# Homepage of Roger Nichols

## Contact Information

Department of Mathematics

The University of Tennessee at Chattanooga

Dept. 6956

615 McCallie Avenue

Chattanooga, Tennessee 37403, USA

E-mail: roger-nichols "at" utc "dot" edu

Office: 329 Lupton Hall

Phone: +1(423) 425-4562

“The best thing for being sad,” replied Merlin, beginning to puff and blow, “is to learn something. That’s the only thing that never fails. You may grow old and trembling in your anatomies, you may lie awake at night listening to the disorder of your veins, you may miss your only love, you may see the world about you devastated by evil lunatics, or know your honour trampled in the sewers of baser minds. There is only one thing for it then—to learn. Learn why the world wags and what wags it. That is the only thing which the mind can never exhaust, never alienate, never be tortured by, never fear or distrust, and never dream of regretting. Learning is the only thing for you. Look what a lot of things there are to learn.” (T. H. White, The Once and Future King)

## About Me

I am a professor in the Department of Mathematics at University of Tennessee at Chattanooga.

Erdős number: 4

Kreĭn number: 3

Two Rogers (ca. May 2006) and a relic (ca. October 1990)

“I have come from Alabama: a fur piece. All the way from Alabama a-walking. A fur piece.” (“Lena” in Light in August by William Faulkner)

## Research Interests

Spectral theory of differential operators and functional analysis

## Book

“Sturm–Liouville Operators, Their Spectral Theory, and Some Applications,” with F. Gesztesy and M. Zinchenko, Colloquium Publications Vol. 67, Amer. Math. Soc., Providence, RI, 2024, 927 pp.

## Research Monograph

“The limiting absorption principle for massless Dirac operators, properties of spectral shift functions, and an application to the Witten index of non-Fredholm operators,” with A. Carey, F. Gesztesy, G. Levitina, F. Sukochev, and D. Zanin, Memoirs of the European Mathematical Society 4, 2023. [PDF]

## Articles in Refereed Journals and Proceedings

44. “Weak convergence of spectral shift functions revisited,” with C. Connard, B. Ingimarson, and A. Paul, Pure Appl. Funct. Anal. 9, No. 4, 1023–1051 (2024). [PDF]

43. “Sturm–Liouville M-functions in terms of Green's functions,” with F. Gesztesy, J. Differential Equations 412, 709–757 (2024). [PDF]

42. “On the spectrum of biharmonic systems,” with L. Kong and M. Wang, J. Math. Sci., doi:10.1007/s10958-024-07233-7. [PDF]

41. “A Bessel analog of the Riesz composition formula,” with C. Fischbacher and F. Gesztesy, Comput. Methods Funct. Theory 24, 547–573, (2024). [PDF]

40. “Donoghue m-functions for singular Sturm–Liouville operators,” with F. Gesztesy, L. Littlejohn, M. Piorkowski, and J. Stanfill, St. Petersburg Math. J. 35 101–138 (2024). [PDF]

39. “Weyl–Titchmarsh M-functions for φ-periodic Sturm–Liouville operators in terms of Green's functions,” with F. Gesztesy, appeared in From Complex Analysis to Operator Theory—A Panorama, M. Brown, F. Gesztesy, P. Kurasov, A. Laptev, B. Simon, G. Stolz, and I. Wood (eds.), Oper. Theory Adv. Appl. 291, Birkhäuser/Springer, Cham, 2023, pp. 573–608. [PDF]

38. “On perturbative Hardy inequalities,” with F. Gesztesy and M. M. H. Pang, J. Math. Phys., Analysis, Geometry 19, 128–149 (2023). [PDF]

37. “Singular fourth-order Sturm–Liouville operators and acoustic black holes,” with B. P. Belinskiy and D. B. Hinton, IMA J. Appl. Math. 87, No. 5, 804–851 (2022). [PDF]

36. “Strict domain monotonicity of the principal eigenvalue and a characterization of lower boundedness for the Friedrichs extension of four-coefficient Sturm–Liouville operators,” with F. Gesztesy, Acta Sci. Math. 88, 189–222 (2022). [PDF]

35. “Multiple weak solutions of biharmonic systems,” with L. Kong, Minimax Theory its Appl. 7, No. 1, 109–118 (2022). [PDF]

