Open Access
Issue
ESAIM: COCV
Volume 32, 2026
Article Number 22
Number of page(s) 50
DOI https://doi.org/10.1051/cocv/2026007
Published online 18 March 2026
  1. J. Dereziński and V. Georgescu, On the domains of Bessel operators. Ann. Henri Poincaré 22 (2021) 3291–3309. [Google Scholar]
  2. S.A. Coon and B.R. Holstein, Anomalies in quantum mechanics: the 1/r2 potential. Am. J. Phys. 70(5) (2002) 513–519. [Google Scholar]
  3. A.M. Essin and D.J. Griffiths, Quantum mechanics of the 1/x2 potential. Am. J. Phys. 74 (2006) 109–117. [Google Scholar]
  4. H. Kalf, U.W. Schmincke, J. Walter and R. Wüst, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, in Spectral Theory and Differential Equations. Lect. Notes in Math. Vol. 448. Springer, Berlin (2006) 182–226. [Google Scholar]
  5. J.L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential. J. Funct. Anal. 173 (2000) 103–153. [Google Scholar]
  6. I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time dependent potentials. Invent. Math. 155 (2004) 451–513. [Google Scholar]
  7. M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978). [Google Scholar]
  8. L. Bruneau, J. Dereziński and V. Georgescu, Homogeneous Schrödinger operators on half-line. Ann. Henri Poincare 12 (2011) 547–590. [Google Scholar]
  9. J. Dereziński and S. Richard, On Schrödinger operators with inverse square potentials on the half-line. Ann. Henri Poincaré 18 (2017) 869–928. [Google Scholar]
  10. N. Anantharaman, M. Leautaud and F. Macià, Wigner measures and observability for the Schrodinger equation on the disk. Invent. Math. 206 (2016) 485–599. [Google Scholar]
  11. N. Anantharaman and F. Macià, Semiclassical measures for the Schrödinger equation on the torus. J. Eur. Math. Soc. 16 (2014) 1253–1288. [Google Scholar]
  12. N. Anantharaman and G. Rivière, Dispersion and controllability for the Schrodinger equation on negatively curved manifolds. Anal. PDE 5 (2012) 313–338. [Google Scholar]
  13. J. Bourgain, N. Burq and M. Zworski, Control for Schrödinger operators on 2-tori: rough potentials. J. Eur. Math. Soc. 15 (2013) 1597–1628. [Google Scholar]
  14. N. Burq and M. Zworski, Geometric control in the presence of a black box. J. Am. Math. Soc. 17 (2004) 443–471. [Google Scholar]
  15. N. Burq and M. Zworski, Control for Schrödinger equations on tori. Math. Res. Lett. 19 (2012) 309–324. [Google Scholar]
  16. N. Burq and M. Zworski, Rough controls for Schrödinger operators on 2-tori. Ann. H. Lebesgue 2 (2019) 331–347. [Google Scholar]
  17. L. Jin, Control for Schröodinger equation on hyperbolic surfaces. Math. Res. Lett. 25 (2018) 1865–1877. [Google Scholar]
  18. S. Huang, G. Wang and M. Wang, Observable sets, potentials and Schrodinger equations. Commun. Math. Phys. 395 (2022) 1297–1343. [CrossRef] [MathSciNet] [Google Scholar]
  19. K. Le Balc'h and J. Martin, Observability estimates for the Schrödinger equation in the plane with periodic bounded potentials from measurable sets, Ann. Inst. Fourier, to appear https://arxiv.org/abs/2304.08050 [Google Scholar]
  20. J. Martin and K. Pravda-Starov, Geometric conditions for the exact controllability of fractional free and harmonic Schrodinger equations. J. Evol. Equ. 21 (2021) 1059–1087. [Google Scholar]
  21. A. Prouf, Observability of Schrodinger equation with subquadratic confining potential in the Euclidean space. Anal. PDE 18 (2025) 1147–1229. [Google Scholar]
  22. M. Täufer, Controllability of the Schroödinger equation on unbounded domains without geometric control condition. ESAIM Control Optim. Calc. Var. 29 (2023) 59. [Google Scholar]
  23. G. Wang, M. Wang and Y. Zhang, Observability and unique continuation inequalities for the Schröodinger equation. J. Eur. Math. Soc. 21 (2019) 3513–3572. [CrossRef] [MathSciNet] [Google Scholar]
  24. P. Jaming, Nazarov's uncertainty principles in higher dimension. J. Approx. Theory 149 (2007) 30–41. [Google Scholar]
  25. V. Havin and B. Jöricke, The Uncertainty Principle in Harmonic Analysis. Springer, Berlin (1994). [Google Scholar]
  26. F.L. Nazarov, Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type. Algebra Anal. 5 (1993) 3–66. [Google Scholar]
  27. S. Huang and A. Soffer, Uncertainty principle, minimal escape velocities, and observability inequalities for Schröodinger equations. Am. J. Math. 143 (2021) 753–781. [Google Scholar]
  28. Z. Li and M. Wang, Observability inequality at two time points for KdV equations. SIAM J. Math. Anal. 53 (2021) 1944–1957. [Google Scholar]
