The functional structure of the monodromy matrix for Harper’s equation
Vladimir Savel'evich Buslaev, Alexander Fedotov · Operator theory · 1994
In this paper we continue our investigation of Harper’s equation: (1.1) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq % aHipqEcaGGOaGaamiEaiabgUcaRiaadIgacaGGPaGaey4kaSIaeqiY % dKNaaiikaiaadIhacqGHsislcaWGObGaaiykaaqaaiaaikdaaaGaey % 4kaSIaci4yaiaac+gacaGGZbGaaGPaVlaadIhacaaMe8UaeqiYdKNa % aiikaiaadIhacaGGPaGaeyypa0JaamyraiabeI8a5jaacIcacaWG4b % Gaaiykaiaac6caaaa!5627! $$ \frac{{\psi (x + h) + \psi (x - h)}}{2} + \cos \,x\;\psi (x) = E\psi (x). $$ Here h is a fixed positive parameter and x ∈ ℝ or x ∈ ℂ. This equation appeared as a model for Bloch electron in a weak constant magnetic field [Ho]. The structure of the spectrum σ h of Harper’s equation on L 2(ℝ) appeared to be very rich and Harper’s equation attracted the attention of both physicists and mathematicians, see, for example, [C-F-K-S].