Final Report Summary - ETAM (A Mathematical Study of Electronic Transport in Aperiodic Media)
We studied the absence of transport in aperiodic media with and without additional energetic disorder. In our setting, we used the so-called Delone-Anderson operators, a class of random Schrödinger operators that serve to model a wide range of Hamiltonians, including the Anderson Hamiltonian and Delone operators. The latter are used to study electron dynamics in purely aperiodic media, as for example, in quasicrystals. In this case, our purpose was to show that the set of configurations associated to localization was a topologically generic set in the space of configurations, an idea that was grounded on the fact that localizing configurations are dense.
Our main tool in the analysis of Delone operators was the Multiscale Analysis (MSA) of Germinet-Klein (JEMS 2013), which is the most advanced technique known to date that allows to prove localization in the most general cases of Anderson Hamiltonians in the continuous setting. We represented Delone operators using auxiliary Delone-Anderson operators with Bernoulli random variables. This representation is valid only when the underlying configuration of impurities maintains a Delone structure, that is, is relatively dense in space. We achieved this by adding a bounded background Delone potential, then the main challenge was to obtain the initial estimate to apply the MSA in the case where the background potential is not periodic. This was done successfully using quantitative unique continuation principles obtained by Rojas-Molina-Veselic' (CMP 2012), and the later improvement by Klein (CMP 2012). Once the MSA could be performed, the next step was to determine the topological structure of the set of configurations for which the operator exhibits localization. This turned out to be very challenging, since we needed to go to the core of the constructive part of the proof of localization by the MSA which is the most technically involved and refined Multiscale method to date. At first, we were able to state that the localizing configurations, at finite scale, were G-delta sets in the space of configurations. However, in the limit when the finite scale grows arbitrarily, the aforementioned G-delta set was reduced to the trivial Delone set of departure. Our objectives concerning this topic needed, therefore, to be revised. Drawing from the theory of rank-one perturbation, in particular the work of Simon (Annals of Math 1995), we explored the possibility that the set of configurations associated with localization is topologically thin. We revised the constructive approach of the MSA and reformulated our goals. We are now confident that we can show that the set of configurations is, at least, topologically thin in the simpler case of the Anderson model. Due to the originality of our result, we expect this to have a high impact in the field of random Schrödinger operators. Later on, we expect to extend these results to Delone-Anderson operators and Delone operators.
We also studied Delone-Anderson operators, which have attracted a lot of interest in the recent years. For these models we were able to settle the question of the existence of the Integrated Density of States (IDS), that had remained open until our work [GMRM15]. We worked in the framework of randomly colored Delone dynamical systems developed by Müller-Richard (Canad. J. Math. 2013). A notion of fundamental importance in this setting is the Delone hull, a dynamical system built by taking all possible spatial translations of one Delone set. We showed that under certain conditions on the Delone set that gives the spatial configuration of impurities, the IDS exists, is independent of the randomness of the medium and, moreover, characterizes the deterministic spectrum of Delone-Anderson operators. The latter result was particularly important in the case of the Landau Hamiltonian perturbed by a Delone-Anderson potential in dimension two, since it showed the almost-sure constancy of the spectrum, a problem that had not been settled through other means. In the case of the negative Laplacian with a Delone-Anderson perturbation, we proved the exponential decay of the IDS at the bottom of the spectrum and showed that, moreover, this yields the initial step needed to perform the Bootstrap MSA (Germinet-Klein, CMP 2001) obtaining, therefore, dynamical localization. Some extra effort was needed to obtain this result uniformly over the Delone hull. We concluded that under certain geometric conditions on the Delone set, which are satisfied by quasicrystals, the ergodic properties of these random operators match those of the well-known Anderson model. On the other hand, dynamical localization holds regardless of the geometric complexity of the underlying Delone set.
During the second part of the project, Dr. Rojas-Molina collaborated with Prof. A. Klein (UC Irvine) and Dr. S. Nguyen (Coastline Community College), authors of the Bootstrap MSA for N-particle models. In [KNRM15] they obtained a characterization of the metal-insulator transport transition for the two-particle Anderson model in the lines of what Germinet-Klein (Duke Math. J. 2004) did for the one-particle case. One part of their proof holds for N-particle systems, while the restriction to two particles comes from the MSA, which presented several technical difficulties. A reformulation of the MSA was needed to proceed with their proof, the main difference with the usual MSA being the use of a pseudo-distance instead of the usual max-norm in the Euclidean space. The involvement of Dr. Rojas-Molina in this work, which resulted in a deeper understanding of the MSA techniques and of multi-particle models, was backed by Prof. Müller and had a positive impact in the development of the other topics treated to the project.
