@article{BeckusPinchover2020, author = {Beckus, Siegfried and Pinchover, Yehuda}, title = {Shnol-type theorem for the Agmon ground state}, series = {Journal of spectral theory}, volume = {10}, journal = {Journal of spectral theory}, number = {2}, publisher = {EMS Publishing House}, address = {Z{\"u}rich}, issn = {1664-039X}, doi = {10.4171/JST/296}, pages = {355 -- 377}, year = {2020}, abstract = {LetH be a Schrodinger operator defined on a noncompact Riemannianmanifold Omega, and let W is an element of L-infinity (Omega; R). Suppose that the operator H + W is critical in Omega, and let phi be the corresponding Agmon ground state. We prove that if u is a generalized eigenfunction ofH satisfying vertical bar u vertical bar <= C-phi in Omega for some constant C > 0, then the corresponding eigenvalue is in the spectrum of H. The conclusion also holds true if for some K is an element of Omega the operator H admits a positive solution in (Omega) over bar = Omega \ K, and vertical bar u vertical bar <= C psi in (Omega) over bar for some constant C > 0, where psi is a positive solution of minimal growth in a neighborhood of infinity in Omega. Under natural assumptions, this result holds also in the context of infinite graphs, and Dirichlet forms.}, language = {en} } @article{BeckusEliaz2021, author = {Beckus, Siegfried and Eliaz, Latif}, title = {Eigenfunctions growth of R-limits on graphs}, series = {Journal of spectral theory / European Mathematical Society}, volume = {11}, journal = {Journal of spectral theory / European Mathematical Society}, number = {4}, publisher = {EMS Press, an imprint of the European Mathematical Society - EMS - Publishing House GmbH, Institut f{\"u}r Mathematik, Technische Universit{\"a}t}, address = {Berlin}, issn = {1664-039X}, doi = {10.4171/JST/389}, pages = {1895 -- 1933}, year = {2021}, abstract = {A characterization of the essential spectrum of Schrodinger operators on infinite graphs is derived involving the concept of R-limits. This concept, which was introduced previously for operators on N and Z(d) as "right-limits," captures the behaviour of the operator at infinity. For graphs with sub-exponential growth rate, we show that each point in sigma(ss)(H) corresponds to a bounded generalized eigenfunction of a corresponding R-limit of H. If, additionally, the graph is of uniform sub-exponential growth, also the converse inclusion holds.}, language = {en} } @misc{BeckusBellissardDeNittis2019, author = {Beckus, Siegfried and Bellissard, Jean and De Nittis, Giuseppe}, title = {Corrigendum to: Spectral continuity for aperiodic quantum systems I. General theory. - [Journal of functional analysis. - 275 (2018), 11, S. 2917 - 2977]}, series = {Journal of functional analysis}, volume = {277}, journal = {Journal of functional analysis}, number = {9}, publisher = {Elsevier}, address = {San Diego}, issn = {0022-1236}, doi = {10.1016/j.jfa.2019.06.001}, pages = {3351 -- 3353}, year = {2019}, abstract = {A correct statement of Theorem 4 in [1] is provided. The change does not affect the main results.}, language = {en} } @article{BeckusBellissardDeNittis2020, author = {Beckus, Siegfried and Bellissard, Jean and De Nittis, Giuseppe}, title = {Spectral continuity for aperiodic quantum systems}, series = {Journal of mathematical physics}, volume = {61}, journal = {Journal of mathematical physics}, number = {12}, publisher = {American Institute of Physics}, address = {Melville, NY}, issn = {0022-2488}, doi = {10.1063/5.0011488}, pages = {19}, year = {2020}, abstract = {This work provides a necessary and sufficient condition for a symbolic dynamical system to admit a sequence of periodic approximations in the Hausdorff topology. The key result proved and applied here uses graphs that are called De Bruijn graphs, Rauzy graphs, or Anderson-Putnam complex, depending on the community. Combining this with a previous result, the present work justifies rigorously the accuracy and reliability of algorithmic methods used to compute numerically the spectra of a large class of self-adjoint operators. The so-called Hamiltonians describe the effective dynamic of a quantum particle in aperiodic media. No restrictions on the structure of these operators other than general regularity assumptions are imposed. In particular, nearest-neighbor correlation is not necessary. Examples for the Fibonacci and the Golay-Rudin-Shapiro sequences are explicitly provided illustrating this discussion. While the first sequence has been thoroughly studied by physicists and mathematicians alike, a shroud of mystery still surrounds the latter when it comes to spectral properties. In light of this, the present paper gives a new result here that might help uncovering a solution.}, language = {en} } @article{BeckusBellissardCornean2019, author = {Beckus, Siegfried and Bellissard, Jean and Cornean, Horia}, title = {Holder Continuity of the Spectra for Aperiodic Hamiltonians}, series = {Annales de l'Institut Henri Poincar{\´e}}, volume = {20}, journal = {Annales de l'Institut Henri Poincar{\´e}}, number = {11}, publisher = {Springer}, address = {Cham}, issn = {1424-0637}, doi = {10.1007/s00023-019-00848-6}, pages = {3603 -- 3631}, year = {2019}, abstract = {We study the spectral location of a strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Holder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.}, language = {en} }