<i>Spectra of Finite Systems</i>

Baltes, H. P.; Hilf, Eberhard R.; Challifour, John L.

doi:10.1063/1.3037672

Cited by 146 publications

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“…The previous numerical study would be still more appealing if we knew how the spectrum scales to other energy regions. The Weyl's law [24] tells us that the density of states associated to the vertical axis in Fig. 3, scales as k. The problem appears with the horizontal axis because the scaling of the density of consecutive avoided crossings ρ a.c is unknown.…”

Section: Numerical Resultsmentioning

confidence: 99%

Characterization of Landau-Zener transitions in systems with complex spectra

Sánchez

Vergini

Wisniacki³

1996

Phys. Rev. E

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This paper is concerned with the study of one-body dissipation effects in idealized models resembling a nucleus. In particular, we study the quantum mechanics of a free particle that collides elastically with the slowly moving walls of a Bunimovich stadium billiard. Our results are twofold. First, we develop a method to solve in a simple way the quantum mechanical evolution of planar billiards with moving walls. The formalism is based on the scaling method [1] which enables the resolution of the problem in terms of quantities defined over the boundary of the billiard.The second result is related to the quantum aspects of dissipation in systems

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Section: Numerical Resultsmentioning

confidence: 99%

Characterization of Landau-Zener transitions in systems with complex spectra

Sánchez

Vergini

Wisniacki³

1996

Phys. Rev. E

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“…Thus, it becomes crucial to be able to check this condition for the concrete operators arising in applications, for instance, for the Laplacian in a bounded domain of R n , say, with Dirichlet or Newmann boundary conditions. However, despite the vast amount of various results on the distribution of eigenvalues and improvements of the classical Weyl asymptotics, see [27,4] and references therein, almost nothing is known about the existence of spectral gaps even in the case of the Laplacian. In particular, for a bounded domain of R 2 , we do not know any examples when the spectral gaps of arbitrarily large size do not exist, but, to the best of our knowledge, there are also no results that such gaps exist for all/generic domains of R 2 .…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

Inertial manifolds and finite-dimensional reduction for dissipative PDEs

Zelik

2014

Proceedings of the Royal Society of Edinburgh: Section A Mathem

243

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Abstract. These notes are devoted to the problem of finite-dimensional reduction for parabolic PDEs. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mané projection theorems. The recent counterexamples which show that the underlying dynamics may be in a sense infinite-dimensional if the spectral gap condition is violated as well as the discussion on the most important open problems are also included.

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“…ρ 1 (E) is trivial and the smoothed ρ S (E) can be found in Ref. [15] for Dirichlet and Neumann BC. A decomposition of the EM case into Dirichlet and Neumann sub-problems, similar to eq.…”

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confidence: 99%

Attractive Casimir Forces in a Closed Geometry

et al. 2005

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We study the Casimir force acting on a conducting piston with arbitrary cross section. We find the exact solution for a rectangular cross section and the first three terms in the asymptotic expansion for small height to width ratio when the cross section is arbitrary. Though weakened by the presence of the walls, the Casimir force turns out to be always attractive. Claims of repulsive Casimir forces for related configurations, like the cube, are invalidated by cutoff dependence.PACS numbers: 03.65. Sq, 03.70.+k, 42.25.Gy In 1948 Casimir predicted a force between conducting plates due to quantum electromagnetic fluctuations [1]. Casimir forces have now been measured precisely [2], and agree with theory to a few percent accuracy. As miniaturization continues, static friction due to the Casimir attraction between components in microelectromechanical systems (MEMS) has become a problem of increasing concern [3]. Since the Casimir force is strongly geometry dependent[4], the question arises whether it can be made repulsive by arranging conductors appropriately. It has been claimed that the parallelepiped of sides a, b, b has positive Casimir energy for aspect ratio in the range 0.408 < a/b < 3.48 [5,6]. If so, a repulsive force would occur for a/b > 0.785, as a is varied at fixed b.However this example, and indeed all claims of repulsive Casimir forces, require elastic deformations of single bodies (a rectangle or a circle in 2-dimensions, a parallelepiped, a cylinder or a sphere in 3-dimensions, typically) treated as perfect, often infinitely thin conductors. It has been shown that, when the conductors are modeled more realistically, the Casimir energies associated with deformation actually depend strongly on material properties such as the plasma frequency and skin depth [7], and diverge in the perfect metal limit where these cutoffs are ignored. In contrast, the forces between rigid bodies remain finite in that limit. Recently Cavalcanti[8] introduced a modification of the rectangle -the "Casimir piston" -that is demonstrably free of cutoff dependence [9]. The 2-dimensional piston studied in Ref. [8] consists of a single rectangle divided in two by a partition (the "piston"). Cavalcanti calculated the Casimir force on the piston due to fluctuations of a scalar field obeying Dirichlet boundary conditions on all surfaces. He found that the force on the piston is always attractive, although substantially weakened with respect to parallel lines.In this Letter and the coming paper [11] we consider the 3-dimensional piston, for both scalar and electromagnetic (EM) fields. We keep careful track of possible cutoff dependences and show that they cancel for the piston configuration [9]. First we give the exact result for pistons of rectangular cross section, expanding the result in powers of the piston-base separation, a, and identify the terms with specific optical paths [12]. Next we consider pistons of arbitrary cross section, where an expansion for small a can be derived. Finally we show that the force on the piston ...

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Spectra of Finite Systems

Cited by 146 publications

References 0 publications

Characterization of Landau-Zener transitions in systems with complex spectra

Characterization of Landau-Zener transitions in systems with complex spectra

Inertial manifolds and finite-dimensional reduction for dissipative PDEs

Attractive Casimir Forces in a Closed Geometry

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