Weakly coupled bound states in quantum waveguides

Abstract

Abstract. We study the eigenvalue spectrum of Dirichlet Laplacians which model quantum waveguides associated with tubular regions outside of a bounded domain. Intuitively, our principal new result in two dimensions asserts that any domain Ω obtained by adding an arbitrarily small "bump" to the tube Ω 0 = R × (0, 1) (i.e., Ω Ω 0 , Ω ⊂ R 2 open and connected, Ω = Ω 0 outside a bounded region) produces at least one positive eigenvalue below the essential spectrum [π 2 , ∞) of the Dirichlet Laplacian −∆ D Ω . For |Ω\Ω 0 | sufficiently small (| . | abbreviating Lebesgue measure), we prove uniqueness of the ground state E Ω of −∆ D Ω and derive the "weak coupling" result E Ω = π 2 − π 4 |Ω\Ω 0 | 2 + O(|Ω\Ω 0 | 3 ) using a Birman-Schwinger-type analysis. As a corollary of these results we obtain the following surprising fact: Starting from the tube Ω 0 with Dirichlet boundary conditions at ∂Ω 0 , replace the Dirichlet condition by a Neumann boundary condition on an arbitrarily small segment (a, b) × {1}, a < b, of ∂Ω 0 . If H(a, b) denotes the resulting Laplace operator in L 2 (Ω 0 ), then H(a, b) has a discrete eigenvalue in [π 2 /4, π 2 ) no matter how small |b − a| > 0 is. §1. Introduction Our goal in this paper is to study the bound state spectra of the Dirichlet Laplacian −∆D Ω for open regions Ω ⊂ R n which are tubes outside of a bounded region (quantum waveguides). (Following the traditional notation in quantum physics, we denote the Laplacian by −∆ as opposed to ∆ in the following.) In particular, let Ω 0 ⊂ R 2 be defined byConsider open connected sets Ω such that:Because of condition (i),Then one of our main goals will be to prove

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“…In particular, Bulla and collaborators have considered in Ref. [7] the problem of an infinite homogeneous waveguide on the region Ω λ = (x, y) ∈ R 2 |0 < y < 1 + λf (x) , (1) obeying the Dirichlet boundary conditions at the border, assuming that f is a C ∞ (R) function of compact support with f ≥ 0. In their calculation λ > 0 is a parameter which controls the deformation of the border (particularly the case λ = 0 reduces to a straight waveguide, with a purely continuum spectrum).…”

Weakly bound states in heterogeneous waveguides

“…Bulla and collaborators [14] have obtained the expression for the fundamental energy of an infinite two-dimensional waveguide, where the upper border is slightly deformed, to second order in the parameter controlling the deformation. Their result proves that a bound state is always present, whenever the deformation corresponds to a local enlargement of the waveguide.…”

Асимптотика Собственных Чисел На Непрерывном Спектре Регулярно Возмущенного Квантового Волновода

“…Результат работы [6] и теорем 1, 8 настоящей работы показывают, что точечный спектр задачи (2) может содержать два собственных числа, одно в дискретном, а другое в непрерывном спектре. Вопрос о построении локального возмущения кван-тового волновода, порождающего большее количество собственных чисел, остается открытым.…”

“…В работах [2]- [5] установлено существование и построена асимптотика собственно-го числа λ ε < λ † для слабо изогнутых волноводов. В статьях [6]- [11] исследованы плоские и многомерные волноводы с регулярными и сингулярными локальными воз-мущениями границы, найдены разнообразные условия существования и отсутствия дискретного спектра, а также указаны асимптотики.…”

“…Дискретный спектр рассматриваемой задачи (2) в волноводе (1) изучен в рабо-те [6], где проверено, что при условии…”