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量子哈密顿量的研究。这些哈密顿量的数学分析提供了必要的了解所调查的系统的物理性质。目前的研究集中在某些类型的量子哈密顿，经常遇到的物理应用，即奇异和复杂的哈密顿。研究这些哈密顿量的传统技术通常失败，需要开发新的方法。 在这个建议中，我专注于特殊函数领域的相互关系，特别是超几何函数领域，以及微分方程与量子哈密顿量的研究。 具有真实的值势的量子哈密顿量在正真实的轴上的零、无穷大或有限个点处具有孤立奇点。另一种量子哈密顿量具有复值势。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.