"Quantum" linearization of Painlev\'{e} equations as a component of their $L,A$ pairs

Abstract

The procedure of the "quantum" linearization of the Hamiltonian ordinary differential equations with one degree of freedom is introduced. It is offered to be used for the classification of integrable equations of the Painleve type. By this procedure and all natural numbers $n$ we construct the solutions $\Psi(\hbar,t,x,n)$ to the non-stationary Shr\"{o}dinger equation with the Hamiltonian $H = (p^2+q^2)/2$ which tend to zero as $x\to\pm\infty$. On the curves $x=q_n (\hbar, t) $ defined by the old Bohr-Sommerfeld quantization rule the solutions satisfy the relation $i\hbar \Psi '_x\equiv p_n (\hbar, t) \Psi $, where $p_n (\hbar, t) = (q_n (\hbar, t)) '_t $ is the classical momentum corresponding to the harmonic $q_n (\hbar, t) $.