34. “The Krein–von Neumann extension revisited,” with G. Fucci, F. Gesztesy, K. Kirsten, L. Littlejohn, and J. Stanfill, Appl. Anal. 101, No. 5, 1593–1616 (2022). [PDF]

33. “The Krein–von Neumann extension of a regular even order quasi-differential operator,” with M. Cho, S. Hoisington, and B. Udall, Opuscula Math. 41, No. 6, 805–841 (2021). [PDF]

32. “Singular Sturm–Liouville operators with extreme properties that generate black holes,” with B. Belinskiy and D. Hinton, Stud. Appl. Math. 147, No. 1, 180–208 (2021). [PDF]

31. “The product formula for regularized Fredholm determinants,” with T. Britz, A. Carey, F. Gesztesy, F. Sukochev, and D. Zanin, Proc. Amer. Math. Soc. Ser. B 8, 42–51 (2021). [PDF]

30. “A survey of some norm inequalities,” with F. Gesztesy and J. Stanfill, Complex Anal. Oper. Theory 15, 23 (2021). [PDF]

29. “On principal eigenvalues of biharmonic systems,” with L. Kong, Commun. Pure Appl. Anal. 20, No. 1, 1–15 (2021) [PDF]

28. “Explicit Krein resolvent identities for singular Sturm–Liouville operators with applications to Bessel operators,” with S. B. Allan, J. H. Kim, G. Michajlyszyn, and D. Rung, Oper. Matrices 14, No. 4, 1043–1099 (2020). [PDF]

27. “On self-adjoint boundary conditions for singular Sturm–Liouville operators bounded from below,” with F. Gesztesy and L. Littlejohn, J. Differential Equations 269, 6448–6491 (2020). [PDF]

26. “Trace ideal properties of a class of integral operators,” with F. Gesztesy, appeared in Integrable Systems and Algebraic Geometry. Volume 1, R. Donagi and T. Shaska (eds.), London Math. Soc. Lecture Note Ser. 458, Cambridge University Press, Cambridge, UK, 2020, pp. 13–37. [PDF]

25. “On absence of threshold resonances for Schrödinger and Dirac operators,” with F. Gesztesy, Discrete Contin. Dyn. Syst. Ser. S 13(12), 3427–3460 (2020). [PDF]

24. “On the global limiting absorption principle for massless Dirac operators,” with A. Carey, F. Gesztesy, J. Kaad, G. Levitina, D. Potapov, and F. Sukochev, Ann. Henri Poincaré 19, No. 7, 1993–2019 (2018). [PDF]

23. “Weak and vague convergence of spectral shift functions of one-dimensional Schrödinger operators with coupled boundary conditions,” with J. Murphy, Methods Funct. Anal. Topology 23, No. 4, 378–403 (2017). [PDF]

22. “On the index of meromorphic operator-valued functions and some applications,” with J. Behrndt, F. Gesztesy, and H. Holden, appeared in Functional Analysis and Operator Theory for Quantum Physics, J. Dittrich, H. Kovarik, and A. Laptev (eds.), Series of Congress Reports, European Mathematical Society, Zürich, 2017. [PDF]

21. “Double operator integral methods applied to continuity of spectral shift functions,” with A. Carey, F. Gesztesy, G. Levitina, D. Potopov, and F. Sukochev, J. Spectr. Theory 6, No. 4, 747–779 (2016). [PDF]

20. “Principal solutions revisited,” with S. Clark and F. Gesztesy, appeared in Stochastic and Infinite Dimensional Analysis, C. C. Bernido, M. V. Carpio-Bernido, M. Grothaus, T. Kuna, M. J. Oliveira, and J. L. da Silva (eds.), Trends in Mathematics, Birkhäuser, Basel, 2016. [PDF]

19. “Dirichlet-to-Neumann maps, abstract Weyl–Titchmarsh M-functions, and a generalized index of unbounded meromorphic operator-valued functions,” with J. Behrndt, F. Gesztesy, and H. Holden, J. Differential Equations 261, 3551–3587 (2016). [PDF]

18. “On stability of square root domains for non-self-adjoint operators under additive perturbations,” with F. Gesztesy and S. Hofmann, Mathematika 62, 111–182 (2016). [PDF]

