A Half-Inverse Problem with Eigenparameter Dependent Boundary Conditions

Abstract

We consider the regular Sturm–Liouville problem

with boundary conditions

where ad − bc > 0, c ≠ 0. In this work, using the Hochstadt and Lieberman's method [15] we show that if q(x) is prescribed on the half interval , then a single spectrum suffices to determine q(x) on [0, 1], the boundary condition at x = 0 and “the asymptotic” boundary condition at x = 1.

Keywords:

• Eigenparameter dependent boundary conditions
• Eigenvalue
• Half-inverse problem
• Sturm–Liouville problem

AMS Subject Classification:

• 34A55
• 34B24
• 45C05

ACKNOWLEDGMENTS

The author would like to thank the referees for their careful reading and valuable comments in improving the original manuscript.

This work was supported by a Grant-in-Aid for Star of Purple and Gold from Nanjing University of Science and Technology (AB 41366) and a Grant-in-Aid for Scientific Research from Nanjing University of Science and Technology (XKF09041).