Determinants of Regular Singular Sturm - Liouville Operators

✍ Scribed by Matthias Lesch

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 1014 KB
• Volume: 194
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19981940110

✦ Synopsis

We consider a regular singular Sturm -Liouville operator on the line segment [0,1]. We impose certain boundary conditions such that we obtain a semibounded self-adjoint operator. It is known (cf. Theorem 1.1 below) that the (-function of this operator ha8 a meromorphic continuation to the whole complex plane with 0 being a regular point. Then, according to [RS] the ( -regularized determinant of L is defined by detc(L) := exp ( -{i(O)) . In this paper we are going to express this determinant in terms of the solutions of the homogeneous differential equation Ly = 0 generalizing earlier work of S. LEVJT and U. SHILANSKY [LS], T. DREWUS and H . DYM [DD], and D. BURGHELEA. L. FRIEDLANDER and T . KAPPELER [BFKl, BFKZ]. More precisely we prove the formula Here 'p, 3 is a certain fundamental system of solutions for the homogeneous equation Ly = 0, W ( 9 , 3 ) denotes their Wronski determinant, and uo, v1 are numbers related to the characteristic roots of the regular singular points 0, 1.