An extension of Weyl’s asymptotic law for eigenvalues

Brownell, F. H.

doi:10.2140/pjm.1955.5.483

Cited by 20 publications

(17 citation statements)

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“…Although the analysis we have presented makes no attempt at proper mathematical rigour, the main result in (3.14) of (3.15) is in agreement with a refinement of Karamata's theorem due to Brownell [32]. (A nice account is contained in Ref.…”

Section: Density Of States Methodssupporting

confidence: 61%

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Bose-Einstein condensation in arbitrarily shaped cavities

Kirsten

Toms

1999

Phys. Rev. E

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We discuss the phenomenon of Bose-Einstein condensation of an ideal non-relativistic Bose gas in an arbitrarily shaped cavity. The influence of the finite extension of the cavity on all thermodynamical quantities, especially on the critical temperature of the system, is considered. We use two main methods which are shown to be equivalent. The first deals with the partition function as a sum over energy levels and uses a Mellin-Barnes integral representation to extract an asymptotic formula. The second method converts the sum over the energy levels to an integral with a suitable density of states factor obtained from spectral analysis. The application to some simple cavities is discussed.

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Section: Density Of States Methodssupporting

confidence: 61%

“…This assumes f << 1. Defining 32) which gives the critical temperature in the bulk (or thermodynamic) limit, 33) and assuming that ξ c = ξ 0 (1 + γ), with γ ≪ 1, to leading order we can approximate (we use the temperature here)…”

Section: Quantum Statistics Of a Free Bose Gas In A D-dimensionalmentioning

confidence: 99%

Bose-Einstein condensation in arbitrarily shaped cavities

Kirsten

Toms

1999

Phys. Rev. E

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“…This makes it easy to compute and universal in its applications to renormalization of ultraviolet divergences. Its formal inverse Laplace transform is an "averaged" spectral density [1][2][3] that is equally local and universal, insensitive to the detailed spacings of the eigenvalues (if the spectrum is discrete at all).The interesting part of renormalization is what is left behind when the counterterms are subtracted. For example, vacuum (Casimir) energy can be calculated from a two-point function (a kernel for the wave equation) by subtracting universal singular terms.…”

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confidence: 99%

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confidence: 99%

Periodic orbits, spectral oscillations, scaling, and vacuum energy: beyond HaMiDeW

Fulling

2002

Nuclear Physics B - Proceedings Supplements

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Oscillations in eigenvalue density are associated with closed orbits of the corresponding classical (or geometrical optics) system. Although invisible in the heat-kernel expansion, these features determine the nonlocal parts of propagators, including the Casimir energy. I review some classic work, discuss the connection with vacuum energy, and show that when the coupling constant is varied with the energy, the periodic-orbit theory for generic quantum systems regains the clarity and simplicity that it always had for the wave equation in a cavity.While quantum gravity people were learning about the heat kernel, atomic physicists were progressing beyond it.The key property of the heat kernel expansion is its locality: the coefficients a n (x) are completely determined by the metric and potentials in a small neighborhood of x. This makes it easy to compute and universal in its applications to renormalization of ultraviolet divergences. Its formal inverse Laplace transform is an "averaged" spectral density [1][2][3] that is equally local and universal, insensitive to the detailed spacings of the eigenvalues (if the spectrum is discrete at all).The interesting part of renormalization is what is left behind when the counterterms are subtracted. For example, vacuum (Casimir) energy can be calculated from a two-point function (a kernel for the wave equation) by subtracting universal singular terms. The wave kernel and the vacuum energy are nonlocal ; they reflect boundary conditions, global topology, presence of periodic or closed classical orbits, and whether the dynamics is chaotic or integrable. This global geometrical information is encoded in the fine structure of the spectrum.More precisely, let T (t, x, y) be the integral kernel of e −t √H , where H is some positive selfadjoint second-order differential operator. (This "cylinder kernel" is analytically more tractable than kernels of wave (e.g., e

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“…Such series can indeed be determined formally by the requirement of consistency with (1.8) (see 3), but the "Tauberian" methods used to establish (1.7) [22] break down when applied to the higher-order terms. Indeed, the series which extends (1.7) is not literally asymptotic to the true eigenvalue distribution [3], [35], [4], [8]; it is valid only in some "averaged" sense [47], [48], [15], [16], [4], [6], [37], [38], [24]. The implications of this situation have not yet been adequately explored.…”

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confidence: 99%

The Local Geometric Asymptotics of Continuum Eigenfunction Expansions I. Overview

Fulling¹

1982

SIAM J. Math. Anal.

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The well-known connection between (a) the asymptotic density of eigenvalues of a differential operator H, and (b) the geometry of the region or manifold where H acts, has a local generalization: There is a connection between (a') the spectral measures or projection kernel describing the proper normalization, relative to a point x 0, of the expansion of an arbitrary function in eigenfunctions of an operator H (possibly with continuous spectrum), and (b') the values of the coefficients (symbol) of H and their derivatives at x 0.Potential applications (especially in general-relativistic quantum field theory) increasingly call for a detailed development of this theory (including calculation of numerical coefficients), which is begun here. The small-time expansion of the Green function of the heat operator, + H, is used to define the "mean" or "effective" expansion of the spectral measures. For the case of one independent and one dependent variable, the spectral and heat expansions are worked out in detail to a rather high order. They are shown to be related to (and obtainable from) existing phase-integral expansions with local amplitude functions (WKB-FrSman expansions) of the eigenfunctions.

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An extension of Weyl’s asymptotic law for eigenvalues

Cited by 20 publications

References 2 publications

Bose-Einstein condensation in arbitrarily shaped cavities

Bose-Einstein condensation in arbitrarily shaped cavities

Periodic orbits, spectral oscillations, scaling, and vacuum energy: beyond HaMiDeW

The Local Geometric Asymptotics of Continuum Eigenfunction Expansions I. Overview

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