Quantitative unique continuation properties of elliptic PDEs with variable 2nd order coefficients and applications in control theory, Anderson localization, and photonics

• Applicant: Professor Dr. Ivan Veselic
• Subject Area: Mathematics
• Term: from 2020 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 441959487

• Final Report Year: 2025

Final Report Abstract

The project was devoted to the study unique continuation estimates and uncertainty principles for functions in the range of spectral projectors of elliptic differential operators on unbounded and large bounded domains. We originally intended to study mostly elliptic differential equations with variable second order coefficients that are used to model condensed matter physics. For various reasons in the course of the project the focus turned to second order elliptic differential equations motivated by kinetic theory of gases. For a number of such models we established uncertainty principles in the form of spectral inequalities, as well as observability and null-controllability estimates. During the reporting period the PI and team members associated with the project published 15 papers concerned with uncertainty relations, unique continuation, observability and related topics.

Publications

• An abstract Logvinenko-Sereda type theorem for spectral subspaces. Journal of Mathematical Analysis and Applications, 500(1), 125149.
  Egidi, Michela & Seelmann, Albrecht
  
• Protecting points from operator pencils. Journal of Operator Theory, 85(2), 383-389.
  Seelmann, Albrecht; Taufer, Matthias & Veseli, Kresimir
  
• Quantum Hamiltonians with Weak Random Abstract Perturbation. II. Localization in the Expanded Spectrum. Journal of Statistical Physics, 182(1).
  Borisov, Denis; Täufer, Matthias & Veselić, Ivan
  
• The Laplacian on Cartesian products with mixed boundary conditions. Archiv der Mathematik, 117(1), 87-94.
  Seelmann, Albrecht
  
• Unifying the treatment of indefinite and semidefinite perturbations in the subspace perturbation problem. Operators and Matrices(3), 1181-1188.
  Seelmann, Albrecht
  
• Wegner Estimate for Random Divergence-Type Operators Monotone in the Randomness. Mathematical Physics, Analysis and Geometry, 24(3).
  Dicke, Alexander
  
• On a Minimax Principle in Spectral Gaps. Complex Analysis and Operator Theory, 16(3).
  Seelmann, Albrecht
  
• Spectral inequalities for Schrödinger operators and parabolic observability, 2022. PhD thesis, Technische Universität Dortmund.
  Alexander Dicke
  
• The Reflection Principle in the Control Problem of the Heat Equation. Journal of Dynamical and Control Systems, 28(3), 635-655.
  Egidi, Michela & Seelmann, Albrecht
  
• Uncertainty principles with error term in Gelfand–Shilov spaces. Archiv der Mathematik, 119(4), 413-425.
  Dicke, Alexander & Seelmann, Albrecht
  
• A unified observability result for non-autonomous observation problems. Archiv der Mathematik, 122(2), 227-239.
  Gabel, Fabian & Seelmann, Albrecht
  
• Control problem for quadratic parabolic differential equations with sparse sensor sets of finite volume or anisotropically decaying density. ESAIM: Control, Optimisation and Calculus of Variations, 29, 80.
  Dicke, Alexander; Seelmann, Albrecht & Veselić, Ivan
  
• 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
  
• Relative residual bounds for eigenvalues in gaps of the essential spectrum. Operators and Matrices, (1), 191–203.
  Seelmann, Albrecht
  
• 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
  
• Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability. Comptes Rendus. Mathématique, 362(G12), 1635-1659.
  Alphonse, Paul & Seelmann, Albrecht
  
• 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
  
• Sturm-Liouville problems and global bounds by small control sets and applications to quantum graphs. Journal of Mathematical Analysis and Applications, 535(1), 128101.
  Egidi, Michela; Mugnolo, Delio & Seelmann, Albrecht
  
• Wegner estimate and localisation for alloy-type operators with minimal support assumptions on the single site potential. Random Operators and Stochastic Equations, 32(2), 175-184.
  Täufer, Matthias & Veselić, Ivan
  
• Unique continuation estimates for Baouendi–Grushin equations on cylinders. Pure and Applied Analysis, 7(3), 733-770.
  Alphonse, Paul & Seelmann, Albrecht