# Charles Fulton

**Emeritus Faculty** | Mathematical Sciences

## Contact Information

## Personal Overview

Spectral theory of ordinary differential operators

Mathematical software for Sturm-Liouville problems

Numerical linear algebra

Mathematical software linear algebra problems on vector and parallel computers

Research in all of the above areas has been supported by six National Science Foundation Grants over the last 30 years.

## Educational Background

B.A. University of Redlands, 1965

M.S. University of Minnesota, 1967

D.Sc. Rheinisch-Westfalische Technische Hochschule Aachen, Germany, 1973

## Professional Experience

Recent Conference talks on current research have been given in sessions Dr. Fulton has organized at:

(i) 9th Meeting of American Institute of Mathematical Sciences (AIMS) in Orlando, Florida, July 2012

(ii) World Congress of Nonlinear Analysts (WCNA-2008) in Orlando, Florida, in July 2008

## Selected Publications

# C. Fulton, D. Pearson, and s. Pruess, Estimating Spectral Density Functions for Sturm-Liouville Problems with two singular endpoints, IMA Jour. of Numerical Analysis, published on-line Oct 17, 2013. Download link:

# C. Fulton, D. Pearson, and S. Pruess, Characterization of the Spectral Density Function for a one-sided tridiagonal Jacobi Matrix Operator, Discrete and Continuous Dynamical Systems, Supplement 2013, (2013) pp. 247-257 Download link:

http://aimsciences.org/journals/displayPaperPro.jsp?paperID=9210

# C. Fulton, H. Langer, and A. Luger, Mark Krein's Directing Functionals and Singular Potentials, Math. Nachr. 285 (14-15) (2012), 1791-1798 (also available as arXiv 1110.6709v1).

# C. Fulton and H. Langer,Sturm-Liouville Operators with singularities and generalized Nevanlinna functions, Complex Anal. and Oper. Theory 4 (2010), 179-243.

# C. Fulton, D. Pearson and S. Pruess, Titchmarsh Weyl theory for tridiagonal Jacobi matrices and computation of their spectral functions, Chap 15 (pp. 167-174) in Advaces in Mathematical Problems in Engineering Aerospace and Sciences,ed. S. Sivasundaram, Cambridge Scientific Publishers, Ltd, Dec 2009

# C. Fulton, Titchmarsh-Weyl m-functions for Second-order Sturm-Liouville Problems with two singular endpoints Math. Nachr. 281 (2008), 1418-1475.

# C. Fulton, D. Pearson, and S. Pruess, New characterizations of spectral density functions for singular Sturm-Liouville problems, J. Comput. Appl. Math 212 (2) (2008), pp. 194-213.

# C. Fulton, D. Pearson, and S. Pruess, Efficient calculation of spectral density functions for specific classes of singular Sturm Liouville problems, J. Comput. Appl. Math (2008) 212 (2), pp. 150-178.

# G.W. Howell, J. Demmel, C. Fulton, S. Hammarling, K. Marmol, ACM Trans. on Math. Softw. 32 (3) (2008), pp.~13-46-Liouville problems, J. Comput. Appl. Math 212 (2) (2008), pp. 150-178.

# C. Knoll, C. Fulton, Using a computer algebra system to simplify expressions for Titchmarsh-Weyl m-functions associated with the Hydrogen Atom on the half line, Florida Institute of Technology Research Report, 2007. (Available as arXiv 0812.4974)

## Research

Current research activity of Dr. Fulton is in the area of theory and computation of spectral density functions (and corresponding spectral functions) for Sturm-Liouville (Schroedinger) problems having absolutely continuous spectra; also for Discrete Schroedinger problems (that is, tridiagaonl Jacobi matrices) when absolutely continuous spectrum is known to occur. Theoretical work, obtaining connection formulas for SL problems with two singular endpoints have been obtained in recent years, in papers published in 2008 (Math. Nachr), 2010 (Complex Anal and Oper Theory), and 2012 (Math. Nachr.); the most recent of these papers provides sufficient conditions under which an endpoint of Weyl's Limit Point type necessarily allows definition of a single (principal) solution of the SL equation for all values of the eigenparameter which is entire in the eigenvalue parameter; and this solution is therefore possible to use in a version of the eigenfunction expansion involving any classification at the other singular (or regular) endpoint (thus ensuring simple spectrum). Further problems having LP endpoints satisfying these conditions are currently being investigated. In Dr. Fulton's most recent paper (2013 in IMA Jour Numer Anal) algorithms for computation of the spectral density functions for problems with two singular endpoints were given and the numerical computations were performed on several examples from recent papers (2010 and 2008) to verify high accuracy and (compared to the SLEDGE softwarte package) high speed. Some newer methods based on the "Value Distribution" theory of Pearson and Breimesser are currently being pursued, and these methods give rise to many new mathematical formulas; and, moreover, will be applicable to all problems known to have, in some interval, absolutely continuous spectrum (e.g. cases with a.c. spectrum filling a half line, and cases such as periodic and quasi-periodic potentials where band spectra occur). These newer methods also have discrete analogs for tridiagonal Jacobi matrix operators.