# Transactions of the American Mathematical Society

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## Szegő limit theorems for the harmonic oscillatorHTML articles powered by AMS MathViewer

by A. J. E. M. Janssen and Steven Zelditch
Trans. Amer. Math. Soc. 280 (1983), 563-587 Request permission

## Abstract:

Let $H = - \frac {1}{2}{d^2}/d{x^2} + \frac {1}{2}{x^2}$ be the harmonic oscillator Hamiltonian on ${L^2}( {\mathbf {R}})$, and let $A$ be a selfadjoint $DO$ of order $O$ in the Beals-Fefferman class with weights $\varphi = 1,\Phi (x,\xi ) = {(1 + |\xi {|^2} + |x {|^2})^{1/2}}$. Form the measure $\mu (f) = {\lim _{\lambda \to \infty }}(1/{\text {rank}}\;{\pi _\lambda }) {\text {tr}} f({\pi _\lambda } A{\pi _\lambda })$ where ${\pi _\lambda } A{\pi _\lambda }$ is the compression of $A$ onto the span of the Hermite functions with eigenvalue less than or equal to $\lambda$. Then one has the following Szegö limit theorem: $\mu (f) = \lim \limits _{T \to \infty } \;\frac {1} {{2 \pi T}}\;\int _{H(x,\xi ) \leqslant T} {f(a(x,\xi ))\;dx} \;d\xi \qquad {\text {for}}\ f \in C({\mathbf {R}}).$ For the special case where $f(x) = x$, this will be proved for a considerably wider class of operators by employing the Weyl correspondence. Moreover, by using estimates on Wigner functions of Hermite functions we are able to prove the full Szegö theorem for a fairly general class of multiplication operators.
References
• Richard Askey and Stephen Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695–708. MR 182834, DOI 10.2307/2373069
• V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187–214. MR 157250, DOI 10.1002/cpa.3160140303
• Richard Beals, A general calculus of pseudodifferential operators, Duke Math. J. 42 (1975), 1–42. MR 367730
• N. G. de Bruijn, A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence, Nieuw Arch. Wisk. (3) 21 (1973), 205–280. MR 482163
• Ingrid Daubechies, On the distributions corresponding to bounded operators in the Weyl quantization, Comm. Math. Phys. 75 (1980), no. 3, 229–238. MR 581947, DOI 10.1007/BF01212710
• H. J. Groenewold, On the principles of elementary quantum mechanics, Physica 12 (1946), 405–460. MR 18562, DOI 10.1016/S0031-8914(46)80059-4
• Victor Guillemin, Some classical theorems in spectral theory revisited, Seminar on Singularities of Solutions of Linear Partial Differential Equations (Inst. Adv. Study, Princeton, N.J., 1977/78) Ann. of Math. Stud., vol. 91, Princeton Univ. Press, Princeton, N.J., 1979, pp. 219–259. MR 547021
• A. Grossmann, G. Loupias, and E. M. Stein, An algebra of pseudodifferential operators and quantum mechanics in phase space, Ann. Inst. Fourier (Grenoble) 18 (1968), no. fasc. 2, 343–368, viii (1969) (English, with French summary). MR 267425, DOI 10.5802/aif.305
• Roger Howe, Quantum mechanics and partial differential equations, J. Functional Analysis 38 (1980), no. 2, 188–254. MR 587908, DOI 10.1016/0022-1236(80)90064-6
• L. Hörmander, The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32 (1979), 359-443.
• A. J. E. M. Janssen, Positivity of weighted Wigner distributions, SIAM J. Math. Anal. 12 (1981), no. 5, 752–758. MR 625830, DOI 10.1137/0512063
• A. J. E. M. Janssen, Bargmann transform, Zak transform, and coherent states, J. Math. Phys. 23 (1982), no. 5, 720–731. MR 655886, DOI 10.1063/1.525426
• —, Application of the Wigner distribution to harmonic analysis of generalized stochastic processes, MC-tract 114, Amsterdam, 1979.
• Giancarlo Mauceri, The Weyl transform and bounded operators on $L^{p}(\textbf {R}^{n})$, J. Functional Analysis 39 (1980), no. 3, 408–429. MR 600625, DOI 10.1016/0022-1236(80)90035-X
• Jaak Peetre, The Weyl transform and Laguerre polynomials, Matematiche (Catania) 27 (1972), 301–323 (1973). MR 340675
• Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
• Barry Simon, Functional integration and quantum physics, Pure and Applied Mathematics, vol. 86, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 544188
• G. Szegö, Orthogonal polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1975.
• Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, No. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463, DOI 10.1515/9781400886104
• André Unterberger, Oscillateur harmonique et opérateurs pseudo-différentiels, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 3, xi, 201–221 (French, with English summary). MR 552965
• A. Voros, An algebra of pseudodifferential operators and the asymptotics of quantum mechanics, J. Functional Analysis 29 (1978), no. 1, 104–132. MR 496088, DOI 10.1016/0022-1236(78)90049-6
• H. Weyl, The theory of groups and quantum mechanics, Dover, New York, 1950.
• Harold Widom, Eigenvalue distribution theorems for certain homogeneous spaces, J. Functional Analysis 32 (1979), no. 2, 139–147. MR 534671, DOI 10.1016/0022-1236(79)90051-X
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Additional Information
• © Copyright 1983 American Mathematical Society
• Journal: Trans. Amer. Math. Soc. 280 (1983), 563-587
• MSC: Primary 35S05; Secondary 81C10
• DOI: https://doi.org/10.1090/S0002-9947-1983-0716838-1
• MathSciNet review: 716838