论文标题
特征函数和韦格纳估计值的唯一延续
Unique continuation for the gradient of eigenfunctions and Wegner estimates for random divergence-type operators
论文作者
论文摘要
我们证明了差异型操作员的特征函数的梯度，即形式的$ - \ mathrm {div} a \ nabla $，其中矩阵$ a $ a $均匀椭圆形。该证明使用椭圆二阶运算符的唯一延续原理，并在$ l^2 $ norm上使用与严格阳性特征值相对应的$ l^2 $ norm。 作为一个应用程序，我们证明了特征值提起估计，使我们能够证明对随机散布型运算符的Wegner估计值。在这里，我们的方法使我们能够摆脱限制性覆盖条件，这在先前证明此类模型的韦格纳估计值的证据中至关重要。
We prove a scale-free quantitative unique continuation estimate for the gradient of eigenfunctions of divergence-type operators, i.e. operators of the form $-\mathrm{div}A\nabla$, where the matrix function $A$ is uniformly elliptic. The proof uses a unique continuation principle for elliptic second order operators and a lower bound on the $L^2$-norm of the gradient of eigenfunctions corresponding to strictly positive eigenvalues. As an application, we prove an eigenvalue lifting estimate that allows us to prove a Wegner estimate for random divergence-type operators. Here our approach allows us to get rid of a restrictive covering condition that was essential in previous proofs of Wegner estimates for such models.
