Colloquium

Seminar information archive ~12/07Next seminarFuture seminars 12/08~

Organizer(s) ASUKE Taro, TERADA Itaru, HASEGAWA Ryu, MIYAMOTO Yasuhito (chair)
URL https://www.ms.u-tokyo.ac.jp/seminar/colloquium_e/index_e.html

2010/10/08

16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Ryu Sasaki (Yukawa Institute for Theoretical Physics, Kyoto University)
Exceptional Jacobi polynomials as solutions of a Schroedinger
(Sturm-Liouville) equation with $3 +¥ell$ ($¥ell=1,2,¥ldots) regular
singularities (JAPANESE)
[ Abstract ]
Global solutions of Fuchsian differential equations with more than 3 (hypergeometric) or four (Heun) regular singularities had been virtually unkown. Here I present a complete set of eigenfunctions of a Schroedinger (Sturm-Liouville) equation with $3 + ¥ell$ ($¥ell=1,2,¥ldots$) regular singularities. They are deformations of the Darboux-P¥" oschl-Teller potential with the Hamiltonian (Schroedinger operator) ¥[ ¥mathcal{H}=-¥frac{d^2}{dx^2}+¥frac{g(g-1)}{¥sin^2x}+¥frac{h(h-1)} {¥cos^2x}¥] The eigenfunctions consist of the {¥em exceptional Jacobi polynomials} $¥{P_{¥ell,n}(¥eta)¥}$, $n=0,1,2,¥ldots$, with deg($P_{¥ell,n}$)$=n+¥ell$. Thus the restriction due to Bochner's theorem does not apply. The confluent limit produces two sets of the exceptional Laguerre polynomials for $¥ell=1,2,¥ldots$. Similar deformation method provides the exceptional Wilson and Askey-Wilson polynomials for $¥ell=1,2,¥ldots$.