Milivoje Lukic A First Course in Spectral Theory

Milivoje Lukić


Department of Mathematics
Rice University

Office:  HBH 418



  • 2023-         Professor, Rice University
  • 2018-2023 Associate Professor, Rice University
  • 2016-2018 Assistant Professor, Rice University
  • 2014-2016 Postdoctoral Fellow, University of Toronto
  • 2011-2014 Evans Instructor, Rice University


  • Ph.D., Mathematics, California Institute of Technology, Pasadena, CA, 2011
  • M.Sc., Physics, California Institute of Technology, Pasadena, CA, 2010
  • B.Sc., Astrophysics, University of Belgrade, Belgrade, Serbia, 2007
  • B.Sc., Mathematics, University of Belgrade, Belgrade, Serbia, 2006

Research interests

Analysis and mathematical physics. In particular:

  • direct and inverse spectral theory of Schrödinger operators and canonical systems
  • connections between spectral theory and potential theory and harmonic analysis
  • KdV, NLS, and other nonlinear integrable partial differential equations
  • universality phenomena in orthogonal polynomials and random matrices

I am an editor for the Journal of Spectral Theory and for the New York Journal of Mathematics.

I co-organize the Analysis Seminar at Rice University.

Research articles

  1. Spectral properties of Schrödinger operators with locally \(H^{-1}\) potentials (with S. Sukhtaiev, X. Wang)
    J. Spectr. Theory , accepted [arXiv:2206.07079]
  2. Limit-Periodic Dirac Operators with Thin Spectra (with B. Eichinger, J. Fillman, E. Gwaltney)
    J. Funct. Anal. 283 (2022), 109711 [arXiv]
  3. The Deift Conjecture: A Program to Construct a Counterexample (with D. Damanik, A. Volberg, P. Yuditskii)
  4. An approach to universality using Weyl m-functions (with B. Eichinger, B. Simanek)
  5. Asymptotics of Chebyshev rational functions with respect to subsets of the real line (with B. Eichinger, G. Young)
    Constr. Approx., accepted [arXiv:2101.01744]
  6. Stahl-Totik regularity for Dirac operators (with B. Eichinger, E. Gwaltney)
  7. Reflectionless canonical systems, II. Almost periodicity and character-automorphic Fourier transforms (with R. Bessonov, P. Yuditskii)
    Adv. Math. 444 (2024), Paper No. 109636. [arXiv:2011.05266]
  8. Reflectionless canonical systems, I. Arov gauge and right limits (with R. Bessonov, P. Yuditskii)
    Integr. Equ. Oper. Theory 94 (2022), 4 [arXiv]
  9. Orthogonal rational functions with real poles, root asymptotics, and GMP matrices (with B. Eichinger, G. Young)
    Trans. Amer. Math. Soc. Ser. B 10 (2023), 1-47 [arXiv]
  10. Stahl-Totik regularity for continuum Schrödinger operators (with B. Eichinger)
    Anal. PDE , accepted [arXiv:2001.00875]
  11. Uniqueness of solutions of the KdV-hierarchy via Dubrovin-type flows (with G. Young)
    J. Funct. Anal. 279 (2020), 108705 [arXiv]
  12. Ergodic Schrödinger Operators in the Infinite Measure Setting (with M. Boshernitzan, D. Damanik, J. Fillman)
    J. Spectr. Theory 11 (2021), 873-902 [arXiv]
  13. Spectral edge behavior for eventually monotone Jacobi and Verblunsky coefficients
    J. Spectr. Theory 9 (2019), 1115-1155 [arXiv]
  14. \(\ell^2\) bounded variation and absolutely continuous spectrum of Jacobi matrices (with Y. Last)
    Comm. Math. Phys. 359 (2018), 101-119 [arXiv]
  15. Almost Periodicity in Time of Solutions of the Toda Lattice (with I. Binder, D. Damanik, T. VandenBoom)
    C. R. Math. Rep. Acad. Sci. Canada 40 (2018), 1-28 [arXiv]
  16. Almost Periodicity in Time of Solutions of the KdV Equation (with I. Binder, D. Damanik, M. Goldstein)
    Duke Math. J. 167 (2018), 2633-2678 [arXiv]
  17. Limit-Periodic Continuum Schrödinger Operators with Zero Measure Cantor Spectrum (with D. Damanik, J. Fillman)
    J. Spectr. Theory 7 (2017), 1101-1118 [arXiv]
  18. Spectral Homogeneity of Limit-Periodic Schrödinger Operators (with J. Fillman)
    J. Spectr. Theory 7 (2017), 387-406 [arXiv]
  19. Generalized Prüfer variables for perturbations of Jacobi and CMV matrices (with D. C. Ong)
    J. Math. Anal. Appl. 444 (2016), 1490-1514 [arXiv]
  20. Characterizations of Uniform Hyperbolicity and Spectra of CMV Matrices (with D. Damanik, J. Fillman, W. Yessen)
    Discrete Contin. Dyn. Syst. Ser. S 9 (2016), 1009-1023 [arXiv]
  21. The Isospectral Torus of Quasi-Periodic Schrödinger Operators via Periodic Approximations (with D. Damanik, M. Goldstein)
    Invent. Math. 207 (2017), 895-980 [arXiv]
  22. A Multi-Scale Analysis Scheme on Abelian Groups with an Application to Operators Dual to Hill's Equation (with D. Damanik, M. Goldstein)
    Trans. Amer. Math. Soc. 369 (2017), 1689-1755 [arXiv]
  23. The Spectrum of a Schrödinger Operator With Small Quasi-Periodic Potential is Homogeneous (with D. Damanik, M. Goldstein)
    J. Spectr. Theory 6 (2016), 415-427 [arXiv]
  24. New Anomalous Lieb-Robinson Bounds in Quasi-Periodic XY Chains (with D. Damanik, M. Lemm, W. Yessen)
    Phys. Rev. Lett. 113 (2014), 127202 [arXiv]
  25. On Anomalous Lieb-Robinson Bounds for the Fibonacci XY Chain (with D. Damanik, M. Lemm, W. Yessen)
    J. Spectr. Theory 6 (2016), 601-628 [arXiv]
  26. Quantum Dynamics of Periodic and Limit-Periodic Jacobi and Block Jacobi Matrices with Applications to Some Quantum Many Body Problems (with D. Damanik, W. Yessen)
    Comm. Math. Phys. 337 (2015), 1535-1561 [arXiv]
  27. Uniform Hyperbolicity for Szegő Cocycles and Applications to Random CMV Matrices and the Ising Model (with D. Damanik, J. Fillman, W. Yessen)
    Int. Math. Res. Not. 2015 (2015), 7110-7129 [arXiv]
  28. On higher-order Szegő theorems with a single critical point of arbitrary order
    Constr. Approx. 44 (2016), 283-296 [arXiv]
  29. Wigner-von Neumann type perturbations of periodic Schrödinger operators (with D. C. Ong)
    Trans. Amer. Math. Soc. 367 (2015), 707-724 [arXiv]
  30. Square-summable variation and absolutely continuous spectrum
    J. Spectr. Theory 4 (2014), 815-840 [arXiv]
  31. On a conjecture for higher-order Szegő theorems
    Constr. Approx. 38 (2013), 161-169 [arXiv]
  32. A class of Schrödinger operators with decaying oscillatory potentials
    Comm. Math. Phys. 326 (2014), 441-458 [arXiv]
  33. Schrödinger operators with slowly decaying Wigner-von Neumann type potentials
    J. Spectr. Theory 3 (2013), 147-169 [arXiv]
  34. Derivatives of \(L^p\) eigenfunctions of Schrödinger operators
    Math. Model. Nat. Phenom. 8 (2013), 170-174 [arXiv]
  35. Orthogonal polynomials with recursion coefficients of generalized bounded variation
    Comm. Math. Phys. 306 (2011), 485-509 [arXiv]

