Computing the spectral function for singular Sturm-Liouville problems

Algorithms for computing Sturm-Liouville spectral density functions are developed based on several mathematical characterizations. Convergence and error bounds are derived and methods are tested on several examples. The results are compared with those from the existing software package SLEDGE.

[1]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[2]  Steven Pruess,et al.  Mathematical software for Sturm-Liouville problems , 1993, TOMS.

[3]  M. Naderi,et al.  Think globally... , 2004, HIV prevention plus!.

[4]  Steven Pruess,et al.  Using the SLEDGE package on Sturm-Liouville problems having nonempty essential spectra , 1996, TOMS.

[5]  C. D. Boor,et al.  Good approximation by splines with variable knots. II , 1974 .

[6]  S. Pruess,et al.  Eigenvalue and Eigenfunction Asymptotics for Regular Sturm-Liouville Problems , 1994 .

[7]  Yu Safarov,et al.  The Asymptotic Distribution of Eigenvalues of Partial Differential Operators , 1996 .

[8]  A. Iserles,et al.  Lie-group methods , 2000, Acta Numerica.

[9]  S. Pruess.  Solving linear boundary value problems by approximating the coefficients , 1973 .

[10]  A. Iserles,et al.  On the Implementation of the Method of Magnus Series for Linear Differential Equations , 1999 .

[11]  B. Simon,et al.  Almost periodic Schrödinger operators II. The integrated density of states , 1983 .

[12]  D. Pearson,et al.  Quantum scattering and spectral theory , 1988 .

[13]  Jaroslav Kautsky,et al.  Equidistributing Meshes with Constraints , 1980 .

[14]  M. Eastham.  The Asymptotic Nature of Spectral Functions in Sturm–Liouville Problems with Continuous Spectrum , 1997 .

[15]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[16]  Arieh Iseries,et al.  Think globally, act locally: solving highly-oscillatory ordinary differential equations , 2002 .

[17]  Arieh Iserles,et al.  On the Global Error of Discretization Methods for Highly-Oscillatory Ordinary Differential Equations , 2002 .

[18]  P. Deift,et al.  Almost periodic Schrödinger operators , 1983 .

[19]  Steven Pruess,et al.  Estimating the Eigenvalues of Sturm–Liouville Problems by Approximating the Differential Equation , 1973 .

[20]  A. G. Kefalas,et al.  Think globally, act locally , 1998 .

[21]  J. Pryce.  Numerical Solution of Sturm-Liouville Problems , 1994 .

[22]  N. Levinson.  A simplified proof of the expansion theorem for singular second order linear differential equations , 1951 .

[23]  L. Shampine,et al.  Fundamentals of Numerical Computing , 1997 .

[24]  Yoshimi Saito,et al.  Eigenfunction Expansions Associated with Second-order Differential Equations for Hilbert Space-valued Functions , 1971 .

[25]  H. Munthe-Kaas,et al.  Computations in a free Lie algebra , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[26]  Steven Pruess,et al.  The computation of spectral density functions for singular Sturm-Liouville problems involving simple continuous spectra , 1998, TOMS.

[27]  A. Hinz,et al.  Sturm-Liouville Theory , 2005 .

[28]  E. C. Titchmarsh,et al.  Reviews , 1947, The Mathematical Gazette.

[29]  Steven Pruess,et al.  Performance of the Sturm-Liouville Software Package SLEDGE , 1991 .

[30]  B. M. Brown,et al.  Spectral concentration and rapidly decaying potentials , 1997 .