|
Inverse Eigenvalue Problems on Directed Graphs
Author(s):
Robert
Carlson
Journal:
Trans. Amer. Math. Soc.
351
(1999),
4069-4088.
MSC (1991):
Primary 34L05
Posted:
July 1, 1999
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
The differential operators  and  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  . Second, the compactness of isospectral sets for  is established by computation of the residues of the zeta function.
References:
- 1.
- L. Ahlfors, Complex analysis, McGraw-Hill, New York, 1966. MR 32:5844
- 2.
- R. Carlson, Expansions associated with non-self-adjoint boundary-value problems, Proceedings of the American Mathematical Society 73 (1979), no. 2, 173-179. MR 80e:47040
- 3.
- F. Chung, Spectral graph theory, American Mathematical Society, 1997. MR 97k:58183
- 4.
- F. Chung and S. Sternberg, Laplacian and vibrational spectra for homogeneous graphs, Journal of Graph Theory 16 (1992), no. 6, 605-627. MR 93j:58135
- 5.
- E.A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955. MR 16:1022b
- 6.
- E. Davies, Large deviations for heat kernels on graphs, Journal of the London Mathematical Society (2) 47 (1993), 65-72. MR 94f:58135
- 7.
- Y. Colin de Verdiere, Spectre du Laplacien et longueurs des geodesiques periodiques II, Compositio Mathematica 27 (1973), no. 2, 159-184. MR 50:1293
- 8.
- L. Dikii, Trace formulas for Sturm-Liouville operators, American Mathematical Society Translations 18 (1958), 81-115. MR 23:A1874
- 9.
- P. Exner and P. Seba, Schroedinger operators on unusual manifolds, Ideas and methods in quantum and statistical physics (Oslo 1988) (S. Albeverio, J. Fenstad, H. Holden, and T. Lindstrom, eds.), 1992, pp. 227-253. MR 94a:81026
- 10.
- N. Gerasimenko and B. Pavlov, Scattering problems on noncompact graphs, Theoretical and Mathematical Physics 74 (1988), no. 3, 230-240. MR 90f:47010
- 11.
- P. Gilkey, The index theorem and the heat equation, Publish or Perish, Boston, 1974. MR 56:16704
- 12.
- T. Kato, Perturbation theory for linear operators, Springer, New York, 1995. MR 96a:47025
- 13.
- B.M. Levitan and V.V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press, Cambridge, 1982. MR 84g:34004
- 14.
- H. McKean and P. van Moerbeke, The spectrum of Hill's equation, Inventiones Math. 30 (1975), 217-274. MR 53:936
- 15.
- B. Mohar and W. Woess, A survey on spectra of infinite graphs, Bull. London Math. Soc. 21 (1989), 209-234. MR 90d:05162
- 16.
- J. Poschel and E. Trubowitz, Inverse spectral theory, Academic Press, Orlando, 1987. MR 89b:34061
- 17.
- M. Reed and B. Simon, Methods of modern mathematical physics, 1, Academic Press, New York, 1972. MR 58:12429a
- 18.
- M. Shubin, Pseudodifferential operators and spectral theory, Springer-Verlag, Berlin, 1987. MR 88c:47105
- 19.
- P. Sy and T. Sunada, Discrete Schrödinger operators on a graph, Nagoya Math Journal 125 (1992), 141-150. MR 93c:58232
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(1991):
34L05
Retrieve articles in all Journals with MSC
(1991):
34L05
Additional Information:
Robert
Carlson
Affiliation:
Department of Mathematics, University of Colorado at Colorado Springs, Colorado Springs, Colorado 80933
Email:
carlson@vision.uccs.edu
DOI:
10.1090/S0002-9947-99-02175-3
PII:
S 0002-9947(99)02175-3
Keywords:
Inverse eigenvalue problem,
graph spectral theory,
zeta function
Received by editor(s):
May 13, 1996
Received by editor(s) in revised form:
April 7, 1997
Posted:
July 1, 1999
Copyright of article:
Copyright
1999,
American Mathematical Society
|
Comments: webmaster@ams.org