Our main tool in the analysis of Delone operators was the Multiscale Analysis (MSA) of Germinet-Klein (JEMS 2013), which is the most advanced technique known to date that allows to prove localization in the most general cases of Anderson Hamiltonians in the continuous setting. We represented Delone operators using auxiliary Delone-Anderson operators with Bernoulli random variables. This representation is valid only when the underlying configuration of impurities maintains a Delone structure, that is, is relatively dense in space. We achieved this by adding a bounded background Delone potential, then the main challenge was to obtain the initial estimate to apply the MSA in the case where the background potential is not periodic. This was done successfully using quantitative unique continuation principles obtained by Rojas-Molina-Veselic' (CMP 2012), and the later improvement by Klein (CMP 2012). Once the MSA could be performed, the next step was to determine the topological structure of the set of configurations for which the operator exhibits localization. This turned out to be very challenging, since we needed to go to the core of the constructive part of the proof of localization by the MSA which is the most technically involved and refined Multiscale method to date. At first, we were able to state that the localizing configurations, at finite scale, were G-delta sets in the space of configurations. However, in the limit when the finite scale grows arbitrarily, the aforementioned G-delta set was reduced to the trivial Delone set of departure. Our objectives concerning this topic needed, therefore, to be revised. Drawing from the theory of rank-one perturbation, in particular the work of Simon (Annals of Math 1995), we explored the possibility that the set of configurations associated with localization is topologically thin. We revised the constructive approach of the MSA and reformulated our goals. We are now confident that we can show that the set of configurations is, at least, topologically thin in the simpler case of the Anderson model. Due to the originality of our result, we expect this to have a high impact in the field of random Schrödinger operators. Later on, we expect to extend these results to Delone-Anderson operators and Delone operators.
We also studied Delone-Anderson operators, which have attracted a lot of interest in the recent years. For these models we were able to settle the question of the existence of the Integrated Density of States (IDS), that had remained open until our work [GMRM15]. We worked in the framework of randomly colored Delone dynamical systems developed by Müller-Richard (Canad. J. Math. 2013). A notion of fundamental importance in this setting is the Delone hull, a dynamical system built by taking all possible spatial translations of one Delone set. We showed that under certain conditions on the Delone set that gives the spatial configuration of impurities, the IDS exists, is independent of the randomness of the medium and, moreover, characterizes the deterministic spectrum of Delone-Anderson operators. The latter result was particularly important in the case of the Landau Hamiltonian perturbed by a Delone-Anderson potential in dimension two, since it showed the almost-sure constancy of the spectrum, a problem that had not been settled through other means. In the case of the negative Laplacian with a Delone-Anderson perturbation, we proved the exponential decay of the IDS at the bottom of the spectrum and showed that, moreover, this yields the initial step needed to perform the Bootstrap MSA (Germinet-Klein, CMP 2001) obtaining, therefore, dynamical localization. Some extra effort was needed to obtain this result uniformly over the Delone hull. We concluded that under certain geometric conditions on the Delone set, which are satisfied by quasicrystals, the ergodic properties of these random operators match those of the well-known Anderson model. On the other hand, dynamical localization holds regardless of the geometric complexity of the underlying Delone set.
During the second part of the project, Dr. Rojas-Molina collaborated with Prof. A. Klein (UC Irvine) and Dr. S. Nguyen (Coastline Community College), authors of the Bootstrap MSA for N-particle models. In [KNRM15] they obtained a characterization of the metal-insulator transport transition for the two-particle Anderson model in the lines of what Germinet-Klein (Duke Math. J. 2004) did for the one-particle case. One part of their proof holds for N-particle systems, while the restriction to two particles comes from the MSA, which presented several technical difficulties. A reformulation of the MSA was needed to proceed with their proof, the main difference with the usual MSA being the use of a pseudo-distance instead of the usual max-norm in the Euclidean space. The involvement of Dr. Rojas-Molina in this work, which resulted in a deeper understanding of the MSA techniques and of multi-particle models, was backed by Prof. Müller and had a positive impact in the development of the other topics treated to the project.