TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 10, Pages 4155–4171 S 0002-9947(01)02765-9 Article electronically published on May 17, 2001 ON THE INVERSE SPECTRAL THEORY OF SCHR ¨ ODINGER AND DIRAC OPERATORS MIKL ´ OS HORV ´ ATH Abstract. We prove that under some conditions finitely many partially known spectra and partial information on the potential entirely determine the potential. This extends former results of Hochstadt, Lieberman, Gesztesy, Simon and others. 1. Introduction Consider the Schr¨ odinger operator −y ′′ + q(x)y = λy on (0,π) (1.1) with the boundary conditions y(0) cos α + y ′ (0) sin α =0, (1.2) y(π) cos β + y ′ (π) sin β =0, (1.3) where q ∈ L 1 (0,π). The eigenvalues of (1.1)–(1.3) form the sequence λ 0 <λ 1 < .... Given a complex value λ define y(x, λ) as the solution of (1.1) with the initial conditions y(0) = sin α, y ′ (0) = − cos α. (1.4) For λ = λ n , y(x, λ n ) is the eigenfunction of (1.1)–(1.3) corresponding to the eigen- value λ n . Introduce the normalizing constants α 2 n =  π 0 y 2 (x, λ n )dx. The spectral function of the problem (1.1)–(1.3) is defined to be (λ)=      ∑ 0<λnλ 1 α 2 n if λ> 0, − ∑ λ<λn0 1 α 2 n if λ< 0. (1.5) Received by the editors February 16, 2000 and, in revised form, June 7, 2000. 1991 Mathematics Subject Classification. Primary 34A55, 34B20; Secondary 34L40, 47A75. Key words and phrases. Inverse spectral theory, m-function, spectral function. Research supported by the Hungarian NSF Grant OTKA T#32374. c 2001 American Mathematical Society 4155 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use