Other publications

  1. A first course in spectral theory
    Graduate Studies in Mathematics, 226. American Mathematical Society, Providence, RI, 2022. xvi+472 pp.
  2. Jacobi and CMV matrices with coefficients of generalized bounded variation
    Operator Theory: Advances and Applications 227 (2013), 117-121
  3. Spectral theory for generalized bounded variation perturbations of orthogonal polynomials and Schrödinger operators
    Ph.D. Dissertation, California Institute of Technology (2011)
  4. Inequalities of Karamata, Schur and Muirhead, and some applications (with Z. Kadelburg, D. Djukić, I. Matić)
    The Teaching of Mathematics 8 (2005), pp. 31-45
  5. Inequalities (with Z. Kadelburg, D. Djukić, I. Matić)
    (in Serbian, math olympiad training textbook, 212 pages) Mathematical Society of Serbia (2003)

Upcoming teaching (Fall 2024)

  • MATH 412: Probability Theory
  • MATH 522: Topics in Analysis

Past teaching at Rice University (2016-)

  • MATH 427/517: Complex Analysis (Spring 2024)
  • MATH 102, Section 4: Single Variable Calculus II (Spring 2023)
  • MATH 425/515: Integration Theory (Fall 2021)
  • MATH 321: Introduction to Analysis (Fall 2021)
  • MATH 412: Probability Theory (Spring 2021)
  • MATH 427/517: Complex Analysis (Spring 2021)
  • MATH 425/515: Integration Theory (Fall 2019)
  • MATH 321: Introduction to Analysis (Fall 2019)
  • MATH 522: Topics in Analysis: Schrödinger operators and the KdV equation (Spring 2019)
  • MATH 222: Honors Calculus IV (Spring 2019)
  • MATH 425/515: Integration Theory (Fall 2017)
  • MATH 102, Section 3: Single Variable Calculus II (Fall 2017)
  • MATH 300: Topics in Undergraduate Math (Fall 2016)
  • MATH 212, Section 1: Multivariable Calculus (Fall 2016)

Past teaching at University of Toronto (2014-2016)

  • MAT236H5: Vector Calculus (Winter 2016)
  • MAT223H5: Linear Algebra I (Fall 2015)
  • MAT212H5: Modeling with Differential Equations in Life Sciences and Medicine (Fall 2014)
  • MAT223H5: Linear Algebra I (Fall 2014)

Past teaching at Rice University (2011-2014)

  • MATH 322: Introduction to Analysis II (Spring 2014)
  • MATH 211, Section 1: Ordinary Differential Equations and Linear Algebra (Spring 2014)
  • MATH 428/518: Topics in Complex Analysis (Fall 2013)
  • MATH 102, Section 3: Single Variable Calculus II (Spring 2013)
  • MATH 370: Calculus on Manifolds (Spring 2013)
  • MATH 211, Section 2: Ordinary Differential Equations and Linear Algebra (Fall 2012)
  • MATH 212, Section 4: Multivariable Calculus (Spring 2012)
  • MATH 381: Introduction to Partial Differential Equations (Fall 2011)
  • MATH 211, Section 5: Ordinary Differential Equations and Linear Algebra (Fall 2011)