Inverse eigenvalue problems on directed graphs

Carlson, Robert

doi:10.1090/S0002-9947-99-02175-3

Inverse eigenvalue problems on directed graphs
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Trans. Amer. Math. Soc. 351 (1999), 4069-4088 Request permission

Abstract:

The differential operators $iD$ and $-D^2 - p$ are constructed on certain finite directed weighted graphs. Two types of inverse spectral problems are considered. First, information about the graph weights and boundary conditions is extracted from the spectrum of $-D^2$. Second, the compactness of isospectral sets for $-D^2 - p$ is established by computation of the residues of the zeta function.

References

Lars V. Ahlfors, Complex analysis: An introduction of the theory of analytic functions of one complex variable, 2nd ed., McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0188405
Robert Carlson, Expansions associated with nonselfadjoint boundary value problems, Proc. Amer. Math. Soc. 73 (1979), no. 2, 173–179. MR 516459, DOI 10.1090/S0002-9939-1979-0516459-4
Fan R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, vol. 92, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. MR 1421568
Fan R. K. Chung and Shlomo Sternberg, Laplacian and vibrational spectra for homogeneous graphs, J. Graph Theory 16 (1992), no. 6, 605–627. MR 1189051, DOI 10.1002/jgt.3190160607
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
E. B. Davies, Large deviations for heat kernels on graphs, J. London Math. Soc. (2) 47 (1993), no. 1, 65–72. MR 1200978, DOI 10.1112/jlms/s2-47.1.65
Yves Colin de Verdière, Spectre du laplacien et longueurs des géodésiques périodiques. I, II, Compositio Math. 27 (1973), 83–106; ibid. 27 (1973), 159–184 (French). MR 348798
L. A. Dikiĭ, Trace formulas for Sturm-Liouville differential operators, Amer. Math. Soc. Transl. (2) 18 (1961), 81–115. MR 0124562, DOI 10.1090/trans2/018/06
Pavel Exner and Petr Šeba, Schroedinger operators on unusual manifolds, Ideas and methods in quantum and statistical physics (Oslo, 1988) Cambridge Univ. Press, Cambridge, 1992, pp. 227–253. MR 1190528
N. I. Gerasimenko and B. S. Pavlov, A scattering problem on noncompact graphs, Teoret. Mat. Fiz. 74 (1988), no. 3, 345–359 (Russian, with English summary); English transl., Theoret. and Math. Phys. 74 (1988), no. 3, 230–240. MR 953298, DOI 10.1007/BF01016616
Peter B. Gilkey, The index theorem and the heat equation, Mathematics Lecture Series, No. 4, Publish or Perish, Inc., Boston, Mass., 1974. Notes by Jon Sacks. MR 0458504
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press, Cambridge-New York, 1982. Translated from the Russian by L. W. Longdon. MR 690064
H. P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math. 30 (1975), no. 3, 217–274. MR 397076, DOI 10.1007/BF01425567
Bojan Mohar and Wolfgang Woess, A survey on spectra of infinite graphs, Bull. London Math. Soc. 21 (1989), no. 3, 209–234. MR 986363, DOI 10.1112/blms/21.3.209
Jürgen Pöschel and Eugene Trubowitz, Inverse spectral theory, Pure and Applied Mathematics, vol. 130, Academic Press, Inc., Boston, MA, 1987. MR 894477
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
M. A. Shubin, Pseudodifferential operators and spectral theory, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. Translated from the Russian by Stig I. Andersson. MR 883081, DOI 10.1007/978-3-642-96854-9
Polly Wee Sy and Toshikazu Sunada, Discrete Schrödinger operators on a graph, Nagoya Math. J. 125 (1992), 141–150. MR 1156908, DOI 10.1017/S0027763000003949