Finite pseudo orbit expansions for spectral quantities of quantum graphs

Band, Ram; Harrison, J. M.; Joyner, Christopher H.

doi:10.1088/1751-8113/45/32/325204

Cited by 28 publications

(61 citation statements)

References 41 publications

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“…Reasons for expecting cancelation of contributions from pseudo-orbits with periods beyond half the Heisenberg time in correlators of spectral determinants were put forth in [17]. More recently, such cancelation has been demonstrated for spectral fluctuations in graphs [18].…”

Section: Summary and Discussionmentioning

confidence: 99%

Chaotic maps and flows: exact Riemann–Siegel lookalike for spectral fluctuations

Braun¹,

Haake²

2012

J. Phys. A: Math. Theor.

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Abstract. To treat the spectral statistics of quantum maps and flows that are fully chaotic classically, we use the rigorous Riemann-Siegel lookalike available for the spectral determinant of unitary time evolution operators F . Concentrating on dynamics without time reversal invariance we get the exact two-point correlator of the spectral density for finite dimension N of the matrix representative of F , as phenomenologically given by random matrix theory. In the limit N → ∞ the correlator of the Gaussian unitary ensemble is recovered. Previously conjectured cancelations of contributions of pseudo-orbits with periods beyond half the Heisenberg time are shown to be implied by the Riemann-Siegel lookalike.

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Section: Summary and Discussionmentioning

confidence: 99%

Chaotic maps and flows: exact Riemann–Siegel lookalike for spectral fluctuations

Braun¹,

Haake²

2012

J. Phys. A: Math. Theor.

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“…The resonance condition (4.2) can be alternatively expressed by contributions of the irreducible pseudo-orbits similarly as for quantum graphs [22][23][24]. This expression is just yet another way how to write the determinant.…”

Section: Pseudo-orbit Expansion For the Resonance Conditionmentioning

confidence: 99%

Asymptotics of Resonances Induced by Point Interactions

Lipovský

Lotoreichik

2017

Acta Phys. Pol. A

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We consider the resonances of the self-adjoint three-dimensional Schrödinger operator with point interactions of constant strength supported on the set X = {xn} N n=1 . The size of X is defined by VX = maxπ∈Π N N n=1 |xn−x π(n) |, where ΠN is the family of all the permutations of the set {1, 2, . . . , N }. We prove that the number of resonances counted with multiplicities and lying inside the disc of radius R behaves asymptotically linearwhere the constant WX ∈ [0, VX ] can be seen as the effective size of X. Moreover, we show that there exist a configuration of any number of points such that WX = VX . Finally, we construct an example for N = 4 with WX < VX , which can be viewed as an analogue of a quantum graph with non-Weyl asymptotics of resonances.

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“…According to (3.3) the cases that give rise to r,t ∈ (0, 1) when n = 5 are (1, 0), (4, 5), (2, 0), (3, 5), (2, 1) and (3,4). Each case is discussed in the sequel.…”

Section: Ordermentioning

confidence: 99%