17. “Some applications of almost analytic extensions to operator bounds in trace ideals,” with F. Gesztesy, Methods Funct. Anal. Topology 21, No. 2, 151–169 (2015). [PDF]

16. “A Jost–Pais-type reduction of (modified) Fredholm determinants for semi-separable operators in infinite dimensions,” with F. Gesztesy, appeared in Recent Advances in Schur Analysis and Stochastic Processes - A Collection of Papers Dedicated to Lev Sakhnovich, D. Alpay and B. Kirstein (eds.), Operator Theory: Advances and Applications 244, 287–314 (2015). [PDF]

15. “On factorizations of analytic operator-valued functions and eigenvalue multiplicity questions,” with F. Gesztesy and H. Holden, Integral Eq. and Operator Th. 82, No. 1, 61–94 (2015). [PDF]

14. “On a problem in eigenvalue perturbation theory,” with F. Gesztesy and S. Naboko, J. Math. Anal. Appl. 428, No. 1, 295–305 (2015). [PDF]

13. “Inverse spectral problems for Schrödinger-type operators with distributional matrix-valued potentials,” with J. Eckhardt, F. Gesztesy, A. Sakhnovich, and G. Teschl, Differential Integral Equations 28, No. 5–6, 505–522 (2015). [PDF]

12. “Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions II,” with F. Gesztesy, M. Mitrea, and E. M. Ouhabaz, Proc. Amer. Math. Soc. 143, No. 4, 1635–1649 (2015). [PDF]

11. “Stability of square root domains associated with elliptic systems of PDEs on nonsmooth domains,” with F. Gesztesy and S. Hofmann, J. Differential Equation 258, 1749–1764 (2015). [PDF]

10. “Supersymmetry and Schrödinger-type operators with distributional matrix-valued potentials,” with J. Eckhardt, F. Gesztesy, and G. Teschl, J. Spectr. Theory 4, No. 4, 715–768 (2014). [PDF]

9. “Boundary data maps and Krein's resolvent formula for Sturm–Liouville operators on a finite interval,” with S. Clark, F. Gesztesy, and M. Zinchenko, Oper. Matrices 8, No. 1, 1–71 (2014). [PDF]

8. “Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions,” with F. Gesztesy and M. Mitrea, J. Anal. Math. 122, 229–287 (2014). [PDF]

7. “On square root domains for non-self-adjoint Sturm–Liouville operators,” with F. Gesztesy and S. Hofmann, Methods Funct. Anal. Topology 19, No. 3, 227–259 (2013). [PDF]

6. “Inverse spectral theory for Sturm–Liouville operators with distributional potentials,” with J. Eckhardt, F. Gesztesy, and G. Teschl, J. London Math. Soc. (2) 88, 801–828 (2013). [PDF]

5. “Weyl–Titchmarsh theory for Sturm–Liouville operators with distributional potentials,” with J. Eckhardt, F. Gesztesy, and G. Teschl, Opuscula Math. 33, No. 3, 467–563 (2013). [PDF]

4. “Simplicity of eigenvalues in Anderson-type models,” with G. Stolz and S. Naboko, Ark. Mat. 51, 157–183 (2013). [PDF]

3. “An abstract approach to weak convergence of spectral shift functions and applications to multi-dimensional Schrödinger operators,” with F. Gesztesy, J. Spectr. Theory 2, No. 3, 225–266 (2012). [PDF]

2. “Weak convergence of spectral shift functions for one-dimensional Schrödinger operators,” with F. Gesztesy, Math. Nachr. 285, No. 14–15, 1799–1838 (2012). [PDF]

1. “Spectral properties of discrete random displacement models,” with G. Stolz, J. Spectr. Theory 1, No. 2, 123–153 (2011). [PDF]

## Preprints

2. “A generalized Birman–Schwinger principle and applications to one-dimensional Schrödinger operators with distributional coefficients,” with F. Gesztesy.

1. “On eigenvalue multiplicities of regular Sturm–Liouville operators,” with F. Gesztesy and M. Zinchenko.

“This went on at any odd hour, if necessary, with a floor rug over his shoulders, with the fine quiet of the scholar which is nearest of all things to heavenly peace.” (F. Scott Fitzgerald, Tender is the Night)