Generalised Quantum Waveguides
Generalised Quantum Waveguides
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广义量子波导
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
S. Teufel
中科院分区：
文献类型：
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作者：
Stefan Haag;J. Lampart;S. Teufel
我们研究了一般量子波导，并在这些空间上建立了拉普拉斯量的显式有效哈密顿量。传统的量子波导是$${mathbb{R}^3}$$ R3中曲线的$${epsilon}$$ λ管状邻域，我们感兴趣的对象是该管在渐近极限$${epsilonll1}$$ λ≪1处的狄利克雷拉普拉斯函数。我们通过考虑一个完整的d维子流形$${Bsubsetmathbb{R}^{d+k}}$$ B∧Rd+k上的光纤束M来推广这一点，其中光纤微胚于$${Fsubsetmathbb{R}^k}$$ F∧Rk，其总空间嵌入到B的$${epsilon}$$ λ邻域中。从这个角度来看，B扮演曲线的角色，F扮演传统量子波导的圆盘形截面的角色。我们的方法，除其他外，允许波导的横截面F沿着B变形，也研究在这种波导的边界上的拉普拉斯。通过应用纤维束上Schrödinger算符的绝热极限的最新结果，我们特别表明，对于小能量，M上的拉普拉斯算子的动力学和谱由与正常拉普拉斯算子基态带相关的绝热近似反映。我们给出了在各种情况下L2(B)上相应有效算子的显式公式，从而改进和扩展了$${mathbb{R}^3}$$ R3中量子波导和量子层的许多已知结果。
We study general quantum waveguides and establish explicit effective Hamiltonians for the Laplacian on these spaces. A conventional quantum waveguide is an $${epsilon}$$ϵ -tubular neighbourhood of a curve in $${mathbb{R}^3}$$R3 and the object of interest is the Dirichlet Laplacian on this tube in the asymptotic limit $${epsilonll1}$$ϵ≪1. We generalise this by considering fibre bundles M over a complete d-dimensional submanifold $${Bsubsetmathbb{R}^{d+k}}$$B⊂Rd+k with fibres diffeomorphic to $${Fsubsetmathbb{R}^k}$$F⊂Rk, whose total space is embedded into an $${epsilon}$$ϵ -neighbourhood of B. From this point of view, B takes the role of the curve and F that of the disc-shaped cross section of a conventional quantum waveguide. Our approach allows, among other things, for waveguides whose cross sections F are deformed along B and also the study of the Laplacian on the boundaries of such waveguides. By applying recent results on the adiabatic limit of Schrödinger operators on fibre bundles we show, in particular, that for small energies the dynamics and the spectrum of the Laplacian on M are reflected by the adiabatic approximation associated with the ground state band of the normal Laplacian. We give explicit formulas for the accordingly effective operator on L2(B) in various scenarios, thereby improving and extending many of the known results on quantum waveguides and quantum layers in $${mathbb{R}^3}$$R3.
约束量子系统的有效哈密顿量
DOI：
10.1090/memo/1083
发表时间：
2013
期刊：
影响因子：
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作者：
Teufel;Wachsmuth
通讯作者：
Wachsmuth