Closed geodesics and flat tori in spectral theory on symmetric spaces

 Access to full text

 Full (PDF)

1. [BB] R. Balian, C. Bloch, Ann. Phys. 63 ( 1971); Ann. Phys. 64 ( 1971); Ann. Phys. 69 ( 1972); Ann. Phys. 85 ( 1974); MR284729
2. [Bo] G. Borg, Umkerhrung der Sturm-Liouvillischen Eigebnvertanfgabe Bestimung der differentialgleichung die Eigenverte, Acta Math. 78 ( 1946), 1-96. Zbl0063.00523MR15185
3. [Ch] J. Chazarain, Formule de Poisson pour les variétés Riemanniens, Inv. Math. 24 ( 1974), 65-82. Zbl0281.35028MR343320
4. [Co] Y. Colin de Verdière, 1) Spectre conjoint d'operatuers pseudodifferentials qui commutent, Math. Z, 171, 51-73 ( 1980). Zbl0478.35073MR566483
5. Y. Colin de Verdière 2) Sue les longuers des trajectoires periodic d'un billiard, Seminaire Sud-Rhodanien de Geometrie 111 ( 1983), 122-140. Zbl0599.58039
6. Y. Colin de Verdière 3) Quasi-modes sur les variétés riemanniennes compactes, Inv. Math., 43 ( 1977), 15-52. Zbl0449.53040MR501196
7. [DG] J. Duistermaat, V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Inv. Math. 29 ( 1975), 39-79. Zbl0307.35071MR405514
8. [DKV] J. Duistermaat, J. Kolk, V. Varadarajan, 1) Spectra of compact locally symmetric manifolds of negative curvature, Inv. Math. 52 ( 1979), 27-93. Zbl0434.58019MR532745
9. J. Duistermaat, J. Kolk, V. Varadarajan 2) Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes of real semisimple Lie groups, Comp. Math. 49 ( 1983), 309-398. Zbl0524.43008MR707179
10. [ERT] G. Eskin, J. Ralston, E. Trubowitz, The multidimensional Inverse spectral Problem with a periodic potential, Contem. Math. 27 ( 1984). Zbl0545.58044MR741038
11. [Gi] P.B. Gilkey, 1) Invariance theory, the heat equation and Atiyah-Singer index theorem, Publish or Perish (Boston), 1985. Zbl0565.58035MR783634
12. P.B. Gilkey 2) The spectral geometry of a Riemannian manifold, J. Diff. Geom., 10 ( 1975), 601-618. Zbl0316.53035MR400315
13. [Gui] V. Guillemin, 1) Spectral theory on S2: some open questions, Adv. Math. 42 ( 1981), 283-290. Zbl0479.58020MR642394
14. V. Guillemin 2) Some spectral results for the Laplacian on the n-sphere, Adv. Math. 27 ( 1978), 273-2S6. Zbl0433.35052MR478245
15. V. Guillemin 3) A Szego-type Theorem for symmetric spaces, Seminar on microlocal analysis, Ann. Math. Studies, 93, Princeton Univ. Press, 1979. Zbl0473.53045MR560312
16. V. Guillemin 4) Symplectic spinors and partial differential equations, CNRS Colloque International de Géométrie Symplectique et de Physique Mathématique, 1974, 217-252. Zbl0341.58014MR461591
17. [GK] V. Guillemin, D. Kazdan, Some Inverse spectral results for negatively curved manifolds, Proc. Sym. Pure Appl. Math.36, ( 1980), 153-180. Zbl0456.58031
18. [Gur] D. Gurarie, 1) Asymptotic inverse spectral problem for anharmonic oscillators, Corn. Math. Phys. 112 ( 1987), 491-502. Zbl0637.35065MR908550
19. D. Gurarie 2) Inverse spectral Problem for the 2-sphere Schrödinger operators with zonal potentials, Letters Math. Phys. 16 ( 1988), 313-323; Zbl0693.35124MR974478
20. D. Gurarie 3) Two-sphere Schrödinger operators with odd potentials, Inverse Problems, 6 ( 1990); 371-378; Zbl0728.35028MR1057031
21. D. Gurarie 4) Zonal Schrödinger operators on the n-sphere: inverse spectral problem and rigidity, Corn. Math. Phys., ( 1990); 571-603. Zbl0798.34085MR1065897
22. D. Gurarie 5) Multiparameter averaging and inverse spectral problem for anharmonic oscillators in ℝn, Comp. & Math, with Appl., 22, no. 4/5, 1991, 81-92. Zbl0756.35115MR1127216
