UNIQUENESS OF THE SOLUTION OF HALF INVERSE PROBLEM FOR THE IMPULSIVE STURM LIOUVILLE OPERATOR

Abstract

Abstract. The half-inverse spectral problem for an impulsive Sturm–Liouville operator consists in reconstruction of this operator from itsspectrum and half of the potential. In this study, the spectrum of the im-pulsive Sturm–Liouville problem is given and by using the Hochstadt andLieberman’s method we show that if q(x) is prescribed on0, π2 , thenonly one spectrum is sufficient to determine q(x) on the interval (0,π) forthis problem 1. IntroductionAssume that Q(x) is a real-valued function in L 2 (0,π),αand βare realnumbers such that 0 0. Define ρ(x) =ˆ1, x π2 anddenote by L= L(Q(x),ρ(x),β) a Sturm–Liouville problem in L 2 (0,π) that isgiven by the differential equation(1) ly:= −y ′′ (x) +Q(x)y(x) = λρ(x)y(x), x∈ (0,π2)∪ (π2,π)with the boundary conditions(2) U(y) := y ′ (0) = 0,(3) V(y) := y ′ (π) = 0,and the jump conditions(4)y(π2+0) = βy(π2−0) ,y ′ (π2+0) = β −1 y ′ (π2−0),where λis the spectral parameter. Received October 25, 2011; Revised February 22, 2012.2010 Mathematics Subject Classification. 34A55, 34B24, 34L05, 45C05.Key words and phrases. Sturm-Liouville operator, determination of the potential, discon-tinuous condition, half inverse problem.