×

A semi-classical trace formula for several commuting operators. (English) Zbl 0927.35138

The authors study the joint spectrum of \(k\) semiclassical pseudodifferential operators \(A_1(h),\dots\), \(A_k(h)\) in \(\mathbb{R}^n\), \(k\leq n\), which commute to each other. Under suitable assumptions, in a compact neighborhood \(K\subset \mathbb{R}^k\) of a fixed level of energy \(E\in\mathbb{R}^k\), the joint spectrum consists of finitely many joint eigenvalues \(\lambda(h)= (\lambda_1(h),\dots, \lambda_k(h))\), and the trace corresponding to a fixed \(\zeta\) function in \(\mathbb{R}^k\) is defined by \[ \text{Tr}_\zeta(h)= \sum_{\lambda(h)\in K} \zeta((E- \lambda(h))/h). \] The authors relate the asymptotic behaviour as \(h\to 0\) of \(\text{Tr}_\zeta(h)\) to the periodic orbits of the joint Hamiltonian flow of the principal symbol \(a_0\) lying on the energy surface \(\Sigma= \{a_0= E\}\). As relevant reference, we quote V. Petkov and G. Popov [Ann. Inst. Henri Poincaré, Phys. Théor. 68, No. 1, 17-83 (1998)], concerning the case \(k= 1\).
Reviewer: L.Rodino (Torino)

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
35Q40 PDEs in connection with quantum mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arnold V., Singularities of Differentiable Maps 2 (1985)
[2] Abraham R., Foundations of mechanics Second editon (1978)
[3] DOI: 10.1215/S0012-7094-95-07823-5 · Zbl 0849.58067 · doi:10.1215/S0012-7094-95-07823-5
[4] DOI: 10.1007/BF02099074 · Zbl 0729.35093 · doi:10.1007/BF02099074
[5] Charbonel A.-M., Asymptotic Analysis 78 pp 227– (1988)
[6] Charbonnel A.-M., Ann. Inst. Henri Poincaré 56 pp 187– (1992)
[7] DOI: 10.1080/0360530800882148 · Zbl 0437.70014 · doi:10.1080/0360530800882148
[8] Colin de Verdière Y., Comp. Math. 27 pp 159– (1973)
[9] Colin de Verdière Y. Théorie spectrale et méthodes semi-classiques Institute Fourier Grenoble 1992
[10] DOI: 10.1215/S0012-7094-79-04608-8 · Zbl 0411.35073 · doi:10.1215/S0012-7094-79-04608-8
[11] Dozias S., Opérateurs h-pseudo-différentiels à flot périodique et asymptotique semi-classique (1994)
[12] DOI: 10.1002/cpa.3160270205 · Zbl 0285.35010 · doi:10.1002/cpa.3160270205
[13] DOI: 10.1007/BF01405172 · Zbl 0307.35071 · doi:10.1007/BF01405172
[14] DOI: 10.1016/0022-1236(91)90114-K · Zbl 0739.58020 · doi:10.1016/0022-1236(91)90114-K
[15] DOI: 10.1007/BF01393968 · Zbl 0686.58040 · doi:10.1007/BF01393968
[16] Guriev T., Trudy Mathem. Inst. Steklov 179 pp 35– (1988)
[17] DOI: 10.1063/1.1665596 · doi:10.1063/1.1665596
[18] DOI: 10.5802/aif.844 · Zbl 0451.35022 · doi:10.5802/aif.844
[19] DOI: 10.1016/0022-1236(83)90034-4 · Zbl 0524.35103 · doi:10.1016/0022-1236(83)90034-4
[20] DOI: 10.1002/cpa.3160320304 · Zbl 0388.47032 · doi:10.1002/cpa.3160320304
[21] Hörmander L., The Analysis of Linear Partial Differential Operators III, IV (1985)
[22] Marvizi Sh., J. Differ Geom 77 pp 475– (1982)
[23] DOI: 10.1016/0034-4877(92)90019-W · Zbl 0794.58046 · doi:10.1016/0034-4877(92)90019-W
[24] Uribe T. Paul et A., C. R. Acad. Sci. Paris, t 313 pp 217– (1991)
[25] Petkov V., Ann. Inst. henri Poincaré 68 pp 17– (1998)
[26] DOI: 10.1007/BF02571890 · Zbl 0816.58008 · doi:10.1007/BF02571890
[27] DOI: 10.1080/03605308508820382 · Zbl 0574.35067 · doi:10.1080/03605308508820382
[28] DOI: 10.1007/BF03025724 · Zbl 0804.53068 · doi:10.1007/BF03025724
[29] Popov G., Math. Phys.
[30] Safarov Yu., Zap. Nauchn. Sem. Leningrad. Otdel Mat. Inst. Steklov (LOMI) 152 pp 94– (1986)
[31] Safarov Yu., Izv. AN SSSR, Ser. Mat 52 pp 1230– (1988)
[32] Sjöstrand J., Asymptotic Analysis 6 pp 29– (1992)
[33] DOI: 10.1007/BF02097000 · Zbl 0791.58102 · doi:10.1007/BF02097000
[34] DOI: 10.1007/BF01075524 · Zbl 0351.32011 · doi:10.1007/BF01075524
[35] DOI: 10.1080/03605309208820840 · Zbl 0749.58062 · doi:10.1080/03605309208820840
[36] Zonna D., Thèse de trpoème cycle (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.