[Paper Review] Schrödinger operators on armchair nanotubes. II

Andrey Badanin, Jochen Brüning+1 more|ArXiv.org|Jul 26, 2007

Spectral Theory in Mathematical Physics22 references3 citations

TL;DR

This paper analyzes the spectral properties of Schrödinger operators with periodic potentials on armchair nanotube graphs, focusing on the absolutely continuous spectrum. It characterizes the spectrum's structure, including gap endpoints via periodic/antiperiodic eigenvalues and resonances, and derives asymptotic behavior of gaps at high energy, revealing that resonance gaps correspond to non-real values of the Lyapunov function.

ABSTRACT

We consider the Schrödinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe the absolutely continuous spectrum of the Schrödinger operator: 1) the multiplicity, 2) endpoints of the gaps, they are given by periodic or antiperiodic eigenvalues or resonances (branch points of the Lyapunov function), 3) resonance gaps, where the Lyapunov function is non-real. We determine the asymptotics of the gaps at high energy.

Motivation & Objective

• To characterize the absolutely continuous spectrum of Schrödinger operators on quasi-1D armchair nanotube graphs.
• To determine the endpoints of spectral gaps in terms of periodic or antiperiodic eigenvalues and resonances (branch points of the Lyapunov function).
• To analyze the asymptotic behavior of spectral gaps at high energy.
• To identify resonance gaps where the Lyapunov function is non-real.
• To establish the multiplicity of the absolutely continuous spectrum and its dependence on the potential's symmetry.

Proposed method

• Modeling the nanotube as a periodic graph with a fundamental cell composed of six edges arranged in a hexagonal lattice.
• Defining the Schrödinger operator $\mathscr{H} = -\Delta + \mathscr{V}_q$ on the graph $\Gamma^N$, with $q \in L^2_{\text{even}}(0,1)$.
• Using the transfer matrix method and the Lyapunov function to analyze spectral properties and locate gap endpoints.
• Applying Kirchhoff boundary conditions at vertices to ensure self-adjointness of the operator.
• Deriving asymptotic expansions for eigenvalues and spectral gaps using perturbation theory and Fourier coefficient analysis.
• Employing the function $F(\lambda)$ and its associated $F_-$ to determine resonance conditions and gap structure.

Experimental results

Research questions

• RQ1 What determines the endpoints of spectral gaps in the absolutely continuous spectrum of Schrödinger operators on armchair nanotubes?
• RQ2 How do periodic and antiperiodic eigenvalues relate to the spectral gap structure?
• RQ3 What is the asymptotic behavior of spectral gaps at high energy for these operators?
• RQ4 Under what conditions do resonance gaps—where the Lyapunov function is non-real—arise?
• RQ5 How does the symmetry of the potential ($q \in L^2_{\text{even}}(0,1)$) affect the spectral multiplicity and gap structure?

Key findings

• The absolutely continuous spectrum consists of intervals separated by gaps, with endpoints determined by periodic or antiperiodic eigenvalues or resonances (branch points of the Lyapunov function).
• Resonance gaps occur precisely when the Lyapunov function is non-real, indicating a transition in spectral type.
• At high energy, the asymptotic behavior of the gaps is characterized by $\lambda_{2,2n}^{0,\pm} = (\pi n)^2 \pm \sqrt{\frac{2}{3}q_{sn}^2 + q_{cn}^2} + o(n^{-1})$, where $q_{sn}, q_{cn}$ are Fourier coefficients of the potential.
• For even potentials, $F_- = 0$, leading to $F_{k,1}(r) = F_{k,2}(r) = -\frac{s_k^2 + 1}{2} \in (-1, -\frac{1}{2}]$, which determines the sign of the spectral function $v_k$.
• When the potential is a delta-like perturbation $q_\varepsilon = \frac{1}{\varepsilon}\delta(t - \frac{1}{2} - c_k\varepsilon - \varepsilon^2)$, $v_k(\lambda_{1,2n-1}^{0,\pm}, q_\varepsilon) > 0$ for small $\varepsilon$, implying the existence of embedded eigenvalues.
• For $k \neq N/2$, the condition $v_k(\lambda_{1,2n-1}^{0,\pm}) \geq 0$ holds if and only if $F_{k,1}(r_{k,n}^{\pm}) = F_{k,2}(r_{k,n}^{\pm}) \leq -1$, and $v_k < 0$ if the values lie in $(-1, -\frac{1}{2}]$.