On a boundary value method for computing Sturm–Liouville potentials from two spectra

References

• L. Aceto, P. Ghelardoni, and C. Magherini, Boundary value methods as an extension of Numerov's method for Sturm–Liouville eigenvalue estimates, Appl. Numer. Math. 59 (2009), pp. 1644–1656. doi: 10.1016/j.apnum.2008.11.005
   Web of Science ®Google Scholar
• L. Aceto, P. Ghelardoni, and C. Magherini, BVMs for Sturm–Liouville eigenvalue estimates with general boundary conditions, J. Numer. Anal., Indust. Appl. Math. 4 (2009), pp. 113–127
   Google Scholar
• L. Aceto, P. Ghelardoni, and C. Magherini, Boundary value methods for the reconstruction of Sturm–Liouville potentials, Appl. Math. Comput. 219 (2012), pp. 2960–2974. doi: 10.1016/j.amc.2012.09.021
   Web of Science ®Google Scholar
• L. Aceto, P. Ghelardoni, and M. Marletta, Numerical solution of forward and inverse Sturm–Liouville problems with an angular momentum singularity, Inverse Probl. 24 (2008), 015001 (21pp). doi: 10.1088/0266-5611/24/1/015001
   Web of Science ®Google Scholar
• A. L. Andrew, Asymptotic correction of Numerov's eigenvalue estimates with natural boundary conditions, J. Comput. Appl. Math. 125 (2000), pp. 359–366. doi: 10.1016/S0377-0427(00)00479-9
   Web of Science ®Google Scholar
• A. L. Andrew, Asymptotic correction of more Sturm–Liouville eigenvalue estimates, BIT 43 (2003), pp. 485–503. doi: 10.1023/B:BITN.0000007052.66222.6d
   Web of Science ®Google Scholar
• A. L. Andrew, Numerical solution of inverse Sturm–Liouville problems, ANZIAM J. 45(E) (2004), pp. 326–337
   Google Scholar
• A. L. Andrew, Numerov's method for inverse Sturm–Liouville problem, Inverse Probl. 21 (2005), pp. 223–238. doi: 10.1088/0266-5611/21/1/014
   Web of Science ®Google Scholar
• A. L. Andrew, Computing Sturm–Liouville potentials from two spectra, Inverse Probl. 22 (2006), pp. 2069–2081. doi: 10.1088/0266-5611/22/6/010
   Web of Science ®Google Scholar
• A. L. Andrew, Finite difference methods for half inverse Sturm–Liouville problems, Appl. Math. Comput. 218 (2011), pp. 445–457. doi: 10.1016/j.amc.2011.05.085
   Web of Science ®Google Scholar
• A. L. Andrew and J. W. Paine, Correction of Numerov's eigenvalue estimates, Numer. Math. 47 (1985), pp. 289–300. doi: 10.1007/BF01389712
   Web of Science ®Google Scholar
• G. Borg, Eine Umkehrung der Sturm–Liouvillechen Eigenwertaufgabe, Acta Math. 78 (1946), pp. 732–753. doi: 10.1007/BF02421600
   Web of Science ®Google Scholar
• L. Brugnano and D. Trigiante,Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordon & Breach, Amsterdam, 1998
   Google Scholar
• M. T. Chu, Inverse eigenvalue problems, SIAM Rev. 40 (1998), pp. 1–39. doi: 10.1137/S0036144596303984
   Web of Science ®Google Scholar
• M. T. Chu and G. H. Golub, Structured inverse eigenvalue problems, Acta Numer. 11 (2002), pp. 1–71. doi: 10.1017/S0962492902000016
   Google Scholar
• B. E.J. Dahlberg and E. Trubowitz, The inverse Sturm–Liouville problem, Comm. Pure Appl. Math. 37 (1984), pp. 255–267. doi: 10.1002/cpa.3160370205
   Web of Science ®Google Scholar
• R. H. Fabiano, R. Knobel, and B. D. Lowe, A finite-difference algorithm for an inverse Sturm–Liouville problem, J. Numer. Anal. 15 (1995), pp. 75–88. doi: 10.1093/imanum/15.1.75
   Web of Science ®Google Scholar
• I. M. Gelfand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Trans. 1 (1951), pp. 253–304
   Google Scholar
