Quantum hamiltonians eigenvalue problems and special functions

量子哈密顿：本征值问题和特殊函数在分子、高能和核物理中的许多物理应用都归结为对量子哈密顿的研究。对这些哈密顿量的数学分析提供了对所研究系统的物理性质的必要理解。目前的研究集中在物理应用中经常遇到的某些类型的量子哈密顿量，即奇异哈密顿和复哈密顿。研究这些哈密顿量的传统技术通常是失败的，需要开发新的方法。在这个建议中，我重点讨论了特殊函数场，特别是超几何函数域，以及微分方程与量子哈密顿量研究的相互关系。例如，具有实值位势的量子哈密顿量在正实轴上的零点、无穷点或有限点处隔离了奇点。另一类量子哈密顿算符具有复值位势。这两类哈密顿量在散射理论和超对称性中有着广泛的应用。

Quantum Hamiltonians: eigenvalue problems and  special functions Many of the physical applications in molecular, high energy, and nuclear physics are reduced to the study of quantum Hamiltonians. The mathematical analysis of these Hamiltonians provide the necessary understanding of the physical properties of the system under investigation. The present research concentrates on certain types of quantum Hamiltonians that are frequently encountered in physical applications; namely, Singular and Complex Hamiltonians. The conventional techniques for studying these Hamiltonian usually fail and new methods need to be developed.  In this proposal, I focus  on the interrelationship of the field of special functions, particularly the realm of hypergeometric functions, and differential equations with the study of Quantum Hamiltonians.  The Quantum Hamiltonians with real valued potentials have isolated singularities at zero, infinity or at a finite number of points on the positive real axis. The other kind of Quantum Hamiltonians have complex valued potentials. These two types of Hamiltonian have extensive application in scattering theory and super-symmetry.