Microlocal Analysis and Spectral and Scattering Asymptotics

在头两年，我计划完成该项目，跨越35年，在其目前的形式是“夏普谱渐近及其应用数学物理”。作为总结，我计划重新讨论原子重分子在超强磁场中基态能量的渐近性问题，并考虑磁场包含自生分量的情况（项目的所有其他部分都已完成）。这将导致我的研究专著超过3,000页（同名）。 在剩下的三年里，我计划转向一些新的问题： 积分状态密度的渐近性，目标是导出比直接从局部谱渐近性导出的渐近性更好的剩余估计。这将需要进一步发展的微观局部分析和长期传播的奇异性。 散射理论与共振（散射极）分布在复平面上。 这就需要把早先发展的自伴算子的技巧推广到耗散算子。

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.