Extension Theory and Localization of Resonances for Domains of Trap Type

Popov, I. Yu.

doi:10.1070/sm1992v071n01abeh001394

Cited by 31 publications

(21 citation statements)

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“…The weak logarithmic singularity in (2.7) on η = 0 was observed previously for the sphere in [23] (see page 247 of [23]) and for general domains in [40], [32], and [44]. Our calculation in Appendix A has identified the regular part of the singularity structure for G s in (2.7), which is needed below to obtain a three-term expansion for the MFPT.…”

Section: Narrow Escape From a Spherical Domainmentioning

confidence: 86%

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

Cheviakov¹,

Ward²,

Straube³

2010

Multiscale Model. Simul.

170

263

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Abstract. The mean first passage time (MFPT) is calculated for a Brownian particle in a spherical domain in R 3 that contains N small nonoverlapping absorbing windows, or traps, on its boundary. For the unit sphere, the method of matched asymptotic expansions is used to derive an explicit three-term asymptotic expansion for the MFPT for the case of N small locally circular absorbing windows. The third term in this expansion, not previously calculated, depends explicitly on the spatial configuration of the absorbing windows on the boundary of the sphere. The threeterm asymptotic expansion for the average MFPT is shown to be in very close agreement with full numerical results. The average MFPT is shown to be minimized for trap configurations that minimize a certain discrete variational problem. This variational problem is closely related to the well-known optimization problem of determining the minimum energy configuration for N repelling point charges on the unit sphere. Numerical results, based on global optimization methods, are given for both the optimum discrete energy and the arrangements of the centers {x 1 , . . . , x N } of N circular traps on the boundary of the sphere. A scaling law for the optimum discrete energy, valid for N 1, is also derived.Key words. narrow escape, mean first passage time, matched asymptotic expansions, surface Neumann Green's functions, discrete variational problem, logarithmic switchback terms AMS subject classifications. 35B25, 35C20, 35P15, 35J05, 35J08DOI. 10.1137/100782620 1. Introduction. The narrow escape problem concerns the motion of a Brownian particle confined in a bounded domain Ω ∈ R d (d = 2, 3) whose boundary ∂Ω = ∂Ω r ∪ ∂Ω a is almost entirely reflecting (∂Ω r ), except for small absorbing windows, or traps, labeled collectively by ∂Ω a , through which the particle can escape. Denoting the trajectory of the Brownian particle by X(t), the mean first passage time (MFPT) v(x) is defined as the expectation value of the time τ taken for the Brownian particle to become absorbed somewhere in ∂Ω a starting initially fromThe calculation of v(x) becomes a narrow escape problem in the limit when the measure of the absorbing set |∂Ω a | = O(ε d−1 ) is asymptotically small, where 0 < ε 1 measures the dimensionless radius of an absorbing window. Since the MFPT diverges as ε → 0, the calculation of the MFPT v(x) constitutes a singular perturbation problem.The narrow escape problem has many applications in biophysical modeling (see [2], [16], [19], [39] and the references therein). For the case of a two-dimensional domain, the narrow escape problem has been studied with a variety of analytical methods in [19], [42], [43], [20], and Part I of this paper [30]. In this paper, we use

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Section: Narrow Escape From a Spherical Domainmentioning

confidence: 86%

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

Cheviakov¹,

Ward²,

Straube³

2010

Multiscale Model. Simul.

170

263

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“…This procedure gives us a model of point-like windows in the plane (see [12], [13]). The type of the chosen self-adjoint extension is related with the type of physical coupling through the windows.…”

Section: Resultsmentioning

confidence: 99%

Chain of point-like potentials in ℛ³and infiniteness of the number of bound states

Boitsev

Popov

Sokolov

2014

J. Phys.: Conf. Ser.

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Abstract. Infinite chain of point-like potentials having the Hamiltonian with infinite number of eigenvalues below the continuous spectrum is constructed. The background of the model is the theory of self-adjoint extensions of symmetric operators in the Hilbert space. The analogous example of the Hamiltonian is obtained for the system of three-dimensional waveguides coupled through point-like windows. IntroductionThe problem of existence of infinite number of eigenvalues of the Schrödinger operator. It appears in different physical situations. Particularly, in three-particle problem it is known as the Efimov effect [1], [2], [3]. Strict mathematical justification for the existence of infinitely many bound states below the continuous spectrum for short-range perturbations of a much larger class of spinorbit Hamiltonians, which includes, in particular, the above Rashba Hamiltonian was obtained in [4]. The authors uses min-max principle in the spirit of [5]. In the present paper we construct an operator having infinitely many different eigenvalues below the continuous spectrum. The result can be interested from the point of view of functional model due to close relation between the properties of the characteristic functions and the operator spectrum [6] More precisely, we construct a chain of zero-range potentials with specific intensities which gives us the operator in question. This type of potential has useful advantage-it allows us to construct the spectral equation in an explicit form, correspondingly, the spectral analysis of the operator is simpler. Point-like interaction is introduced by a conventional way [8], [7]. The construction is based on the theory of self-adjoint extensions of symmetric operators. It is widely used for the description of various quantum systems.Let us describe briefly the model operator construction. Consider a chain of n delta-potentials in R 3 . The positions of the potentials are x m . First, we restrict the Laplace operator onto the set of smooth functions vanishing at the points x m . The obtained operator is symmetric non-self-adjoint. To construct a self-adjoint extension, it is more convenient to deal with the corresponding restriction of the adjoint operator instead of the extension of the symmetric operator. There are several ways of extensions descriptions, e.g., boundary triplets method ([9], von Neumann formulas ([11]), Krein resolvent formula ([10]). We will use here a variant of the second approach which allows one in the case of semi-boundedness of the Hamiltonian to present

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“…where by L some logarithmic term is denoted, see 1) [Mfaddeev], [IP90] and B s (λ) is a term containing the spectral information, [Opening]. Summarising (2.1) and (2.2) we may estimate the Green function for the Neumann Laplacian uniformly for x, y ∈ Ω as: [Mfaddeev] the logarithmic estimate for this term is obtained, |L| ≤ C log 1 |x−as | , in [IP90] it is shown that the term is just proportional to the product of the curvature of the boundary at the point as by the logarithmic singularity log 1 |x−as| .…”

Section: Deficiency Elementsmentioning

confidence: 99%

“…Summarising (2.1) and (2.2) we may estimate the Green function for the Neumann Laplacian uniformly for x, y ∈ Ω as: [Mfaddeev] the logarithmic estimate for this term is obtained, |L| ≤ C log 1 |x−as | , in [IP90] it is shown that the term is just proportional to the product of the curvature of the boundary at the point as by the logarithmic singularity log 1 |x−as| . Similar estimate and even the same leading singular term as in (2.1) in the asymptotics of Green function near each inner point y holds also for Schrödinger operator A with continuous potential.…”

Section: Deficiency Elementsmentioning

confidence: 99%

Scattering on a Compact Domain with Few Semi-Infinite Wires Attached: Resonance Case

et al. 2002

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Extension Theory and Localization of Resonances for Domains of Trap Type

Cited by 31 publications

References 2 publications

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

Chain of point-like potentials in ℛ³and infiniteness of the number of bound states

Scattering on a Compact Domain with Few Semi-Infinite Wires Attached: Resonance Case

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Extension Theory and Localization of Resonances for Domains of Trap Type

Cited by 31 publications

References 2 publications

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

Chain of point-like potentials in ℛ3and infiniteness of the number of bound states

Scattering on a Compact Domain with Few Semi-Infinite Wires Attached: Resonance Case

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Chain of point-like potentials in ℛ³and infiniteness of the number of bound states