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Maximal theta functions universal optimality of the hexagonal lattice for Madelung-like lattice energies

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Abstract

We present two families of lattice theta functions accompanying the family of lattice theta functions studied by Montgomery in [H. Montgomery, Minimal theta functions. Glasgow Mathematical Journal, 30 (1988), 75–85]. The studied theta functions are generalizations of the Jacobi theta-2 and theta-4 functions. Contrary to Montgomery’s result, we show that, among lattices, the hexagonal lattice is the unique maximizer of both families of theta functions. As an immediate consequence, we obtain a new universal optimality result for the hexagonal lattice among two-dimensional alternating charged lattices and lattices shifted by the center of their unit cell.

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References

  1. A. Aftalion, X. Blanc and F. Nier, Lowest Landau level functional and Bargmann spaces for Bose—Einstein condensates, J. Funct. Anal. 241 (2006), 661–702.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Baernstein II, A minimum problem for heat kernels of flat tori, in Extremal Riemann Surfaces (San Francisco, CA, 1995), American Mathematical Society, Providence, RI, 1997, pp. 227–243.

    Chapter  Google Scholar 

  3. W. Banaszczyk, New bounds in some transference theorems in the geometry of numbers, Math. Ann. 296 (1993), 625–635.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Bardenet, J. Flamant and P. Chainais, On the zeros of the spectrogram of white noise, Appl. Comput. Harmon. Anal. 48 (2020), 682–705.

    Article  MathSciNet  MATH  Google Scholar 

  5. S. N. Bernstein, Sur les fonctions absolument monotones, Acta Math. 52 (1928), 1–66.

    Article  MathSciNet  MATH  Google Scholar 

  6. L. Bétermin, Two-dimensional theta functions and crystallization among Bravais lattices, SIAM J. Math. Anal. 48 (2016), 3236–3269.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Bétermin, Minimal soft lattice theta functions, Constr. Approx. 52 (2020), 115–138.

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Bétermin, M. Faulhuber and H. Knüpfer, On the optimality of the rock-salt structure among lattices and change distributions, Math. Models Methods Appl. Sci. 31 (2021), 293–325.

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Bétermin and H. Knüpfer, On Bom’s conjecture about optimal distribution of charges for an infinite ionic crystal, J. Nonlinear Sci. 28 (2018), 1629–1656.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Bétermin, H. Knüpfer and F. Nolte, Note on crystallization for alternating particle chains. J. Stat. Phys. 181 (2020), 803–815.

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Bétermin and M. Petrache, Dimension reduction techniques for the minimization of theta functions on lattices, J. Math. Phys. 58 (2017), Article no. 071902.

  12. L. Bétermin and M. Petrache, Optimal and non-optimal lattices for non-completely monotone interaction potentials, Anal. Math. Phys. 9 (2019), 2033–2073.

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Borwein, J. M. Borwein and C. Pinner, Convergence of Madelung-like lattice sums, Trans. Amer. Math. Soc. 350 (1998), 3131–3167.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. M. Borwein, M. L. McPhedran, R. C. Wan and I. J. Zucker, Lattice Sums: Then and Now, Cambridge University Press, Cambridge, 2013.

    Book  MATH  Google Scholar 

  15. J.-B. Bost, Theta Invariants of Euclidean Lattices and Infinite Dimensional Hermitian Vector Bundles over Arithmetic Curves, Birkhäuser, Basel, 2020.

    Book  MATH  Google Scholar 

  16. J. W. S. Cassels, On a problem of Rankin about the Epstein zeta-function, Glasg. Math. J. 4 (1959), 73–80.

    MathSciNet  MATH  Google Scholar 

  17. H. Cohn and A. Kumar, Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007), 99–148.

    Article  MathSciNet  MATH  Google Scholar 

  18. H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. S. Viazovska, The sphere packing problem in dimension 24, Ann. Math. (2) 187 (2017), 1035–1068.

    MathSciNet  MATH  Google Scholar 

  19. H. Cohn, A. Kumar, S.D. Miller, D. Radchenko and M.S. Viazovska, Universal optimality of E8and Leech lattices and interpolation formulas, Ann. of Math. (2) 196 (2022), 983–1082.

