Bound-state asymptotic estimates for window-coupled Dirichlet strips and layers

Exner, Pavel; Vugalter, Semjon

doi:10.1088/0305-4470/30/22/023

Cited by 56 publications

(54 citation statements)

References 8 publications

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“…we check that 6) where the constant C is independent of g and δ. In view of the inequality obtained and (2.5) it remains to show that the series…”

Section: Domain Of H ωmentioning

confidence: 99%

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The spectrum of two quantum layers coupled by a window

Борисов¹

2007

J. Phys. A: Math. Theor.

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We consider the Dirichlet Laplacian in a domain two three-dimensional parallel layers having common boundary and coupled by a window. The window produces the bound states below the essential spectrum; we obtain two-sided estimates for them. It is also shown that the eigenvalues emerge from the threshold of essential spectrum as the window passes through certain critical shapes. We prove the necessary condition for the window to be of critical shape. Under an additional assumption we show that this condition is sufficient and obtain the asymptotic expansion for the emerging eigenvalue as well as for the associated eigenfunction.

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“…we check that 6) where the constant C is independent of g and δ. In view of the inequality obtained and (2.5) it remains to show that the series…”

Section: Domain Of H ωmentioning

confidence: 99%

“…The two-dimensional case was studied quite intensively, we refer here to [1], [2], [3], [4], [5], [6], [9] (see also references therein). It was shown that the perturbation by the window(s) is a negative one, i.e., it leads to the presence of the isolated bound states below the essential spectrum; the latter is invariant w.r.t.…”

Section: Introductionmentioning

confidence: 99%

The spectrum of two quantum layers coupled by a window

Борисов¹

2007

J. Phys. A: Math. Theor.

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“…Some results were obtained earlier. Namely, variational estimates for the bound state (not for resonances) close to the threshold are in [3,4], for the combined DirichletÄNeumann case are in [8]. Asymptotics of the bound states for the Dirichlet case is in [5,6].…”

Section: Resonances In Coupled Waveguidesmentioning

confidence: 99%

Quantum computer elements based on coupled quantum waveguides

Gavrilov

Gortinskaya

Pestov

et al. 2007

Phys. Part. Nuclei Lett.

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Possible applications of two weakly coupled quantum waveguides for quantum computation are considered. The approach is based on the resonance phenomena in the system. Two different qubits interpretations are described. Some single-qubit and two-qubits operations are realized in the framework of these interpretations. ·¥¤²μ¦¥´Ò ¢μ §³μ¦´Ò¥ ¶·¨³¥´¥´¨Ö ¤¢ÊÌ¸² ¡μ¸¢Ö § ´´ÒÌ ±¢ ´Éμ¢ÒÌ ¢μ²´μ¢μ¤μ¢ ¤²Ö ±¢ ´Éμ-¢ÒÌ ¢ÒÎ¨¸²¥´¨°.ˆ¸ ¶μ²Ó §Ê¥É¸Ö ³¥Éμ¤, μ¸´μ¢ ´´Ò°´ ·¥ §μ´ ´¸´ÒÌ ÔËË¥±É Ì ¢¸¨¸É¥³¥. ¶¨¸ ´Ò ¤¢¥ · §²¨Î´Ò¥¨´É¥· ¶·¥É Í¨¨±¢ ´Éμ¢μ£μ ¡¨É . ¥ ²¨ §μ¢ ´Ò´¥±μÉμ·Ò¥ μ¤´μ±Ê¡¨Éμ¢Ò¥¨¤¢ÊÌ±Ê-¡¨Éμ¢Ò¥ μ ¶¥· Í¨¨¢ · ³± Ì ¤ ´´ÒÌ¨´É¥· ¶·¥É Í¨°.

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“…However, Exner and Vugalter (1997) used variational techniques to show that for sufficiently small a/d there is just one bound state and that the ground-state eigenvalue, kd, then satisfies (3.18) for some positive c 1 and c 2 . It is clear that for small windows, this will produce values of kd which differ from the upper cut-off by an extremely small amount and this explains why we were unable to compute bound-state energies when a/d is less than about 0.25.…”

Section: Laterally Coupled Planar Waveguidesmentioning

confidence: 99%

“…Exner and Vugalter (1997) considered the case when the hole was sufficiently small so that only one eigenvalue occured below the continuous spectrum. The authors were then able to provide upper and lower asymptotic bounds on the gap between the eigenvalue and the continuous spectrum.…”

Section: Introductionmentioning

confidence: 99%

Bound states in coupled guides. II. Three dimensions

Linton

Ratcliffe

2004

Journal of Mathematical Physics

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We compute bound-state energies in two three-dimensional coupled waveguides, each obtained from the two-dimensional configuration considered in part I by rotating the geometry about a different axis. The first geometry consists of two concentric circular cylindrical waveguides coupled by a finite length gap along the axis of the inner cylinder and the second is a pair of planar layers coupled laterally by a circular hole. We have also extended the theory for this latter case to include the possibility of multiple circular windows. Both problems are formulated using a mode-matching technique, and in the cylindrical guide case the same residue calculus theory as used in I is employed to find the bound-state energies. For the coupled planar layers we proceed differently, computing the zeros of a matrix derived from the matching analysis directly.

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Bound-state asymptotic estimates for window-coupled Dirichlet strips and layers

Cited by 56 publications

References 8 publications

The spectrum of two quantum layers coupled by a window

The spectrum of two quantum layers coupled by a window

Quantum computer elements based on coupled quantum waveguides

Bound states in coupled guides. II. Three dimensions

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