## Szegő limit theorems for the harmonic oscillator

HTML articles powered by AMS MathViewer

- by A. J. E. M. Janssen and Steven Zelditch PDF
- 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, - 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
—, - 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, - 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, - 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

*The Weyl calculus of pseudo-differential operators*, Comm. Pure Appl. Math.

**32**(1979), 359-443.

*Application of the Wigner distribution to harmonic analysis of generalized stochastic processes*, MC-tract 114, Amsterdam, 1979.

*Orthogonal polynomials*, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1975.

*The theory of groups and quantum mechanics*, Dover, New York, 1950.

## 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