Oscillator representations and systems of ordinary differential equations
AUTOR(ES)
Parmeggiani, Alberto
FONTE
The National Academy of Sciences
RESUMO
Using representation-theoretic methods, we determine the spectrum of the 2 × 2 system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}\boldsymbol{{\mathit{Q}}}(\boldsymbol{{\mathit{x}}{\mathrm{,\hspace{.167em}}}{\mathit{D}}}_{\boldsymbol{{\mathit{x}}}})\boldsymbol{{\mathrm{\hspace{.167em}=\hspace{.167em}}}{\mathit{A}}} \left( \boldsymbol{{\mathrm{-}}}\frac{\boldsymbol{{\mathrm{{\partial}}}}^{\boldsymbol{{\mathrm{2}}}}_{\boldsymbol{{\mathit{x}}}}}{\boldsymbol{{\mathrm{2}}}}\boldsymbol{{\mathrm{\hspace{.167em}+\hspace{.167em}}}}\frac{\boldsymbol{{\mathit{x}}}^{\boldsymbol{{\mathrm{2}}}}}{\boldsymbol{{\mathrm{2}}}} \right) \boldsymbol{{\mathrm{\hspace{.167em}+\hspace{.167em}}}{\mathit{B}}} \left( \boldsymbol{{\mathit{x}}{\mathrm{{\partial}}}}_{\boldsymbol{{\mathit{x}}}}\boldsymbol{{\mathrm{\hspace{.167em}+\hspace{.167em}}}}\frac{\boldsymbol{{\mathrm{1}}}}{\boldsymbol{{\mathrm{2}}}} \right) \boldsymbol{{\mathrm{,\hspace{.167em}}}{\mathit{x}}{\mathrm{\hspace{.167em}{\in}\hspace{.167em}{\mathbb{R}},}}}\end{equation*}\end{document} with A, B ∈ Mat2(ℝ) constant matrices such that A = tA > 0 (or <0), B = −tB ≠ 0, and the Hermitian matrix A + iB positive (or negative) definite. We also give results that generalize (in a possible direction) the main construction.