  29. Y. Wang and M. Wang, Observability inequality at two time points for the KdV equation from measurable sets. J. Math. Anal. Appl. 505 (2022) 125643. [Google Scholar]
  30. M. Wang, Z. Li and S. Huang, Unique continuation inequalities for nonlinear Schröodinger equations based on uncertainty principles. Indiana Univ. Math. J. 72 (2021) 133–163. [Google Scholar]
  31. L. Escauriaza, C.E. Kenig, G. Ponce and L. Vega, Unique continuation for Schröodinger evolutions, with applications to profiles of concentration and traveling waves. Commun. Math. Phys. 305 (2011) 487–512. [Google Scholar]
  32. L. Escauriaza, C.E. Kenig, G. Ponce and L. Vega, Uniqueness properties of solutions to Schrödinger equations. Bull. Am. Math. Soc. (N.S.) 49 (2012) 415–442. [Google Scholar]
  33. A.D. Ionescu and C.E. Kenig, Uniqueness properties of solutions of Schröodinger equations. J. Funct. Anal. 232 (2006) 90–136. [Google Scholar]
  34. I. Seo, Global unique continuation from a half space for the Schröodinger equation. J. Funct. Anal. 266 (2014) 85–98. [Google Scholar]
  35. P. Su, C. Sun and X. Yuan, Quantitative observability for one-dimensional Schrodinger equations with potentials, J. Funct. Anal. 228 (2025) 110695. [Google Scholar]
  36. L. Wei, Z. Duan and H. Xu, Uncertainty principle and geometric condition for the observability of Schrodinger equations, J. Fourier Anal. Appl. to appear, https://arxiv.org/abs/2307.09592v4 [Google Scholar]
  37. J. Vancostenoble and E. Zuazua, Hardy inequalities, observability, and control for the wave and Schrüodinger equations with singular potentials. SIAM J. Math. Anal. 41 (2009) 1508–1532. [Google Scholar]
  38. H. Brezis and M. Marcus, Hardy's inequalities revisited. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1997) 217-237. [Google Scholar]
  39. M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness. Academic Press, New York (1975). [Google Scholar]
  40. V.K. Tuan, Uncertainty principles for the Hankel transform. Integral Transforms Spec. Funct. 18 (2007) 369–381. [Google Scholar]
  41. H. Kovařík and F. Truc, Schrödinger operators on a half-line with inverse square potentials. Math. Model. Nat. Phenom. 9 (2014) 170–176. [Google Scholar]
  42. G. Teschl, Mathematical Methods in Quantum Mechanics With Applications to Schrodinger Operators. American Mathematical Society Providence, Rhode Island (2009). [Google Scholar]
  43. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C. (1964). [Google Scholar]
  44. A. Erdelyi, Tables of Integral Transforms. Vol. 2. McGraw-Hill, New York (1954). [Google Scholar]
  45. N.B. Andersen, Real Paley-Wiener theorems for the Hankel transform. J. Fourier Anal. Appl. 12 (2006) 17–25. [Google Scholar]
  46. S. Ghobber and P. Jaming, Strong annihilating pairs for the Fourier-Bessel transform. J. Math. Anal. Appl. 377 (2011) 501–515. [Google Scholar]
  47. L. Grafakos, Classical Fourier Analysis. Graduate Texts in Mathematics. Vol. 249, 3rd edn. Springer, New York (2014). [Google Scholar]
  48. M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis. Academic Press, New York (1980). [Google Scholar]
  49. L. Fanelli, V. Felli, M.A. Fontelos and A. Primo, Time decay of scaling in variant electromagnetic Schrödinger equations on the plane. Commun. Math. Phys. 337 (2015) 1515–1533. [Google Scholar]
  50. C. Miao, X. Su and J. Zheng, The Ws,p-boundedness of stationary wave operators for the Schrödinger operator with the inverse-square potential. Trans. Am. Math. Soc. 376 (2023) 1739–1797. [Google Scholar]
  51. J. Apraiz and L. Escauriaza, Null-control and measurable sets. ESAIM Control Optim. Calc. Var. 19 (2013) 239–254. [Google Scholar]
  52. T.A. Bui and P. D'Ancona, Generalized Hardy operators. Nonlinearity 36 (2023) 171–198. [Google Scholar]
  53. R.L. Frank, K. Merz and H. Siedentop, Equivalence of Sobolev norms involving generalized Hardy operators. Int. Math. Res. Not. 2021 (2019) 2284–2303. [Google Scholar]
  54. R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, Sobolev spaces adapted to the Schrüodinger operator with inverse-square potential. Math. Z. 288 (2018) 1273–1298. [Google Scholar]
  55. M. Konstantin, On scales of Sobolev spaces associated to generalized Hardy operators. Math. Z. 299 (2021) 101–121. [Google Scholar]
  56. J. Le Rousseau and I. Moyano, Null-controllability of the Kolmogorov equation in the whole phase space. J. Differ. Equ. 260 (2016) 3193–3233. [CrossRef] [Google Scholar]

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