Article
Ufa Mathematical Journal
Volume 4, Number 2, pp. 127-136
The "quantum" linearization of the Painleve equations as the component of theier $L-A$ pairs.
Suleimanov B.I.
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The procedure of the ``quantum'' linearization of the hamiltonian ordinary differential equations with one degree of freedom is entered. It is offered to be used for the classification of the integrable equations of Painleve type. For the Hamiltonian $H = (p^2+q^2)/2$ and all natural numbers $n $ the new solutions $ \Psi (\hbar, t, x, n) $ of the non-stationary the Shr\"{o}dinger equation are constructed. The solutions tend to zero at $x\to\pm\infty$. On curves $x=q_n (\hbar, t) $, defined by the old Bohr- Zommerfeld rule, the solutions satisfy the relation \linebreak $i\hbar \Psi ' _x\equiv p_n (\hbar, t) \Psi $. In this relation $p_n (\hbar, t) = (q_n (\hbar, t)) ' _t $ is the classical momentum corresponding to the harmonic $q_n (\hbar, t) $.