Reconstruction of the Sturm-Liouville operators on a graph with $\delta'_s$ couplings
Main Article Content
Abstract
Article Details
References
Borg, G.: Uniqueness theorems in the spectral theory of $y''+(lambda-q(x))y=0$, in Proc. 11th Scandinavian Congress of Mathematicians (Oslo: Johan Grundt Tanums Forlag), 276--287 (1952).
Borman, J. and Kurasov, P.: Symmetries of quantum graphs and the inverse scattering problem, Adv. in Appl. Math. 35, 58--70 (2005).
Browne P. J. and Sleeman B. D.: Inverse nodal problem for Sturm-Liouville equation with eigenparameter dependent boundary conditions, Inverse Problems 12, 377--381 (1996).
Brown, B.~M. and Weikard, R.: A Borg-Levinson theorem for trees, Proc. Royal Soc. London Ser. A 461, 3231--3243 (2005).
Buterin S. A. and Shieh C. T.: Inverse nodal problem for differential pencils, Applied Mathematics Letters 22, 1240--1247 (2009).
Carlson, R.: Large eigenvalues and trace formulas for
matrix Sturm-Liouville problems, SIAM J. Math. Anal. 30, 949--962 (1999).
Chen Y. T., Cheng Y. H., Law C. K. and Tsay J.: $L^1$
convergence of the reconstruction formula for the potential
function, Proc. Amer. Math. Soc. 130, 2319--2324 (2002).
Cheng Y.~H.~, Reconstruction of the Sturm-Liouville
operator on a p-star graph with nodal data, Rocky Mountain Journal of Mathematics, to appear.
Cheon T., Exner P.: An approximation to $delta'$ couplings on graphs, J. Phys. A: Math. Gen. 37, L329--L335 (2004).
Cheng Y.~H., Law C.~K. and Tsay J.: Remarks on a new inverse nodal problem, J. Math. Anal. Appl. 248, 145--155 (2000).
Currie S. and Watson B. A.: Inverse nodal problems for Sturm-Liouville equations on graphs, Inverse Problems 23, 2029--2040 (2007).
Exner, P.: Lattice Kronig-Penney models, Phys. Rev. Lett. 74, 3503--3506 (1995).
Exner, P.: Contact interactions on graph superlattices, J. Phys. A: Math. Gen. 29, 87--102 (1996).
Hald O. H. and McLaughlin J. R.: Solutions of inverse nodal problems, Inverse Problems 5, 307--347 (1989).
Hochstadt H. and Lieberman B.: An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math. 34, 676--680 (1978).
Horv$acute{a}$th, M.: Inverse spectral problems and
closed exponential systems, Ann. Math. 162, 885--918 (2005).
Kostrykin, V. and Schrader, R.: Kirchhoff's rule for
quantum wires, J.~Phys.~A: Math.~Gen. 32, 595--630 (1999).
Kostrykin, V. and Schrader, R.: Quantum wires with
magnetic fluxes, Commun.~Math.~Phys. 237, 161--179 (2003).
Koyunbakan H.: A new inverse problem for the diffusion operator, Applied Mathematics Letters 19, 995--999 (2006).
Kuchment, P.: Quantum graphs I. Some basic structures, Waves Random Media 14, 107--128 (2004).
Kuchment, P.: Quantum graphs II. Some spectral properties of quantum and combinatorial graphs, J.~Phys.~A: Math.~Gen. 38, 4887--4900 (2005).
[ Kurasov, P.: Schr$ddot{o}$dinger operators on graphs and geometry I. Essentially bounded potentials, Report N9, Dept. of Math., Lund Univ. (2007).
Kurasov, P. and Nowaszyk, M.: Inverse spectral problem for quantum graphs, J. Phys. A: Math. Gen. 3, 4901--4915 (2005). Corrigendum: J. Phys. A: Math. Gen. 39, 993 (2006)
Law C. K., Shen C. L. and Yang C. F.: The inverse nodal problem on the smoothness of the potential function, Inverse Problems 15, 253--263 (1999); Errata, 17, 361-364 (2001).
Law C. K. and Tsay J.: On the well-posedness of the inverse nodal problem, Inverse Problems 17, 1493-1512 (2001).
Law C. K. and Yang C. F.: Reconstructing the potential
function and its derivatives using nodal data, Inverse Problems 14 (2), 299-312 (1998).
Levin B.~Ja.: Distribution of zeros of entire functions, AMS Transl. Vol.5, Providence, 1964.
Levitan, B.~M. and Gasymov, M.~G.: Determination of a differential equation by two of its spectra, Usp.~ Mat.~ Nauk 19, 3--63 (1964).
Marchenko, V.~A.: Sturm-Liouville Operators and
Applications (Operator Theory: Advances and Applications,22), Birkh"{a}user, Basel, 1986.
McLaughlin J. R.: Inverse spectral theory using nodal
points as data--a uniqueness result, J. Differential Equations 73, 354--362 (1988).
Naimark, K. and Solomyak, M.: Eigenvalue estimates for the weighted Laplacian on metric trees, Proc.~London~Math.~Soc. 80, 690--724 (2000).
Pivovarchik, V.~N.: Inverse problem for the Sturm-Liouville equation on a simple graph, SIAM J.~Math.~Anal. 32, 801--819 (2000).
Shen C. L.: On the nodal sets of the eigenfunctions of the string equations, SIAM J. Math. Anal. 19, 1419--1424 (1988).
%
Shen C. L. and Shieh C. T.: An inverse nodal problem for vectorial Sturm-Liouville equation, Inverse Problems 16, 349--356 (2000).
Shieh C. T. and Yurko V. A.: Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl. 347, 266--272 (2008).
Simon, B.: A new approach to inverse spectral theory I. Fundamental formalism, Ann.~Math. 150, 1029--1057 (1999).
Yang X. F.: A solution of the inverse nodal problem,
Inverse Problems 13, 203--213 (1997).
Yang C.~F.: Inverse spectral problems for the
Sturm-Liouville operator on a $d$-star graph, J.~Math.~Anal.~Appl., 365, 742--749 (2010).
Yang C.~F., Huang Z.~Y. and Yang X.~P.: Trace formulae for Schr"{o}dinger systems on graphs, Turk.~J.~Math., 34, 181--196 (2010).
Yurko, V.~A: Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems 21, 1075--1086 (2005).
Yurko, V.~A.: Inverse nodal problems for Sturm-Liouville operators on star-type graphs, J.~Inv.~Ill-Posed Problems 16, 715--722 (2008).
Yurko V.~A.: Inverse nodal problems for Sturm-Liouville operators on a star-type graph, Siberian Mathematical Journal 50 (2009), 373--378.