On Schrödinger equations of Okubo type
A system of linear ordinary differential equations of the form (\(A\) and \(B\) are constant matrices) \[ (zI- B)\dot X= AX\tag{1} \] is called a system of Okubo type if the matrix \(B\) is diagonal. Consider a Schrödinger equation of the form \[ i{\partial\Psi(t)\over\partial t}= H(t)\Psi(t)\tag{2} \] with time-dependent Hamiltonian \(H(t)= (H_{ij}(t))\), \(i,j= 1,\dots, N\), where \[ \begin{gathered} H_{11}= \varepsilon(t),\quad H_{12}= V_2,\quad H_{13}= V_3,\dots, H_{1N}= V_N,\\ H_{21}= V_2,\quad H_{31}= V_{31}= V_3,\dots, H_{N1}= V_N,\text{ and }H_{ij}= 0\text{ otherwise},\end{gathered} \] \(\Psi(t)= (\psi_1(t),\dots, \psi_N(t))\) is a wave function, \(V_j\), \(j= 2,\dots, N\), are constants, and the time-dependent part \(\varepsilon\) has the form \(\varepsilon(t)= E_1\tanh(t/T)\) with constant \(E_1\) and \(T\). The authors prove: (2) is reducible to an \(N\)-dimensional Fuchsian system of Okubo type \[ (zI_N- B){d\Phi(z)\over dz}= A\Phi(z) \] with \(B= \text{diag}(i,-i,\dots, -i)\). The Chern class of the canonical extension \(E_0\) defined by system (1) is expressed by the formula \[ c(E_0)= \sum^m_{j=1} \sum^{n_j}_{i=1} [\text{Re\,}\lambda^j_i]+ \sum^n_{i=1} [-\text{Re\,}\lambda^\infty_i], \] where \(\lambda^j_i\) are the eigenvalues of the block \(A_{jj}\), \(j= 1,\dots, m\), \(\infty\). For a real number \(a\), \([a]\) is the largest integer not exceeding \(a\).
scientific article; zbMATH DE number 2093309