for these systems of polynomials by logarithmic potential methods when particle number tends to infinity. The exact case of quadratic Hamiltonian will be considered as an explicit example.

This is a joint work with A. I. Aptekarev and Yu. N. Orlov.