Microlocal Analysis and Spectral and Scattering Asymptotics

在最初的两年里，我计划完成这个项目，时间跨度超过35年，目前的形式是“锐谱渐近及其在数学物理中的应用”。作为总结，我计划重温超强磁场中原子重分子基态能量的渐近性问题，并考虑磁场包括自生分量时的情况(该项目的所有其他部分都已完成)。这将导致我3000多页的研究专著(同名)的发布。*在剩下的三年里，我计划切换到一些新的问题：*国家综合密度的渐近性，目标是用更好的余数估计得出渐近性，而不是直接从局部谱渐近性得出。这将需要进一步发展微观局域分析和奇点的长期传播。*散射理论和复平面上的共振(散射极)分布。*这将需要将早期开发的自伴算符的技术扩展到耗散算符。

During the first two years I plan to finish the project, spanning over 35 years and which in its current form is "Sharp Spectral Asymptotics and their Applications to Mathematical Physics". To wrap it I plan to revisit the problem of the asymptotics of the ground state energy for atoms heavy molecules in the ultra-strong magnetic field, and also consider the case when magnetic field includes self-generated component (all other parts of the project are done). It will result in the releasing my research monograph of over 3,000 pages (with the same name).******During remaining three years I plan to switch to some new problems: ******Asymptotics of the Integrated Density of States with the goal to derive asymptotics with the better remainder estimates than follow directly from local spectral asymptotics. This would require further development of the micro local analysis and long term propagation of singularities.***Scattering Theory and the Distribution of the Resonances (Scattering Poles) ?n the complex plane.***This would require the extension of the technique developed earlier for self-adjoint operators to dissipative operators.