On complex oscillation theory, quasiexact solvability and Fredholm Integral Equations
Abstract
Biconfluent Heun equation (BHE) is a confluent case of the general Heun equation which has one more regular singular points than the Gauss hypergeometric equation on the Riemann sphere $\hat{\mathbb{C}}$. Motivated by a Nevanlinna theory (complex oscillation theory) approach, we have established a theory of \textit{periodic} BHE (PBHE) in parallel with the Lamé equation verses the Heun equation, and the Mathieu equation verses the confluent Heun equation. We have established condition that lead to explicit construction of eigensolutions of PBHE, and their single and double orthogonality, and a related firstorder Fredholmtype integral equation for which the corresponding eigensolutions must satisfy. We have also established a Bessel polynomials analogue at the BHE level which is based on the observation that both the Bessel equation and the BHE have a regular singular point at the origin and an irregular singular point at infinity on the Riemann sphere $\hat{\mathbb{C}}$, and that the former equation has orthogonal polynomial solutions with respect to a complex weight. Finally, we relate our results to an equation considered by Turbiner, Bender and Dunne, etc concerning a quasiexact solvable Schrödinger equation generated by first order operators such that the second order operators possess a finitedimensional invariant subspace in a Lie algebra of $SL_2(\mathbb{C})$
 Publication:

arXiv eprints
 Pub Date:
 October 2013
 arXiv:
 arXiv:1310.5507
 Bibcode:
 2013arXiv1310.5507C
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Primary 30D35;
 34M05;
 33E10;
 Secondary 33C47;
 45B05
 EPrint:
 This paper has been withdrawn by the authors due to a new version with different title "Galoisian approach to complex oscillation theory of Hill equations" and many contents changed