23. D. Gurarie 6) Symmetries and Laplacians: Introduction to Harmonic Analysis, Group representations and applications, North Holland Mathematics Studies Series, 1992. Zbl0787.22001MR1174965
24. D. Gurarie 7) Spectral geometry in higher ranks: closed geodesics and flat tori, Proc. of Symposia in Pure Math., AMS, 1992 (to appear). Zbl0795.58037MR1216631
25. [Gut] M. Gutzwiller, 1) Periodic orbits and classical quantization conditions, J. Math. Phys., 12 ( 1971), 343-358; 
26. M. Gutzwiller 2) Chaos in classical and quantum mechanics, Interdise. Appl. Math., vol. 1 Springer, 1990. Zbl0727.70029MR1077246
27. [He] S. Helgason, Differential geometry and symmetric spaces, Acad. Press, 1962. Zbl0111.18101MR145455
28. [HST] The Selberg trace formula and related topics, D.Hejhal, P.Sarnak, A.Terras, ed. Contem. Math. 53, AMS, 1986. MR853548
29. [IMT] E. Isaacson, H.P. McKean, E. Trubowitz, The Inverse Sturm-Liouville Problem, I, II, Comm. Pure Appl. Math. 36 ( 1983), 767-783; 37 ( 1984), 1-11. Zbl0552.58024MR728263
30. [Ka] M. Kac, Can one hear the shape of the drum? Amer. Math. Monthly 73, ( 1966), 1-23. Zbl0139.05603MR201237
31. [Ke] J. B. Keller, Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems, Ann. Phys. 4 ( 1985), 180-188. Zbl0085.43103MR99207
32. [Ku] R. Kubovara, 1) Band asymptotics of eigenvalues for the Zoll manifold, Lett. Math. Phys., 16 ( 1988). Zbl0669.58033MR958460
33. R. Kubovara 2) Spectrum of the laplacian on vector bundles over C203C0;-manifolds, Diff. Geom. ( 1988), 241-258. Zbl0644.53040
34. [MF] V. Maslov, M. Fedoryuk, Quasiclassical approximations for the equations of Quantum mechanics, Moskwa, Nauka, 1976. Zbl0449.58002MR461590
35. [McS] H.P. McKean, I.M. Singer, Curvature and the eigenvalues of the Laplacian, J. Diff. Geom. 1 ( 1967), 43-69. Zbl0198.44301MR217739
36. [Mc] H.P. McKean, Selberg trace formula as applied to a compact Riemann surface, Comm. Pure Appl. Math. 25 ( 1972), 225-246. MR473166
37. [MN] S. Molchanov, M. Novitski, On spectral invariants of Schrödinger operators on the torus, Math. Phys., Funct. Anal. ( 1986), Kiev, Naukova Dumka, 34-39. Zbl0676.58049MR906076
38. [MP] S. Munakshisundaram, A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Can. J. Math. 1 ( 1949). Zbl0041.42701MR31145
39. [PT] J. Pöschel, E. Trubowitz, Inverse spectral theory, Pure and Appl. Math., v.130, Acad. Press, 1987. Zbl0623.34001MR894477
40. [Sel] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Ind. Math. Soc. 20 ( 1956), 47-87. Zbl0072.08201MR88511
41. [Ur] A. Uribe, 1) A symbol calculus for a class of pseudodifferential operators on Sn, J. Funct. Anal. 59 ( 1984), 535-556. Zbl0561.35082MR769380
42. A. Uribe 2) Band invariants and closed trajectories, Adv. Math. 58 ( 1985), 285-299. Zbl0587.58049MR815359
43. [Vo] A. Voros, 1) The WKB-Maslov method for nonseparable systems, CNRS Colloque International de Géométrie Symplectique et de Physique Mathématique, 237 ( 1974), 277-287. MR467827
44. [Va] V.S. Varadarajan, The eigenvalue problem on negatively curved compact locally symmetric manifolds, Contem. Math., 53 ( 1986), 449-461. Zbl0603.53026MR853572
45. [Wei] A. Weinstein, Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 44 ( 1977), 883-892. Zbl0385.58013MR482878
46. [We] H. Weyl, 1) Uber die asymptotische verteilung der Eigenwerte, Gott. Nach. ( 1911), 110-117. Zbl42.0432.03JFM42.0432.03
47. H. Weyl 2) Ramifications, old and new, of the eigenvalue Problem, Bull. AMS 56 ( 1950), 115-139. Zbl0041.21003MR34940