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Transactions of the American Mathematical Society

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Szegő limit theorems for the harmonic oscillator

Authors: A. J. E. M. Janssen and Steven Zelditch
Journal: Trans. Amer. Math. Soc. 280 (1983), 563-587
MSC: Primary 35S05; Secondary 81C10
MathSciNet review: 716838
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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 + \vert\xi \,{\vert^2} + \vert x\,{\vert^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:

$\displaystyle \mu (f) = \mathop {\lim }\limits_{T \to \infty } \;\frac{1} {{2\,... ...qslant T} {f(a(x,\xi ))\;dx} \;d\xi \qquad {\text{for}}\ f \in C({\mathbf{R}}).$

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.

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  • [AW] R. Askey and S. Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695-708. MR 0182834 (32:316)
  • [Ba] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform. Part I, Comm. Pure Appl. Math. 14 (1961), 187-214. MR 0157250 (28:486)
  • [Be] R. Beals, A general calculus of pseudodifferential operators, Duke Math. J. 42 (1975), 1-42. MR 0367730 (51:3972)
  • [dB] N. G. de Bruijn, A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence, Nieuw Arch. Wisk. 21 (1973), 205-280. MR 0482163 (58:2249)
  • [D] I. Daubechies, On the distributions corresponding to bounded operators in the Weyl quantization, Comm. Math. Phys. 75 (1980), 229-238. MR 581947 (82d:81025)
  • [Gr] H. J. Groenewold, On the principle of elementary quantum mechanics, Physica 21 (1946), 405-460. MR 0018562 (8:301a)
  • [Gu] V. Guillemin, Some classical theorems in spectral theory revisited, Seminar and Singularities of Solutions of Linear Partial Differential Equations (L. Hörmander, editor), Princeton Univ. Press, Princeton, N. J., 1979. MR 547021 (81b:58045)
  • [GLS] A. Grossman, G. Loupias and E. M. Stein, An algebra of pseudo-differential operators and quantum mechanics in phase space, Ann. Inst. Fourier (Grenoble) 18 (1969), 363-368. MR 0267425 (42:2327)
  • [H] R. Howe, Quantum mechanics and partial differential equations, J. Funct. Anal. 38 (1980), 188-254. MR 587908 (83b:35166)
  • [] L. Hörmander, The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32 (1979), 359-443.
  • [Ja] A. J. E. M. Janssen, Positivity of weighted Wigner distributions, SIAM J. Math. Anal. 12 (1981), 752-758. MR 625830 (82j:81023)
  • [Jb] -, Bargmann transform, Zak transform and coherent states, J. Math. Phys. 23 (1982), 720-731. MR 655886 (84h:81041)
  • [Jc] -, Application of the Wigner distribution to harmonic analysis of generalized stochastic processes, MC-tract 114, Amsterdam, 1979.
  • [M] G. Mauceri, The Weyl transform and bounded operators on $ {L^p}({{\mathbf{R}}^n})$, J. Funct. Anal. 39 (1980), 408-429. MR 600625 (82i:44002)
  • [P] J. Peetre, The Weyl transform and Laguerre polynomials, Matematiche (Catania) 27 (1972), 301-323. MR 0340675 (49:5426)
  • [Ru] W. Rudin, Functional analysis, McGraw-Hill, New York, 1973. MR 0365062 (51:1315)
  • [Si] B. Simon, Functional integration and quantum physics, Academic Press, New York, 1979. MR 544188 (84m:81066)
  • [Sz] G. Szegö, Orthogonal polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1975.
  • [T] M. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton, N. J., 1981. MR 618463 (82i:35172)
  • [U] A. Unterberger, Oscillateur harmonique et operateurs pseudo-differentiels, Ann. Inst. Fourier (Grenoble) 29 (1979), 201-221. MR 552965 (81m:58077)
  • [V] A. Voros, An algebra of pseudodifferential operators and the asymptotics of quantum mechanics, J. Funct. Anal. 29 (1978), 104-132. MR 0496088 (58:14697)
  • [We] H. Weyl, The theory of groups and quantum mechanics, Dover, New York, 1950.
  • [Wi] H. Widom, Eigenvalue distribution theorems for certain homogeneous spaces, J. Funct. Anal. 32 (1979), 139-147. MR 534671 (80h:58054)

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