Abstract
We study Schrödinger operators of the form\(H = - \frac{{h^2 }}{2}\Delta + \frac{1}{2}x \cdot A^2 x + V(x)\) on ℝd, whereA 2 is a strictly positive symmetricd×d matrix andV(x) is a continuous real function which is the Fourier transform of a bounded measure. If λ n are the eigenvalues ofH we show that the theta function\(\theta (t) = \sum\limits_n {\exp \left( { - \frac{i}{h}t\lambda _n } \right)}\) is explicitly expressible in terms of infinite dimensional oscillatory integrals (Feynman path integrals) over the Hilbert space of closed trajectories. We use these explicit expressions to give the asymptotic behaviour of θ(t) for smallh in terms of classical periodic orbits, thus obtaining a trace formula for the Schrödinger operators. This then yields an asymptotic expansion of the spectrum ofH in terms of the periodic orbits of the corresponding classical mechanical system. These results extend to the physical case the recent work on Poisson and trace formulae for compact manifolds.
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Albeverio, S., Blanchard, Ph., Høegh-Krohn, R.: Oscillatory integrals and the method of stationary phase in infinitely many dimensions II. Preprint 1980. The results have been announced in Blanchard, Ph.: Asymptotic expansions of Fresnel integrals relative to a non-singular quadratic form. L. Streit, (ed.) Quantum fields-Algebras, Processes, pp. 91–106. Wien: Springer 1980
Albeverio, S., Høegh-Krohn, R.: Invent. Math.40, 59–106 (1977)
Albeverio, S., Høegh-Krohn, R.: Mathematical theory of Feynman path integrals, Lecture Notes in Mathematics Vol. 523. Berlin: Springer 1976
Albeverio, S., Høegh-Krohn, R.: Feynman path integrals and the corresponding method of stationary phase, pp. 3–57 in Feynman path Integrals; Proc. Marseille 1978. Lecture Notes in Physics Vol. 106. Berlin: Springer 1979
Albeverio, S., Blanchard, Ph., Høegh-Krohn, R.: The trace formula for the Schrödinger operators, Materialien des Universitätsschwerpunktes Mathematisierung der Einzelwissenschaften, Materialien XXVII, Bielefeld (Febr. 1980)
Albeverio, S., Blanchard, Ph., Høegh-Krohn, R.: Stationary phase for the Feynman integral and trace formula. In: Functional Integration. J. P. Antoine, E. Tirapegui, (eds.) pp. 23–43. New York: Plenum Press (1980)
Albeverio, S., Blanchard, Ph., Høegh-Krohn, R.: Feynman path integrals, the Poisson formula and the theta function for the Schrödinger operators. In: Trends in Applications of Pure Mathematics to Mechanics, Vol. III, pp. 1–22 R. J. Knops (ed.). New York: Pitman 1981
Amann, H., Zehnder, E.: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Zürich-Bochum Preprint 1979
Baltes, H. P., Hilf, E. R.: Spectra of finite systems, Bibl. Inst. Mannheim 1976
Berger, M. S.: Nonlinearity and functional analysis. New York: Academic Press 1977
Berry, M. V., Mount, K. E.: Rep. Prog. Phys.35, 315–397 (1972)
Chazarain, J.: Spectre d'un Hamiltonien quantique et mécanique classique. Université de Nice, Preprint
Chazarain, J.: Invent. Math.24, 65–82 (1974)
Colin de Verdiére, Y.: Comp. Math.27, 83–106 (1973); Comp. Math.27, 159–184 (1973)
Dashen, R., Hasslacher, B., Neveu, A.: Phys. Rev. D10, 4114 (1974)
De Witt-Morette, C., Maheshwari, A., Nelson, B.: Phys. Rept.50, 256–372 (1979)
Duistermaat, J. J.: Commun. Pure Appl. Math.27, 207–281 (1974)
Duistermaat, J. J., Guillemin, V.: The spectrum of positive elliptic operators and periodic geodesics. Proc. AMS Summer Inst. Differ. Geom., Stanford, 1973
Eckmann, J. P., Sénéor, R.: Arch. Rat. Mech. Anal.61, 153–173 (1976)
Guillemin, V., Weinstein, A.: Bull. Am. Math. Soc.82, 92–95 (1976)
Guillemin, V., Sternberg, S.: Geometric asymptotics Math. Surv.14, Providence: A.M.S. (1977)
Gutzwiller, M. C.: J. Math. Phys.8, 1979–2000 (1967); J. Math. Phys.12 (1971) 343–358
Gutzwiller, M. C.: Classical quantization conditions for a dynamical system with stochastic behaviour, pp. 316–325, Stochastic Behaviour in classical and quantum Hamiltonian Systems, Como, 1977, Casati, G., Ford, J. (eds.) Lecture Notes in Physics93, Berlin, Heidelberg, New York: Springer 1979
Huber, A.: Math. Ann.138, 1–26 (1959)
Leray, J.: Solutions asymptotiques et groupe symplectique. In: Fourier Integral Operators and Partial Differential Equations Nice 1974, J. Chazarain, (ed.) pp. 73–97 Lecture Notes Mathematics. Berlin: Springer 1975
Maslov, V. P.: Théorie des perturbations et méthodes asymptotiques. Trad. J. Lascoux, R. Sénéor, Paris: Dunod-Gauthier-Villars 1972
Maslov, V. P.: J. Comp. Math.1, 744–778 (1961) (transl.)
Molchanov, S. A.: Russ. Math. Surv.30, 1–63 (1975) (transl.)
Rajaraman, R.: Phys. Rep.21C, 227–313 (1975)
Selberg, A.: J. Ind. Math. Soc.20, 47–87 (1956)
Sommerfeld, A.: Atombau und Spektrallinien, 8. bzw. 4. Aufl. Braunschweig: Vieweg 1967–69
Truman, A.: The polygonal path formulation of the Feyman path integral, In: Feyman Path Integrals, Proc. Marseille 1978. Lecture Notes in Physics Vol.106, pp. 73–102. Berlin: Springer 1979
Voros, A.: Ann. Inst. H. Poincaré24, 31–50 (1976)
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Communicated by J. Ginibre
Partially supported by the USP-Mathematisierung, University of Bielefeld (Forschungsprojekt “Unendlich dimensionale Analysis”)
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Albeverio, S., Blanchard, P. & Høegh-Krohn, R. Feynman path integrals and the trace formula for the Schrödinger operators. Commun.Math. Phys. 83, 49–76 (1982). https://doi.org/10.1007/BF01947071
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DOI: https://doi.org/10.1007/BF01947071