Random Schrödinger operators with breather potentials as a paradigmatic model for non-linear influence of randomness

• Applicant: Professor Dr. Ivan Veselic
• Subject Area: Mathematics
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 394221243

• Final Report Year: 2024

Final Report Abstract

The project was mainly devoted to the study of Schrödinger operators with random potentials of breather type. For a class of such models we established Lifschitz asymptotics of the integrated density of states. This was used to deduce spectral and dynamical localization for low lying spectral values. Furthermore, divergence class operators with random coefficients of breather type were studied. For a class of such models a Wegner estimate was proved. The analysis of level sets of stochastic processes played a crucial role in the proofs.

Publications

• Control problem for quadratic differential operators with sensor sets of decaying density via partial harmonic oscillators. Preprint
  Alexander Dicke, Albrecht Seelmann & Ivan Veselí
  
• Lifshitz asymptotics and localization for random breather models. Preprint
  Christoph Schumacher & Ivan Veselí
  
• Wegner Estimate for Random Divergence-Type Operators Monotone in the Randomness. Mathematical Physics, Analysis and Geometry, 24(3).
  Dicke, Alexander
  
• Uncertainty principles with error term in Gelfand–Shilov spaces. Archiv der Mathematik, 119(4), 413-425.
  Dicke, Alexander & Seelmann, Albrecht
  
• Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials. Journal of Differential Equations, 369, 405-423.
  Dicke, Alexander; Rose, Christian; Seelmann, Albrecht & Tautenhahn, Martin
  
• Uncertainty Principle for Hermite Functions and Null-Controllability with Sensor Sets of Decaying Density. Journal of Fourier Analysis and Applications, 29(1).
  Dicke, Alexander; Seelmann, Albrecht & Veselić, Ivan
  
• Unique continuation for the gradient of eigenfunctions and Wegner estimates for random divergence-type operators. Journal of Functional Analysis, 285(7), 110040.
  Dicke, Alexander & Veselić, Ivan
  
• Spectral inequality with sensor sets of decaying density for Schrödinger operators with power growth potentials. Partial Differential Equations and Applications, 5(2).
  Dicke, Alexander; Seelmann, Albrecht & Veselić, Ivan
  
• Spherical Logvinenko–Sereda–Kovrijkine type inequality and null-controllability of the heat equation on the sphere. Archiv der Mathematik, 123(5), 543-556.
  Dicke, Alexander & Veselić, Ivan