A new interpolation formula for the Titchmarsh-Weyl -function

Authors:
Alexei Rybkin and Vu Kim Tuan

Journal:
Proc. Amer. Math. Soc. **137** (2009), 4177-4185

MSC (2000):
Primary 47E05, 65D05

Published electronically:
June 25, 2009

MathSciNet review:
2538578

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For the Titchmarsh-Weyl -function of the half-line Schrödinger operator with Dirichlet boundary conditions we put forward a new interpolation formula which allows one to reconstruct the -function from its values on a certain infinite set of points for a broad class of potentials.

**1.**Jim Agler and John E. McCarthy,*Pick interpolation and Hilbert function spaces*, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR**1882259****2.**Sergei Avdonin, Victor Mikhaylov, and Alexei Rybkin,*The boundary control approach to the Titchmarsh-Weyl 𝑚-function. I. The response operator and the 𝐴-amplitude*, Comm. Math. Phys.**275**(2007), no. 3, 791–803. MR**2336364**, 10.1007/s00220-007-0315-2**3.**Amin Boumenir and Kim Tuan Vu,*The interpolation of the Titchmarsh-Weyl function*, J. Math. Anal. Appl.**335**(2007), no. 1, 72–78. MR**2340306**, 10.1016/j.jmaa.2007.01.054**4.**Amin Boumenir and Vu Kim Tuan,*Sampling eigenvalues in Hardy spaces*, SIAM J. Numer. Anal.**45**(2007), no. 2, 473–483. MR**2300282**, 10.1137/050647335**5.**Fritz Gesztesy and Barry Simon,*A new approach to inverse spectral theory. II. General real potentials and the connection to the spectral measure*, Ann. of Math. (2)**152**(2000), no. 2, 593–643. MR**1804532**, 10.2307/2661393**6.**A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev,*Integrals and series. Vol. 3*, Gordon and Breach Science Publishers, New York, 1990. More special functions; Translated from the Russian by G. G. Gould. MR**1054647****7.**A. G. Ramm,*Recovery of the potential from 𝐼-function*, C. R. Math. Rep. Acad. Sci. Canada**9**(1987), no. 4, 177–182. MR**896977****8.**Barry Simon,*A new approach to inverse spectral theory. I. Fundamental formalism*, Ann. of Math. (2)**150**(1999), no. 3, 1029–1057. MR**1740987**, 10.2307/121061**9.**P. K. Suetin,*Klassicheskie ortogonalnye mnogochleny*, “Nauka”, Moscow, 1979 (Russian). Second edition, augmented. MR**548727**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47E05,
65D05

Retrieve articles in all journals with MSC (2000): 47E05, 65D05

Additional Information

**Alexei Rybkin**

Affiliation:
Department of Mathematics and Statistics, University of Alaska Fairbanks, P.O. Box 756660, Fairbanks, Alaska 99775

Email:
ffavr@uaf.edu

**Vu Kim Tuan**

Affiliation:
Department of Mathematics, University of West Georgia, Carrollton, Georgia 30118

Email:
vu@westga.edu

DOI:
https://doi.org/10.1090/S0002-9939-09-09983-3

Keywords:
Titchmarsh-Weyl $m$-function,
interpolation,
$A$-amplitude,
Hardy space

Received by editor(s):
October 28, 2008

Received by editor(s) in revised form:
April 2, 2009

Published electronically:
June 25, 2009

Additional Notes:
This research was supported in part by the U.S. National Science Foundation under grant DMS 070747

Communicated by:
Nigel J. Kalton

Article copyright:
© Copyright 2009
American Mathematical Society