Asymptotics of the Eigenvalues of Hypoelliptic Operators on a Closed Manifold

Bezyaev, V. I.

doi:10.1070/sm1983v045n02abeh001002

Cited by 3 publications

(5 citation statements)

References 3 publications

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“…The first formalization (see [35,43]) uses the estimates of the trace norm of the operators (~:)2_ ~:; the second one, suggested by Feigin [38] and stated explicitly by Bezjaev [42], is significantly more convenient, since it only uses (12). Here, a further generalization of the scheme [38] is applied.…”

Section: N(t A) = N(o At) N±(t) = N(o At)mentioning

confidence: 99%

“…Scalar hypoeUiptic ~DOs on a closed manifold were studied in [42]. The systems of DO with rapidly growing coefficients were investigated in [38,[44][45][46][47].…”

Section: A Brief Survey Of the Papers Which Have Made Use Of The Asp mentioning

confidence: 99%

“…The approximate spectral projection method (ASP), used for the first time by Tulovskij and Shubin [35] to study the hypoelliptic scalar ~bDO in R n, do not have these defects. Later, the ASP method was also used by Rojtburd [36], Feigin [37][38][39][40][41], Bezjaev [42], H6rmander [43] and Levendorskii [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60] in other situations. This method belongs to the group of variational methods but, with respect to its properties, occupies an intermediate position between Tauberian and other variational methods.…”

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confidence: 99%

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The approximate spectral projection method

Levendorskiĭ¹

1986

Acta Appl Math

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In this paper we develop a general method for investigating the spectral asymptotics for various differential and pseudo-differential operators and their boundary value problems, and consider many of the problems posed when this method is applied to mathematical physics and mechanics. Among these problems are the Schr6dinger operator with growing, decreasing and degenerating potential, the Dirac operator with decreasing potential, the 'quasi-classical' spectral asymptotics for Schr6dinger and Dirac operators, the linearized Navier-Stokes equation, the Maxwell system, the system of reactor kinetics, the eigenfrequency problems of shell theory, and so on. The method allows us to compute the principal term of the spectral asymptotics (and, in the case of Douglis-Nirenberg elliptic operators, also their following terms) with the remainder estimate close to that for the sharp remainder. AMS (MOS) subject classification (1980). 35P30.Key words. Spectral asymptotics, remainder estimate, differential and pseudo-differential operators, boundary value problems. 0. Introduction 0.1. Eigenvalue problems for self-adjoint differential operators arise in many fields of mathematical physics and mechanics. Thus, the eigenvalues of a Schr6dinger operator are the energy levels of a quantum particle, and the eigenvalues of the Maxwell system, of the linearized Navier-Stokes equation, of the shell-theory operator and so on, are the squares of the eigenfrequencies of the corresponding bodies.It is clear that the computation of the eigenvalues of self-adjoint operators is a problem of great importance, but it is usually impossible to compute them exactly, except in some simple cases. Hence, approximate methods of computation are of great interest.One of these methods is the investigation of the asymptotics of the series of eigenvalues An of the operator A, as n--~ ~. If A is an operator with a discrete spectrum, then usually it is more convenient to study the asymptotics of the distribution functions of the series A~ of the positive and negative eigenvalues: N±(t, A) = card{i I + )t~: < t}, as t---~ ~.If the series {An} ~ [a, b), )tn ~ b, and on this interval there are points of a discrete spectrum only, then we can study the asymptotics of the function N((a, b -T), A) = card{/I a < Ai < b -z}, as T---~ +0.

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Section: N(t A) = N(o At) N±(t) = N(o At)mentioning

confidence: 99%

“…Scalar hypoeUiptic ~DOs on a closed manifold were studied in [42]. The systems of DO with rapidly growing coefficients were investigated in [38,[44][45][46][47].…”

Section: A Brief Survey Of the Papers Which Have Made Use Of The Asp mentioning

confidence: 99%

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confidence: 99%

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The approximate spectral projection method

Levendorskiĭ¹

1986

Acta Appl Math

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“…[19, p. 149]): Transfer the Weyl formula to hyperbolic systems. The operator A is outside all classes of operators for which spectral asymptotics is obtained by using the approximate spectral projection method [2][3][4][5]. It is of interest to find out whether we can justify (3.8) by modifying this method.…”

Section: Proposition 33 the Operator T Admits The Representationmentioning

confidence: 99%

“…Sufficiently general methods are available for finding the principal term of the spectral asymptotics for elliptic differential or pseudodifferential operators, the most general and powerful of which is perhaps the approximate spectral projection method [2][3][4][5]. The recent articles [6,7] observe that the spectral problem for hyperbolic differential equations can have important physical applications in connection with the mass spectrum problem for elementary particles.…”

Section: Introductionmentioning

confidence: 98%

Asymptotics of the Distribution of Eigenvalues of a Selfadjoint Second Order Hyperbolic Differential Operator on the Two-Dimensional Torus

Kaplitskii

2010

Sib Math J

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We study the asymptotic properties of the discrete spectrum for general selfadjoint second order hyperbolic operators on the two-dimensional torus. For a broad class of operators with sufficiently smooth coefficients and the principal part coinciding with the wave operator in the light cone coordinates we prove the discreteness of the spectrum and obtain an asymptotic formula for the distribution of eigenvalues. In some cases we can indicate the first two asymptotic terms. We discuss the relations of these questions to analytic number theory and mathematical physics.

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Asymptotic Distribution of Eigenvalues

Levendorskiĭ¹

1983

Math. USSR Izv.

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Asymptotics of the Eigenvalues of Hypoelliptic Operators on a Closed Manifold

Cited by 3 publications

References 3 publications

The approximate spectral projection method

The approximate spectral projection method

Asymptotics of the Distribution of Eigenvalues of a Selfadjoint Second Order Hyperbolic Differential Operator on the Two-Dimensional Torus

Asymptotic Distribution of Eigenvalues

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