# What is absolutely continuous spectrum

@article{Bruneau2016WhatIA, title={What is absolutely continuous spectrum}, author={Laurent Bruneau and Vojkan Jaksic and Yoram Last and Claude-Alain Pillet}, journal={arXiv: Mathematical Physics}, year={2016} }

This note is an expanded version of the author's
contribution to the Proceedings of the ICMP Santiago, 2015, and is based on a
talk given by the second author at the same Congress. It concerns a research
program devoted to the characterization of the
absolutely continuous spectrum of a self-adjoint operator H in terms of the transport
properties of a suitable class of open quantum systems canonically associated to H.

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## References

SHOWING 1-10 OF 25 REFERENCES

### Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators

- Mathematics
- 1996

We provide a short proof of that case of the Gilbert-Pearson theorem that is most often used: That all eigenfunctions bounded implies purely a.c. spectrum. Two appendices illuminate Weidmann's result…

### On the Kotani-Last and Schrodinger conjectures

- Mathematics
- 2012

In the theory of ergodic one-dimensional Schrodinger operators, ac spectrum has been traditionally expected to be very rigid. Two key conjectures in this direction state, on one hand, that ac…

### ON STEKLOV'S CONJECTURE IN THE THEORY OF ORTHOGONAL POLYNOMIALS

- Mathematics
- 1980

This paper constructs an example of a weight function on the interval such that 0$ SRC=http://ej.iop.org/images/0025-5734/36/4/A07/tex_sm_1864_img3.gif/>, , whereas the corresponding sequence of…

### The Thouless Conjecture for a One-Dimensional Chain

- Mathematics, Physics
- 1980

Abstract : The Thouless conjecture relating energy level shifts as a function of boundary conditions to conductance is shown to be incorrect in detail in the one-dimensional chain, though…

### Szegő's Theorem and Its Descendants: Spectral Theory for L 2 Perturbations of Orthogonal Polynomials

- Mathematics
- 2010

This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gabor Szego's classic 1915 theorem and its 1920 extension.…

### Szego's Theorem and Its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials (M. B. Porter Lectures)

- Mathematics
- 2010

This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gbor Szego's classic 1915 theorem and its 1920 extension. Barry…

### Kotani theory for one dimensional stochastic Jacobi matrices

- Mathematics
- 1983

We consider families of operators,Hω, on ℓ2 given by (Hωu)(n)=u(n+1)+u(n−1)+Vω(n)u(n), whereVω is a stationary bounded ergodic sequence. We prove analogs of Kotani's results, including that for a.e.…

### Orthogonal polynomials with exponentially decaying recursion coefficients

- Mathematics
- 2007

We review recent results on necessary and sufficient conditions for
measures on R and ∂D to yield exponential decay of the recursion coefficients of
the corresponding orthogonal polynomials. We…

### Independent electron model for open quantum systems: Landauer-Büttiker formula and strict positivity of the entropy production

- Physics
- 2007

A general argument leading from the formula for currents through an open noninteracting mesoscopic system given by the theory of nonequilibrium steady states to the Landauer-Buttiker formula is…