Research Interests

 
My research focuses primarily on the intersection of analysis, mathematical physics, and operator theory. Key areas include:
  • Partial Differential Equations (PDEs)
  • Mathematical Physics
  • Spectral Theory: Specifically the behavior of Schrödinger operators and Jacobi matrices.
  • Schrödinger Operators: Investigating absolutely continuous spectra.
  • Ergodic Schrödinger Operators: Studying Lyapunov exponents and shift embeddings.

 

Selected Publications

    • Commun. PDE:  Safronov, O. (2001). The Discrete Spectrum of Selfadjoint Operators Under Perturbations of Variable Sign. Commun. in Part. Diff. Eq., 26, (3-4), 629–649 
    • Holt, J.,  Molchanov, S. & Safronov, O. (2008). Cwikel and Quasi-Szegő type estimates for random operators. Commun. Part. Diff. Eq., 33(4), 569–587.
    • Analysis & PDE: Safronov, O. (2023). Eigenvalue bounds for Schrödinger operators with random complex potentials. Analysis & PDE, 16(4), 1033–1060. 
    • Journal of Functional Analysis: Safronov, O.(2001). Spectral shift function in the large coupling constant limit. Journal of Functional Analysis, 182(1), 151–169.
    • Safronov, O. (2006). Multi-dimensional Schrödinger operators with some negative spectrum. Journal of Functional Analysis, 238(1), 327–339.
    • Safronov, O. (2008). Absolutely continuous spectrum of one random elliptic operator. Journal of Functional Analysis, 255(3), 755–767
    • Trans. AMS: Korotyaev, E., & Safronov, O. (2020). Eigenvalue bounds for Stark operators with complex potentials. Trans. Amer. Math. Soc., 373(4), 2697–2717.
    • IMRN: Safronov, O. (2004). The amount of discrete spectrum of a perturbed periodic Schrödinger operator inside a fixed interval (λ₁, λ₂). International Mathematics Research Notices, 2004(9), 411–422.
    • Frank, R. L., & Safronov, O. (2005). Absolutely continuous spectrum of a class of random nonergodic Schrödinger operators. International Mathematics Research Notices, 2005(42), 2559–2577.
    • Safronov, O. (2012). Eigenvalue estimates for the perturbed Anderson model. International Mathematics Research Notices, 2012(18), 4085–4100.
    • Commun. Math. Phys.: Safronov, O. (2005). On the absolutely continuous spectrum of multi-dimensional Schrödinger operators with slowly decaying potentials. Commun. Math. Phys., 254(2), 361–366. 
    • Safronov, O. (2001). The Discrete Spectrum of the Perturbed Periodic Schrödinger Operator. Communications in Mathematical Physics, 218(1), 217–232.
    • Safronov, O.(1998). The Discrete Spectrum in the Gaps of the Continuous One for Non-Sign-Definite Perturbations with a Large Coupling Constant. Communications in Mathematical Physics, 193(1), 233–243
    • Ann. Henri Poincare: Safronov, O. (2017).  Absolutely Continuous Spectrum of a Dirac Operator in the Case of a Positive Mass. Ann. Henri Poincaré18, 1385–1434. 
    • Holt, J., & Safronov, O. (2024). On the number of eigenvalues of the Dirac operator in a bounded interval. Ann. Henri Poincaré, 25(8), 4105–4124.