Physics NotePublished
Solutions for confluent and double-confluent Heun equations and some applications
Bartolomeu Donatila Bonorino Figueiredo
Publication details
- Authors
- Bartolomeu Donatila Bonorino Figueiredo
- Type
- Physics Note
- Status
- Published
- Publication date
- 31/01/2008
- Description
- This paper examines some solutions for confluent and double-confluent Heun equations and their applications to the Schr\"odinger equation with quasi-exactly solvable potentials. In the first place, we review two Leaver's solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation [E. W. Leaver, {\it J. Math. Phys.} {\bf 27}, 1238 (1986)] and introduce an additional expansion in series of irregular confluent hypergeometric functions. Then, we find the conditions under which one of these solutions can be written as a linear combination of the others. In the second place, by means of limiting procedures we generate solutions for the double-confluent equation as well as for special limits of both the confluent and double-confluent equations. In the third place, solutions of the Heun equations are used to solve the one-dimensional Schr\"odinger equation for quasi-exactly solvable potentials. We consider a symmetric and an asymmetric double-Morse potentials which appear in the theory of quantum spin systems [O. B. Zaslavskii and V. V. Ulyanov, {\it Sov. Phys.} JETP {\bf 60}, 991 (1984)], a bottomless volcano-type potential which gives degenerate eigenstates [S. Kar and R. R. Parwani, {\it Europhys. Lett.}, {\bf 80}, 30004 (2007)], and a potential which leads to quasinormal modes, that is, to solutions presenting complex energies [H. T. Cho and C. L. Ho, {\it J. Phys. A: Math. Theor.} {\bf 40}, 1325 (2007)].
- Year
- 2008
- Linked to a thesis
- No
- Linked to a grant
- No