Opuscula Math. 28, no. 2 (2008), 137-150

Opuscula Mathematica

# A first-order spectral phase transition in a class of periodically modulated Hermitian Jacobi matrices

Irina Pchelintseva

Abstract. We consider self-adjoint unbounded Jacobi matrices with diagonal $$q_n = b_{n}n$$ and off-diagonal entries $$\lambda_n = n$$, where $$b_{n}$$ is a $$2$$-periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum of the operator is either purely absolutely continuous or discrete. We study the situation where the spectral phase transition occurs, namely the case of $$b_{1}b_{2} = 4$$. The main motive of the paper is the investigation of asymptotics of generalized eigenvectors of the Jacobi matrix. The pure point part of the spectrum is analyzed in detail.

Keywords: Jacobi matrices, spectral phase transition, absolutely continuous spectrum, pure point spectrum, discrete spectrum, subordinacy theory, asymptotics of generalized eigenvectors.

Mathematics Subject Classification: 47A10, 47B36.

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• Irina Pchelintseva
• St. Petersburg University, Institute of Physics, Department of Mathematical Physics, Ulianovskaia 1, 198904, St. Petergoff, St. Petersburg, Russia
• Revised: 2007-12-06.
• Accepted: 2007-12-06.