

Complex Approximations, Orthogonal Polynomials and Applications Workshop
June 10, 2021 15:30–15:55, Sochi






On the asymptotics of orthogonal measure for special polynomials in the problem of radiation scattering
M. A. Lapik^{} ^{} Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow

Number of views: 
This page:  17 

Abstract:
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 nonclassical
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.
Language: English
Website:
https://us02web.zoom.us/j/8618528524?pwd=MmxGeHRWZHZnS0NLQi9jTTFTTzFrQT09
^{*} Zoom conference ID: 861 852 8524 , password: caopa

