Semiclassical asymptotics of a model problem (Q1326906)

The theory of semi-classical approximation, of differential operators in the last two decades has tackled using heavy geometrical theory. However, a good deal of detailed information has been available classically in low dimensional cases. Here, the authors discuss the semi-classical asymptotics of a differential operator \(H\) associated to the classical Hamiltonian \[ H(p_1, p_2, q_1, q_2)= \sum^2_{k, j= 1} {1\over 2} \omega_{kj}(p_k+ iq_k) (p_j- iq_j)+ \sum^2_{k, j= 1} V_{k, j} |p_k+ iq_k|^2 |p_j- iq_j|^2, \] where \(\omega= (\begin{smallmatrix} 2\\ -1\end{smallmatrix} \begin{smallmatrix} - 1\\ 2\end{smallmatrix})\) and \(V\) is a symmetric, real positive definite matrix, via the correspondence principle \[ (p_k \psi)( x_1, x_2)= - ih {\partial\psi\over \partial x_k} (x_1, x_2),\quad (q_k \psi) (x_1, x_2)= x_k \psi(x_1, x_2). \] Using the contact transformation \(p_k= \sqrt{2I_k}\cos \phi_k\), \(q_k= \sqrt{2I_k} \sin \phi_k\) the Hamiltonian may be rewritten in the form: \[ H(I_1, I_2, \varphi_1, \varphi_2)= \sum^2_{k, j= 1} V_{kj} I_j I_k+ \sum^2_{k, j= 1} \omega_{kj} \sqrt{I_j I_k} \cos(\varphi_k- \varphi_j). \] A trajectory \(\Lambda'(E)\), \(\varphi_1= \varphi_2= \Omega t\), \(I_1= I_{10}\), \(I_2= I_{20}\), satisfies the associated Hamiltonian equations if the relations \[ \Omega I_1= 2I_2- \sqrt{I_1 I_2}+ 2V_{11} I^2_1+ 2V_{12} I_1 I_2,\quad \Omega I_2= 2I_2- \sqrt{I_1 I_2}+ 2V_{21} I_1 I_2+ 2V_{22} I^2_2 \] are satisfied. An explicit calculation involving the first variational equation associated to the Hamiltonian system enables one to calculate a basis of the space of vectors skew-orthogonal to \((\dot I, \dot \varphi)\): \[ \left\{\begin{pmatrix} \delta I\\ \delta\varphi\end{pmatrix},\;\begin{pmatrix} \dot I\\ \dot \varphi\end{pmatrix}\right\}= 0. \] The eigenvalues and eigenvectors then being calculated by well-known formulae of semi-classical approximation theory as \(h\downarrow 0\). At energy \(E\) with \(\Omega\), \(I_1\), \(I_2\) related as above and \(\beta(E)\) given by \[ \beta(E)= \sqrt{ 4\sqrt{I_1(E) I_2(E)} (V_{11}+ V_{22}- 2V12)+{(I_1(E)+I_2(E))^2\over I_1(E) I_2(E)}} \] one has \[ {I_1(E)+ I_2(E)\over h}= l_1+ \left( {\beta(E)\over \Omega(E)}+ 2\right) \left(\nu_1+ {1\over 2}\right),\quad l, v_1= 0, 1,2,\dots\;. \] Specific formulae are given for the asymptotic eigenfunctions near the turning points.