On Schrödinger equations of Okubo type
type: Item
instance of: scholarly article
title: On Schrödinger equations of Okubo type
zbMATH Open document ID: 1062.34103
DOI: 10.1023/B:JODS.0000024120.32465.44
published in: Journal of Dynamical and Control Systems
publication date: 2004-08-19T00:00:00Z
review text: 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 \(a\).
reviewed by: Masaaki Yoshida
Mathematics Subject Classification ID: 34M99
Mathematics Subject Classification ID: 34A26
Mathematics Subject Classification ID: 34C14
zbMATH DE Number: 2093309
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: Fuchsian system
zbMATH Keywords: monodromy
zbMATH Keywords: exponents
zbMATH Keywords: holomorphic vector bundle
zbMATH Keywords: Chern class
zbMATH Keywords: Birkhoff factorization
zbMATH Keywords: partial indices
zbMATH Keywords: resolvent
author: Grigory Giorgadze
author: Giorgi Khimshiashvili
MaRDI profile type: Publication
full work available at URL: b:jods.0000024120.32465.44
OpenAlex ID: W2060376287
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description: scientific article; zbMATH DE number 2093309