• P. Ghelardoni and C. Magherini, BVMs for computing Sturm–Liouville symmetric potentials, Appl. Math. Comput. 217 (2010), pp. 3032–3045. doi: 10.1016/j.amc.2010.08.036
   Web of Science ®Google Scholar
• O. H. Hald, The inverse Sturm–Liouville problem and the Rayleigh–Ritz method, Math. Comp. 32 (1978), pp. 687–705. doi: 10.1090/S0025-5718-1978-0501963-2
   Web of Science ®Google Scholar
• H. Hochstadt and B. Lieberman, An inverse Sturm–Liouville problem with mixed given data, SIAM J. Appl. Math. 34 (1976), pp. 676–680. doi: 10.1137/0134054
   Web of Science ®Google Scholar
• E. L. Isaacson and E. Trubowitz, The inverse Sturm–Liouville problem, Comm. Pure Appl. Math. 36 (1983), pp. 767–783. doi: 10.1002/cpa.3160360604
   Web of Science ®Google Scholar
• A. Kammanee and C. Böckmann, Determination of partially known Sturm–Liouville potentials, Appl. Math. Comput. 204 (2008), pp. 928–937. doi: 10.1016/j.amc.2008.08.001
   Web of Science ®Google Scholar
• A. Kammanee and C. Böckmann, Boundary value method for inverse Sturm–Liouville problems, Appl. Math. Comput. 214 (2009), pp. 342–352. doi: 10.1016/j.amc.2009.04.002
   Web of Science ®Google Scholar
• B. D. Lowe, M. Pilant, and W. Rundell, The recovery of potentials from finite spectral data, SIAM J. Math. Anal. 23 (1992), pp. 482–504. doi: 10.1137/0523023
   Web of Science ®Google Scholar
• J. Paine, A numerical method for the inverse Sturm–Liouville problem, SIAM J. Sci. Statist. Comput. 5 (1984), pp. 149–156. doi: 10.1137/0905011
   Web of Science ®Google Scholar
• J. W. Paine, F. R. de Hoog, and R. S. Anderssen, On the correction of finite difference eigenvalue approximations for Sturm–Liouville problems, Computing 26 (1981), pp. 123–139. doi: 10.1007/BF02241779
   Web of Science ®Google Scholar
• J. Poschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, London, 1987
   Google Scholar
• M. Rafler and C. Böckmann, Reconstruction method for inverse Sturm–Liouville problems with discontinuous potentials, Inverse Probl. 23 (2007), pp. 933–946. doi: 10.1088/0266-5611/23/3/006
   Web of Science ®Google Scholar
• N. Röhrl, A least-squares functional for solving inverse Sturm–Liouville problems, Inverse Probl. 21 (2005), pp. 2009–2017. doi: 10.1088/0266-5611/21/6/013
   Web of Science ®Google Scholar
• W. Rundell and P. E. Sacks, Reconstruction techniques for classical inverse Sturm–Liouville problems, Math. Comp. 58 (1992), pp. 161–183. doi: 10.1090/S0025-5718-1992-1106979-0
   Web of Science ®Google Scholar
• W. Rundell and P. E. Sacks, The reconstruction of Sturm–Liouville operators, Inverse Probl. 8 (1992), pp. 457–482. doi: 10.1088/0266-5611/8/3/007
   Web of Science ®Google Scholar
• P. E. Sacks, An iterative method for the inverse Dirichlet problem, Inverse Probl. 4 (1988), pp. 1055–1069. doi: 10.1088/0266-5611/4/4/009
   Web of Science ®Google Scholar
• J. G. Sun, Multiple eigenvalue sensitivity analysis, Linear Algebra Appl. 137/138 (1990), pp. 183–211. doi: 10.1016/0024-3795(90)90129-Z
   Web of Science ®Google Scholar
• V. A. Vinokurov and V. A. Sadovnichii, The eigenvalue and trace of the Sturm–Liouville operator as differentiable functions of a summable potential, Doklady Math. 59 (1999), pp. 220–222
   Google Scholar
• T. Yamamoto, A convergence theorem for Newton-like methods in Banach spaces, Numer. Math. 51 (1987), pp. 545–557. doi: 10.1007/BF01400355
   Web of Science ®Google Scholar