We prove and apply two theorems: first, a quantitative, scale-free unique continuation estimate for functions in a spectral subspace of a Schrödinger operator on a bounded or unbounded domain; second, a perturbation and lifting estimate for edges of the essential spectrum of a self-adjoint operator under a semi-definite perturbation. These two results are combined to obtain lower and upper Lipschitz bounds on the function parametrizing locally a chosen edge of the essential spectrum of a Schrödinger operator in dependence of a coupling constant. Analogous estimates for eigenvalues, possibly in gaps of the essential spectrum, are exhibited as well.

中文翻译：

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无界域上薛定ding算子的谱带边缘的唯一延续和提升

我们证明并应用两个定理：首先，对有界或无界域上Schrödinger算子的光谱子空间中函数的定量，无标度的唯一连续估计；第二，在半确定摄动下，自伴算子基本谱边的摄动和提升估计。将这两个结果结合起来，可获得根据耦合常数局部参数化Schrödinger算子的基本谱的选定边的函数的Lipschitz上下边界。还显示了可能在必要光谱的间隙中的特征值的类似估计。