Professor

Department of Mathematics

Rice University

Office: HBH 418

E-mail: milivoje.lukic@rice.edu

- 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

Analysis and mathematical physics. In particular:

- direct and inverse spectral theory of SchrÃ¶dinger 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.

- Necessary and sufficient conditions for universality limits
(with B. Eichinger, H. Woracek)

[arXiv:2409.18045] - On point spectrum of Jacobi matrices generated by iterations of quadratic polynomials
(with B. Eichinger, P. Yuditskii)

[arXiv:2405.19470] - Modified Jost solutions of Schrödinger operators with locally \(H^{-1}\) potentials
(with X. Wang)

[arXiv:2405.17715] - Spectral properties of Schrödinger operators with locally \(H^{-1}\) potentials
(with S. Sukhtaiev, X. Wang)

J. Spectr. Theory 14 (2024), 59-120 [arXiv:2206.07079] - Limit-Periodic Dirac Operators with Thin Spectra
(with B. Eichinger, J. Fillman, E. Gwaltney)

J. Funct. Anal. 283 (2022), 109711 [arXiv] - The Deift Conjecture: A Program to Construct a Counterexample
(with D. Damanik, A. Volberg, P. Yuditskii)

[arXiv:2111.09345] - An approach to universality using Weyl m-functions
(with B. Eichinger, B. Simanek)

[arXiv:2108.01629] - Asymptotics of Chebyshev rational functions with respect to subsets of the real line
(with B. Eichinger, G. Young)

Constr. Approx. 59, 541-581 (2024) [arXiv] - Stahl-Totik regularity for Dirac operators
(with B. Eichinger, E. Gwaltney)

[arXiv:2012.12889] - Reflectionless canonical systems, II. Almost periodicity and character-automorphic Fourier transforms
(with R. Bessonov, P. Yuditskii)

Adv. Math. 444 (2024), Paper No. 109636. [arXiv] - Reflectionless canonical systems, I. Arov gauge and right limits
(with R. Bessonov, P. Yuditskii)

Integr. Equ. Oper. Theory 94 (2022), 4 [arXiv] - 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] - Stahl-Totik regularity for continuum Schrödinger operators
(with B. Eichinger)

Anal. PDE , accepted [arXiv:2001.00875] - Uniqueness of solutions of the KdV-hierarchy via Dubrovin-type flows
(with G. Young)

J. Funct. Anal. 279 (2020), 108705 [arXiv] -
Ergodic Schrödinger Operators in the Infinite Measure Setting
(with M. Boshernitzan, D. Damanik, J. Fillman)

J. Spectr. Theory 11 (2021), 873-902 [arXiv] - Spectral edge behavior for eventually monotone Jacobi and Verblunsky coefficients

J. Spectr. Theory 9 (2019), 1115-1155 [arXiv] -
\(\ell^2\) bounded variation and absolutely continuous spectrum of Jacobi matrices
(with Y. Last)

Comm. Math. Phys. 359 (2018), 101-119 [arXiv] -
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] -
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] -
Limit-Periodic Continuum Schrödinger Operators with Zero Measure Cantor Spectrum
(with D. Damanik, J. Fillman)

J. Spectr. Theory 7 (2017), 1101-1118 [arXiv] -
Spectral Homogeneity of Limit-Periodic Schrödinger Operators
(with J. Fillman)

J. Spectr. Theory 7 (2017), 387-406 [arXiv] -
Generalized Prüfer variables for perturbations of Jacobi and CMV matrices
(with D. C. Ong)

J. Math. Anal. Appl. 444 (2016), 1490-1514 [arXiv] -
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] -
The Isospectral Torus of Quasi-Periodic Schrödinger Operators via Periodic Approximations
(with D. Damanik, M. Goldstein)

Invent. Math. 207 (2017), 895-980 [arXiv] -
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] -
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] -
New Anomalous Lieb-Robinson Bounds in Quasi-Periodic XY Chains
(with D. Damanik, M. Lemm, W. Yessen)

Phys. Rev. Lett. 113 (2014), 127202 [arXiv] -
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] -
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] -
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] - On higher-order Szegő theorems with a single critical point of arbitrary order

Constr. Approx. 44 (2016), 283-296 [arXiv] - Wigner-von Neumann type perturbations of periodic Schrödinger operators
(with D. C. Ong)

Trans. Amer. Math. Soc. 367 (2015), 707-724 [arXiv] - Square-summable variation and absolutely continuous spectrum

J. Spectr. Theory 4 (2014), 815-840 [arXiv] - On a conjecture for higher-order Szegő theorems

Constr. Approx. 38 (2013), 161-169 [arXiv] - A class of Schrödinger operators with decaying oscillatory potentials

Comm. Math. Phys. 326 (2014), 441-458 [arXiv] - Schrödinger operators with slowly decaying Wigner-von Neumann type potentials

J. Spectr. Theory 3 (2013), 147-169 [arXiv] - Derivatives of \(L^p\) eigenfunctions of Schrödinger operators

Math. Model. Nat. Phenom. 8 (2013), 170-174 [arXiv] - Orthogonal polynomials with recursion coefficients of generalized bounded variation

Comm. Math. Phys. 306 (2011), 485-509 [arXiv]

- A first course in spectral theory

Graduate Studies in Mathematics, 226. American Mathematical Society, Providence, RI, 2022. xvi+472 pp. - Jacobi and CMV matrices with coefficients of generalized bounded variation

Operator Theory: Advances and Applications 227 (2013), 117-121 - Spectral theory for generalized bounded variation perturbations of orthogonal polynomials and Schrödinger operators

Ph.D. Dissertation, California Institute of Technology (2011) -
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 -
Inequalities (with Z. Kadelburg, D. Djukić, I. Matić)

(in Serbian, math olympiad training textbook, 212 pages) Mathematical Society of Serbia (2003)

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

- 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)

- 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)

- 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)