Quantum hamiltonians eigenvalue problems and special functions

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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.