    MathSciNet  MATH  Google Scholar 

  20. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, New York, 1998.

    Google Scholar 

  21. R. Coulangeon and A. Schürmann, Energy minimization, periodic sets and spherical designs, Int. Math. Res. Not. IMRN 2012 (2012), 829–848.

    Article  MathSciNet  MATH  Google Scholar 

  22. B. Delaunay, Sur la sphère vide, Izv. Ross. Akad. Nauk Ser. Mat. 6 (1934), 793–800.

    MATH  Google Scholar 

  23. P. H. Diananda, Notes on two lemmas concerning the Epstein zeta-function, Glasg. Math. J. 6 (1964), 202–204.

    MathSciNet  MATH  Google Scholar 

  24. V. Ennola, A lemma about the Epstein zeta function, Glasg. Math. J. 6 (1964), 198–201.

    MathSciNet  MATH  Google Scholar 

  25. A.-M. Ernvall-Hytönen and E. V. Vesalainen, On the secrecy gain of l-modular lattices, SIAM J. Discrete Math. 32 (2018), 1441–1457.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Faulhuber, A short note on the frame set of odd functions, Bull. Aust. Math. Soc. 98 (2018), 481–493.

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Faulhuber, Minimal frame operator norms via minimal theta functions, J. Fourier Anal. Appl. 24 (2018), 545–559.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Faulhuber, The Strohmer and Beaver conjecture for Gaussian Gabor systems: a deep mathematical problem (?), in Proceedings of the 13th International Conference on Sampling Theory and Applications (SampTA), IEEE, New York, 2019, https://doi.org/10.1109/SampTA45681.2019.9030963.

    Google Scholar 

  29. M. Faulhuber, An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and aWeltkonstante”-or-how Ramanujan split temperatures, Ramanujan J. 54 (2021), 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Faulhuber, Extremal determinants of Laplace—Beltrami operators for rectangular tori, Math. Z. 297 (2021), 175–195.

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Faulhuber and S. Steinerberger, Optimal Gabor frame bounds for separable lattices and estimates for Jacobi theta functions, J. Math. Anal. Appl. 445 (2017), 407–422.

    Article  MathSciNet  MATH  Google Scholar 

  32. M. Faulhuber and S. Steinerberger, An extremal property of the hexagonal lattice, J. Stat. Phys. 177 (2019), 285–298.

    Article  MathSciNet  MATH  Google Scholar 

  33. P. Flandrin, On spectrogram local maxima, in 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, New York, 2017, pp. 3979–3983.

    Google Scholar 

  34. G. B. Folland, Harmonic Analysis in Phase Space. Princeton University Press, Princeton, NJ, 1989.

    Book  MATH  Google Scholar 

  35. T. Gannon, Lattices and Theta Functions. PhD thesis, McGill University, Montreal, 1991.

    Google Scholar 

  36. M. A. de Gosson, Symplectic Methods in Harmonic Analysis and in Mathematical Physics, Birkhauser/Springer, Basel, 2011.

    Book  MATH  Google Scholar 

  37. K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, MA, 2001.

    Book  MATH  Google Scholar 

  38. T. C. Hales, A proof of the Kepler conjecture, Ann. of Math. (2) 162 (2005), 1065–1185.

    Article  MathSciNet  MATH  Google Scholar 

  39. T. C. Hales, M. Adams, G. Bauer, T. D. Dang, J. Harrison, L. T. Hoang, C. Kaliszky, V. Magron, S. McLaughlin, T. T. Nguyen, Q. T. Nguyen, A. Nipkow, T. H. A. Ta, N. T. Tran, T. D. Trieu, J. Urban, V. Ky and R. Zumkeller. A formal proof of the Kepler conjecture, Forum Math. Pi 5 (2017), Article no. e2.

  40. A. Henn, The hexagonal lattice and the Epstein zeta function, in Dynamical Systems, Number Theory and Applications World Scientific, Singapore, 2016, pp. 127–140.

