Inverse problem for a damped Stieltjes string from parts of spectra

Abstract

We solve the following inverse problem for boundary value problems generated by the difference equations describing the motion of a Stieltjes string (a thread with beads). Given are certain parts of the spectra of two boundary value problems with two different Robin conditions at the left end and the same damping condition at the right end. From these two partial spectra, the difference of the Robin parameters, the damping constant, and the total length of the string, find the values of the point masses, and of the lengths of the intervals between them. We establish necessary and sufficient conditions for two sets of complex numbers to be the eigenvalues of two such boundary value problems and give a constructive solution of the inverse problem.

Keywords:

• inverse problem
• Stieltjes string
• Robin boundary condition
• damping
• continued fraction
• Hermite–Biehler polynomial

AMS Subject Classifications:

• Prim. 39A60
• Sec. 15A29
• 30B70
• 30H15
• 70F17

Acknowledgements

C. Tretter gratefully acknowledges a guest professorship of the Knut och Alice Wallenbergs Stiftelse and thanks the Matematiska institutionen at Stockholms universitet for the kind hospitality.