I have been asking myself if one has different analytical representations (analytical extensions in complex functions) of some expressions containing hypergeometric function 2_F_1(a, b; c; z) derived from a Hamiltonian in quantum physics for a 1-D movement of a particle, what is the importance of the multiples set of eigenvalues "E" when choosing a specific analytical extension or continuation of a vital function immersed in the results like that specified in a spectral determinant for the Hamiltonian and its eigenvalues or det(E-H) =0. I am working on this problem and determined a set of eigenvalues "E" from a general solution of the Time-Independent Schrödinger equation in 1-D movement, but I suspect that depending on the analytical extension or analytical continuation of certain functions in the spectral determinant det(E-H) one can represent multiple spectral values of E (eigenvalues). So the analytical extension can represent multiple values of energy in quantum problems.
It would be useful to start with some simple exercises, in order to understand what can and what can't happen.
The spectrum of the Hamiltonian can be expressed as En=/(the label n will be over the integers, if the spectrum is discrete), where |ψn> are the eigenstates.
So it suffices to check how the states transform and how the Hamiltonian transforms under the proposed transformations, to see how the energies transform.
Landau Lifshitz (non relativistic QM) has examples with Hypergeometric functions.
Im not sure that you properly interprate how QM calculations are carried out, but this reference should help.
Juan Weisz Good morning, I have derived a Quantum Hamiltonian whose det(E-H) (basically, the spectral determinant of that) has roots "E" when det(E-H) = 0, so I am interested in det(E-H) as it has hypergeometric functions in its structure, which demands analytical continuation of complex functions in order to calculate several sets of eigenvalues "E". That is my interest
When you diagonalize H the eigenvalues (real if H is Hermitian) appear along the diagonal with equal number to the size of H.
I dont understand your alusion to analytical continuation or several sets.
However Hpergeometric functions confluent or not are an aide to solve some quantum problems (see Landau-Lifshitz). YoU have degenerate roots?
Carlos Hernan Lopez Zapata , in addition to the book of Landau, you may also refer to https://www.researchgate.net/project/Solutions-of-the-Schroedinger-equation-in-terms-of-the-Heun-functions, where there exists a considerable amount of work regarding solutions to stationary Schrodinger equation that are expressed in terms of hypergeometric functions.
Spiros Konstantogiannis Thanks a lot, I will check it. Have a nice day!
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