We prove weighted uniform estimates for the resolvent of the Laplace operator in Schatten spaces, on non-trapping asymptotically conic manifolds of dimension $n\ge 3$, generalizing a result of Frank and Sabin, obtained in the Euclidean setting. As an application of these estimates we establish Lieb-Thirring type bounds for eigenvalues of Schr\"odinger operators with complex potentials on non-trapping asymptotically conic manifolds, extending those of Frank, Frank and Sabin, and Frank and Simon proven in the Euclidean setting. In particular, our results are valid for the metric Schr\"odinger operator in the Euclidean space, with a metric being a sufficiently small compactly supported perturbation of the Euclidean one. To the best of our knowledge, these are the first Lieb-Thirring type bounds for non-self-adjoint elliptic operators, with principal part having variable coefficients.

中文翻译：

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具有非陷阱度量的非自伴随薛定谔算子的特征值界限

我们证明了 Schatten 空间中拉普拉斯算子的解算符的加权均匀估计，在维度为 $n\ge 3$ 的非陷阱渐近圆锥流形上，概括了在欧几里德设置中获得的 Frank 和 Sabin 的结果。作为这些估计的应用，我们为 Schr\"odinger 算子的特征值建立 Lieb-Thirring 类型边界，在非捕获渐近圆锥流形上具有复势，扩展了 Frank、Frank 和 Sabin 以及 Frank 和 Simon 在欧几里得设置中证明的那些特别是，我们的结果对于欧几里得空间中的度量 Schr\"odinger 算子是有效的，度量是欧几里得的一个足够小的紧凑支持的扰动。据我们所知，这些是非自伴随椭圆算子的第一个 Lieb-Thirring 类型边界，