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Inverse problems for impulsive Sturm-Liouville operator with spectral parameter linearly contained in boundary conditions. (English) Zbl 1181.34019

The following boundary-value problem for the impulsive Sturm-Liouville operator with a discontinuity at d(0,π 2), namely

-y '' +qy=λy,in(0,π){d}

subject to

y ' (0)=0,lim xd y(x)=αlim xd y(x),lim xd y ' (x)=1 αlim xd y ' (x),

where α + {1}, qL 2 (0,π), and another boundary condition at x=π containing the spectral parameter λ linearly, namely

λ(y ' (π)+Hy(π))=H 1 y ' (π)+H 2 y(π),

where HH 1 >H 2 , is investigated.

Uniqueness of the sixtet (q(x),H,H 1 ,H 2 ,d,α) is established from the knowledge of either: (i) the Weyl function; (ii) from the spectral data {λ n ,α n { n0 ; or (iii) from the two spectra {λ n ,μ n } n0 .

34A55Inverse problems of ODE
34B24Sturm-Liouville theory
34L05General spectral theory for OD operators