Spectral Asymptotics and Dynamics

我将完成这个项目，时间跨度超过30年，目前的形式是“锐谱渐近论及其在数学物理中的应用”。为了包装它，我计划 (1)重做许多(主要是我自己的)旧结果，目标--更普遍、更尖锐的结果； (2)证明以前无法证明的全新结果。 实现这些目标还有赖于对相应动力学(经典和“量子”)的研究。 现在都有货 (A)卓越的微局部分析技术。 (B)上级变分估计(特别是在临界和超临界情况下)； (C)经典动力系统理论的较好结果。 我还计划重温磁场中重分子基态能量的渐近性问题，特别是在磁场超强的情况下，以及相关的过度电荷问题，并考虑磁场是自生的或自生和外加的情况。

I am going to finish the project, spanning over thirty years and which in its current form is "Sharp Spectral Asymptotics and their Applications to Mathematical Physics". To wrap it I plan (1) To redo many of (mainly my own) old results, the goal - more general and more sharp results; (2) To prove completely new results, which could not be proven before. Achievement of these goals also relies upon a study of the corresponding dynamics (classical and "quantum"). Now are available (a) superior microlocal analysis technique. (b) superior variational estimates (especially in the critical and super-critical cases); (c) better results in the theory of classical dynamical systems. I also plan to revisit the problem of the asymptotics of the ground state energy for heavy molecules in the magnetic field, especially in the case when magnetic field is ultra-strong and the related problems of the excessive charge, and also consider the case when magnetic field is either self-generated or a sum of self-generated and external.