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A Gutzwiller trace formula for large Hermitian matrices. (English) Zbl 1375.81104
MSC:
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
39A12 Discrete version of topics in analysis
47B39 Linear difference operators
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[1] Bouzouina, A.; De Bièvre, S., Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Comm. Math. Phys., 178, 83-105, (1996) · Zbl 0876.58041
[2] Boutet de Monvel, L.; Guillemin, V., The Spectral Theory of Toeplitz Operators, 99, (1981), Princeton University Press, Princeton, NJ · Zbl 0469.47021
[3] Boutet de Monvel, L.; Sjöstrand, J., Journées: Équations aux Dérivées Partielles de Rennes, Sur la singularité des noyaux de Bergman et de Szegő, 123-164, (1976), Soc. Math. France, Paris · Zbl 0344.32010
[4] Bodmann, B. G.; Klauder, J. R., Path integral quantization for a toroidal phase space, Coherent States, Quantization and Gravity, Proceedings of the XVIIth Workshop on Geometric Methods in Physics, 3-10, (2001), Warsaw University Press
[5] Borthwick, D.; Paul, T.; Uribe, A., Semiclassical spectral estimates for Toeplitz operators, Ann. Inst. Fourier (Grenoble), 48, 1189-1229, (1998) · Zbl 0920.58059
[6] Colin de Verdière, Y., Spectre du laplacien et longueurs des géodésiques périodiques. I, Compos. Math., 27, 83-106, (1973) · Zbl 0272.53034
[7] Colin de Verdière, Y., Spectre du laplacien et longueurs des géodésiques périodiques. II, Compos. Math., 27, 159-184, (1973) · Zbl 0281.53036
[8] Charles, L., Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators, Comm. Partial Differential Equations, 28, 1527-1566, (2003) · Zbl 1038.53086
[9] Esposti, M. D.; Graffi, S., The Mathematical Aspects of Quantum Maps, 618, (2003), Springer-Verlag, Berlin
[10] Dixon, A. L.; Ferrar, W. L., On the summation formulae of voronoï and Poisson, Quart. J. Math., 8, 66-74, (1937) · JFM 63.0134.04
[11] de Faria, E.; de Melo, W., Mathematical Aspects of Quantum Field Theory, 127, (2010), Cambridge University Press, Cambridge · Zbl 1200.81001
[12] Duistermaat, J. J.; Guillemin, V. W., The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., 29, 39-79, (1975) · Zbl 0307.35071
[13] Duistermaat, J. J., Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appl. Math., 27, 207-281, (1974) · Zbl 0285.35010
[14] Duistermaat, J. J., On the Morse index in variational calculus, Adv. Math., 21, 173-195, (1976) · Zbl 0361.49026
[15] Duistermaat, J. J., Fourier Integral Operators, 130, (1996), Birkhäuser Boston, Inc., Boston, MA · Zbl 0841.35137
[16] Folland, G. B., Harmonic Analysis in Phase Space, 122, (1989), Princeton University Press, Princeton, NJ · Zbl 0682.43001
[17] Faure, F.; Tsujii, M., Prequantum transfer operator for symplectic Anosov diffeomorphism, Astérisque, 375, ix+222, (2015) · Zbl 1417.37012
[18] Gat, O.; Avron, J. E., Semiclassical analysis and the magnetization of the Hofstadter model, Phys. Rev. Lett., 91, 186801, (2003)
[19] Grafakos, L., Classical Fourier Analysis, 249, (2008), Springer, New York · Zbl 1220.42001
[20] Grigis, A.; Sjöstrand, J., Microlocal Analysis for Differential Operators, 196, (1994), Cambridge University Press, Cambridge · Zbl 0804.35001
[21] Guillemin, V.; Sternberg, S., Semi-Classical Analysis, (2013), International Press, Boston, MA · Zbl 1298.58001
[22] Gutzwiller, M. C., Periodic orbits and classical quantization conditions, J. Math. Phys., 12, 343-358, (1971)
[23] Harper, P. G., The general motion of conduction electrons in a uniform magnetic field, with application to the diamagnetism of metals, Proc. Phys. Soc. London Sect. A, 68, 879-892, (1955) · Zbl 0065.23707
[24] Hannay, J. H.; Berry, M. V., Quantization of linear maps on a torus — fresnel diffraction by a periodic grating, Phys. D, 1, 267-290, (1980) · Zbl 1194.81107
[25] Hörmander, L., The Analysis of Linear Partial Differential Operators. III — Pseudodifferential Operators, 274, (1985), Springer-Verlag, Berlin
[26] Hörmander, L., The Analysis of Linear Partial Differential Operators. IV — Fourier Integral Operators, 275, (1985), Springer-Verlag, Berlin · Zbl 0612.35001
[27] Lancaster, P., Theory of Matrices, (1969), Academic Press, New York-London · Zbl 0186.05301
[28] Lee, J. M., Introduction to Smooth Manifolds, (2013), Springer, New York · Zbl 1258.53002
[29] Ligabò, M., Torus as phase space: Weyl quantization, dequantization and Wigner formalism, J. Math. Phys., 57, 082110, (2016) · Zbl 1351.81066
[30] Meinrenken, E., Semiclassical principal symbols and gutzwiller’s trace formula, Rep. Math. Phys., 31, 279-295, (1992) · Zbl 0794.58046
[31] Meinrenken, E., Trace formulas and the Conley-Zehnder index, J. Geom. Phys., 13, 1-15, (1994) · Zbl 0791.53040
[32] E. Meinrenken, Symplectic geometry, Lecture Notes, University of Toronto (2000); http://www.math.toronto.edu/mein/teaching/LectureNotes/sympl.pdf.
[33] Paoletti, R., Local asymptotics for slowly shrinking spectral bands of a Berezin-Toeplitz operator, Int. Math. Res. Not. IMRN, 2011, 5, 1165-1204, (2011) · Zbl 1230.47055
[34] R. Paoletti, Local scaling asymptotics for the Gutzwiller trace formula in Berezin-Toeplitz quantization (2016); arXiv:1601.02128. · Zbl 1398.32023
[35] Petkov, V.; Popov, G., Semi-classical trace formula and clustering of eigenvalues for Schrödinger operators, Ann. Inst. H. Poincaré Phys. Théor., 68, 17-83, (1998) · Zbl 0919.35095
[36] Rothe, H. J., Lattice Gauge Theories, (2012), World Scientific Publishing, Hackensack, NJ · Zbl 1255.81001
[37] Reed, M.; Simon, B., Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, (1975), Academic Press, New York-London · Zbl 0308.47002
[38] Schlichenmaier, M., Berezin-Toeplitz quantization for compact Kähler manifolds, A review of results, Adv. Math. Phys., 2010, 927280, (2010) · Zbl 1207.81049
[39] Sadovski, D. A.; Zhilinski, B. I., Quantum monodromy and its generalizations and molecular manifestations, Mol. Phys., 104, 2595-2615, (2006)
[40] Zelditch, S., Index and dynamics of quantized contact transformations, Ann. Inst. Fourier (Grenoble), 47, 305-363, (1997) · Zbl 0865.47018
[41] Zelditch, S., The Breadth of Symplectic and Poisson Geometry, 232, Quantum maps and automorphisms, 623-654, (2005), Birkhäuser Boston, Boston, MA · Zbl 1077.53076
[42] Zworski, M., Semiclassical Analysis, 138, (2012), American Mathematical Society, Providence, RI · Zbl 1252.58001
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