一维谐振子的Fulling-Kuchment定理
A Fulling-Kuchment theorem for the 1D harmonic oscillator
摘要：
证明了存在一对“非等谱”一维半经典薛定谔算子，其谱模h^ \infty一致.特别地，它们的所有半经典迹不变量都是相同的。我们的证明是基于一个想法的Fulling-Kuchment和阿达玛的变分公式适用于适当的扰动的谐振子。保留字：逆谱问题，半经典薛定谔算子，迹不变量，阿达玛变分公式，谐振子，彭罗斯蘑菇，斯特姆-刘维理论。
英文摘要：
We prove that there exists a pair of "non-isospectral" 1D semiclassical Schrödinger operators whose spectra agree modulo h^\infty. In particular, all their semiclassical trace invariants are the same. Our proof is based on an idea of Fulling-Kuchment and Hadamard's variational formula applied to suitable perturbations of the harmonic oscillator. Keywords: Inverse spectral problems, semiclassical Schrödinger operators, trace invariants, Hadamard's variational formula, harmonic oscillator, Penrose mushroom, Sturm-Liouville theory.
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