四阶周期算子的谱渐近性
Spectral asymptotics for periodic fourth order operators
摘要：
我们考虑具有实周期势$V$的实直线上的算子${d^4dt^4}+V$。这个算子的谱是绝对连续的，并且由间隔隔开的区间组成。我们定义了一个李雅普诺夫函数，这是解析的两个单黎曼曲面。在每一个薄片上，李雅普诺夫函数具有与标量情形相同的性质，但它有分支点，我们称之为共振。我们证明了存在的真正的以及非真正的共振特定的潜力。我们确定的周期和反周期的频谱和共振在高能量的渐近性。我们证明了存在两种类型的带隙：1）稳定带隙，其中端点是周期和反周期本征值，2）不稳定（共振）带隙，其中端点是共振（即，在频谱底部以上的李雅普诺夫函数的实分支点）。我们还表明，周期谱和反周期谱一起决定我们的运营商的频谱。最后，我们表明，对于小的潜力$V\ne 0$的频谱在最低频带的多重数为4和底部的频谱是一个共振，而不是周期（或反周期）的本征值。
英文摘要：
We consider the operator ${d^4dt^4}+V$ on the real line with a real periodic potential $V$. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a Lyapunov function which is analytic on a two sheeted Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove the existence of real as well as non-real resonances for specific potentials. We determine the asymptotics of the periodic and anti-periodic spectrum and of the resonances at high energy. We show that there exist two type of gaps: 1) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, 2) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function above the bottom of the spectrum). We also show that the periodic and anti-periodic spectrum together determine the spectrum of our operator. Finally, we show that for small potentials $V\ne 0$ the spectrum in the lowest band has multiplicity 4 and the bottom of the spectrum is a resonance, and not a periodic (or anti-periodic) eigenvalue.
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