TY - JOUR

T1 - Monodromization and a $$ \mathcal{P} \mathcal{T} $$-Symmetric Nonself-Adjoint Quasi-Periodic Operator

AU - Федотов, Александр Александрович

AU - Борисов, Денис

PY - 2023/9/1

Y1 - 2023/9/1

N2 - Abstract: We study the operator acting in L_2(\mathbb{R}) by the formula (\mathcal{A} \psi)(x)=\psi(x+\omega)+\psi(x-\omega)+ \lambda e^{-2\pi i x} \psi(x) , where x\in\mathbb R is a variable, and \lambda>0 and \omega\in(0,1) are parameters. It is related to the simplest quasi-periodic operator introduced by P. Sarnak in 1982. We investigate \mathcal{A} using the monodromization method, the Buslaev–Fedotov renormalization approach, which arose when trying to extend the Bloch–Floquet theory to difference equations on \mathbb{R} . Within this approach, the analysis of \mathcal{A} turns out to be very natural and transparent. We describe the geometry of the spectrum and calculate the Lyapunov exponent.

AB - Abstract: We study the operator acting in L_2(\mathbb{R}) by the formula (\mathcal{A} \psi)(x)=\psi(x+\omega)+\psi(x-\omega)+ \lambda e^{-2\pi i x} \psi(x) , where x\in\mathbb R is a variable, and \lambda>0 and \omega\in(0,1) are parameters. It is related to the simplest quasi-periodic operator introduced by P. Sarnak in 1982. We investigate \mathcal{A} using the monodromization method, the Buslaev–Fedotov renormalization approach, which arose when trying to extend the Bloch–Floquet theory to difference equations on \mathbb{R} . Within this approach, the analysis of \mathcal{A} turns out to be very natural and transparent. We describe the geometry of the spectrum and calculate the Lyapunov exponent.

UR - https://www.mendeley.com/catalogue/c2eb524c-c49c-3d90-a59a-5901ca3ac749/

U2 - 10.1134/s1061920823030032

DO - 10.1134/s1061920823030032

M3 - Article

VL - 30

SP - 294

EP - 309

JO - Russian Journal of Mathematical Physics

JF - Russian Journal of Mathematical Physics

SN - 1061-9208

IS - 3

ER -