紧流形上具有复势的广义薛定谔算子的预解界与特征值估计
Resolvent bounds and eigenvalue estimates of generalized Schrödinger operators with complex potentials on compact manifolds
- Institut für Analysis und Zahlentheorie, TU Graz（格拉茨工业大学分析与数论研究所）
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AI总结：
本文将复势薛定谔算子的谱界推广至伪微分算子，通过预解原理结合Sogge谱簇界，得到外部与内部区域的预解估计，并在Zoll流形上讨论最优性。
AI中文摘要：
我们将Cuenin关于紧流形上具有复势的薛定谔算子的谱界推广到一般的伪微分框架中。更确切地说，我们研究算子\(P+V\)，其中\(P\)是一个正阶的正自伴椭圆经典伪微分算子，\(V\)是复值函数。主要的分析输入是一个预解原理，表明\(P\)的谱簇估计蕴含沿适当复曲线的\(L^p\)-\ (L^{p'}\)预解估计。结合Sogge的谱簇界，这给出了外部区域的预解估计，推广了Krupchyk和Uhlmann的结果；我们还证明了内部区域的直接预解界。在Zoll流形上，我们讨论了所得谱界的最优性。
英文摘要：
We extend Cuenin's compact-manifold spectral bounds for Schrödinger operators with complex potentials to a general pseudodifferential setting. More precisely, we study operators \(P+V\), where \(P\) is a positive self-adjoint elliptic classical pseudodifferential operator of positive order and \(V\) is complex-valued. The main analytic input is a resolvent principle showing that spectral cluster estimates for \(P\) imply \(L^p\)-\(L^{p'}\) resolvent estimates along suitable complex curves. Combined with Sogge's spectral cluster bounds, this yields exterior-region resolvent estimates extending those of Krupchyk and Uhlmann; we also prove direct resolvent bounds in the interior region. On Zoll manifolds, we discuss the sharpness of the resulting spectral bounds.