    Chapter  MATH  Google Scholar 

  41. A. J. E. M. Janssen, Some Weyl—Heisenberg frame bound calculations, Indag. Math. (N.S.) 7 (1996), 165–183.

    Article  MathSciNet  MATH  Google Scholar 

  42. A. J. E. M. Janssen and T. Strohmer, Hyperbolic secants yield Gabor frames, Appl. Comput. Harmon. Anal. 12 (2002), 259–267.

    Article  MathSciNet  MATH  Google Scholar 

  43. J. Jorgenson and S. Lang, The ubiquitous heat kernel, in Mathematics Unlimited—2001 and Beyond, Springer, Berlin-Heidelberg, 2001, pp. 655–683.

    Chapter  Google Scholar 

  44. S. Luo and J. Wei, On minima of sum of theta functions and Mueller—Ho Conjecture, Arch. Ration. Mech. Anal. 243 (2022), 139–199.

    Article  MathSciNet  MATH  Google Scholar 

  45. E. Madelung, Das elektrische Feld in Systemen von regelmässig angeordneten Punktladungen, Phys. Z. 19 (1918), 524–533.

    MATH  Google Scholar 

  46. J. Martinet, Perfect Lattices in Euclidean Spaces, Springer, Berlin-Heidelberg, 2003.

    Book  MATH  Google Scholar 

  47. H. L. Montgomery. Minimal theta functions, Glasg. Math. J. 30 (1988), 75–85.

    Article  MathSciNet  MATH  Google Scholar 

  48. D. Mumford, Tata Lectures on Theta I, Birkhäuser, Boston, MA, 1983.

    Book  MATH  Google Scholar 

  49. S. Nonnenmacher and A. Voros, Chaotic eigenfunctions in phase space, J. Stat. Phys. 92 (1998), 431–518.

    Article  MathSciNet  MATH  Google Scholar 

  50. B. Osgood, R. Phillips and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148–211.

    Article  MathSciNet  MATH  Google Scholar 

  51. R A. Rankin, A minimum problem for the Epstein zeta-function, Glasg. Math. J. 1 (1959), 149–158.

    MathSciNet  MATH  Google Scholar 

  52. E. Sandier and S. Serfaty, From the Ginzburg—Landau model to vortex lattice problems, Comm. Math. Phys. 313 (2012), 635–743.

    Article  MathSciNet  MATH  Google Scholar 

  53. J.-P. Serre, A Course in Arithmetic, Springer, New York, 1973.

    Book  MATH  Google Scholar 

  54. E. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, Princeton, NJ, 2003.

    MATH  Google Scholar 

  55. T. Strohmer and S. Beaver, Optimal OFDM design for time-frequency dispersive channels, Communications, IEEE Transactions 51 (2003), 1111–1122.

    Article  Google Scholar 

  56. M. P. Tosi, Cohesion if ionic solids in the Born model, Solid State Phys. 16 (1964), 1–120.

    Article  Google Scholar 

  57. M. S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 187 (2017), 991–1015.

    MathSciNet  MATH  Google Scholar 

  58. E.T. Whittaker and G.N. Watson, A Course of Modern Analysis. Cambridge University Press, Cambridge, 1969.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institut Camille Jordan, Université Claude Bernard Lyon 1, 21 avenue Claude Bernard, 69622, Villeurbanne Cedex, France

    Laurent Bétermin

  2. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Markus Faulhuber

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  1. Laurent Bétermin
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  2. Markus Faulhuber
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Correspondence to Markus Faulhuber.

Additional information

Laurent Bétermin was part of the Faculty of Mathematics, University of Vienna, Austria at the time of writing and was supported by the Vienna Science and Technology Fund (WWTF) MA14-009 as well as the Austrian Science Fund (FWF) project F65.

Markus Faulhuber was with the Department of Mathematics, RWTH Aachen University, Germany at the time of writing and was partially supported by the Vienna Science and Technology Fund (WWTF) VRG12-009 and the Austrian Science Fund (FWF) P33217 and TAI6.

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Bétermin, L., Faulhuber, M. Maximal theta functions universal optimality of the hexagonal lattice for Madelung-like lattice energies. JAMA 149, 307–341 (2023). https://doi.org/10.1007/s11854-022-0254-z

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  • DOI: https://doi.org/10.1007/s11854-022-0254-z

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