In this paper, we consider the Schrodinger equation, $$Hu = - {u^"} + \left({V\left(x \right) + {V_0}\left(x \right)} \right)u = Eu,$$ where V0(x) is 1-periodic and V(x) is a decaying perturbation. By Floquet theory, the spectrum of H0 = − ∇2 + V0 is purely absolutely continuous and consists of a union of closed intervals (often referred to as spectral bands). Given any finite set of points $$\left\{{{E_j}} \right\}_{j = 1}^N$$ in any spectral band of H0 obeying a mild non-resonance condition, we construct smooth functions $$V\left(x \right) = {{O\left(1 \right)} \over {1 + \left| x \right|}}$$ such that H = H0 + V has eigenvalues $$\left\{{{E_j}} \right\}_{j = 1}^N$$ . Given any countable set of points {Ej} in any spectral band of Ho obeying the same non-resonance condition, and any function h(x) > 0 going to infinity arbitrarily slowly, we construct smooth functions $$\left| {V\left(x \right)} \right| \le {{h\left(x \right)} \over {1 + \left| x \right|}}$$ such that H = H0 + V has eigenvalues {Ej}. On the other hand, we show that there is no eigenvalue of H = H0 + V embedded in the spectral bands if $$V\left(x \right) = {{o\left(1 \right)} \over {1 + \left| x \right|}}$$ as x goes to infinity. We prove also an analogous result for Jacobi operators.

中文翻译：

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嵌入到扰动周期算子谱带中的特征值的急剧谱跃迁

我们构造平滑函数 $$\left| {V\left(x \right)} \right| \le {{h\left(x \right)} \over {1 + \left| x \right|}}$$ 使得 H = H0 + V 具有特征值 {Ej}。另一方面，如果 $$V\left(x \right) = {{o\left(1 \right)} \over {1 + \左| x \right|}}$$ 当 x 趋于无穷大时。我们也证明了雅可比算子的类似结果。