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Local realization of spectral clusters

Anass Nifa

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Abstract

Let λ\lambda be an isolated eigenvalue of multiplicity mm of a real Schrödinger operator on a compact connected manifold of dimension at least two, with a fixed closed, Dirichlet, or Neumann realization. Every symmetric matrix sufficiently close to λI\lambda I is the corresponding spectral-cluster matrix of a small scalar-potential perturbation supported in any prescribed nonempty interior open set. The perturbation depends 1/21/2-Hölder continuously on the matrix, vanishes at λI\lambda I, and is real analytic away from that value. Sections with these properties exist for positive Laplace eigenvalues in the full conformal class of a closed manifold and, in dimension at least three, for supported metric perturbations preserving the volume density pointwise.

For potentials and conformal factors, every nonzero scalar contraction of the intrinsic cokernel Hessian has infinite positive and negative index. The first nonzero transverse diagonal term of the contracted eigenfunction kernel is homogeneous and harmonic. On a closed manifold, the multiplicity locus in the full metric space or a full conformal class is a C1C^1 embedded submanifold exactly where the cluster derivative is onto. Unrestricted metric stability holds through multiplicity seven and can fail at multiplicity eight. For potentials and conformal metrics, the projection connection has arbitrary prescribed curvature at the reference along a compatible matrix germ. In particular, its local holonomy on the exact fixed-eigenvalue level is SO(m)\operatorname{SO}(m).

For a finite effective symmetry group, the matrix target is the self-adjoint commutant. In corank one, the same index condition gives continuous relative lifts over compact spaces of finite covering dimension. On each fixed harmonic space ⨁j=2Nℋj(S2)\bigoplus_{j=2}^N\mathcal H_j(S^2), the maximum width of the first N+1N+1 positive clusters is bounded below by a positive multiple of the squared L2L^2 norm of every sufficiently small conformal perturbation.

MSC 2020

58J50
Spectral problems; spectral geometry; scattering theory on manifolds
35P05
General topics in linear spectral theory for PDEs
35S05
Pseudodifferential operators as generalizations of partial differential operators
58C40
Spectral theory; eigenvalue problems on manifolds
53C29
Issues of holonomy in differential geometry
57R22
Topology of vector bundles and fiber bundles

Mathematics Subject Classification 2020

Record

Version DOI
10.5281/zenodo.23224622
All versions
10.5281/zenodo.22859621
Subjects
spectral cluster; Laplace operator; Schrödinger operator; local realization; multiplicity; holonomy; spectral rigidity
Language
English

Cite this paper

Anass Nifa. Local realization of spectral clusters. Preprint, Zenodo, v2 (2026). doi:10.5281/zenodo.23224622

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References

As listed in the manuscript bibliography.

  1. [1]

    A. Agrachev, F. Boarotto, and A. Lerario, Homotopically invisible singular curves, Calc. Var. Partial Differential Equations 56 (2017), Article 105. DOI: 10.1007/s00526-017-1203-z; arXiv:1603.08937.

    DOI · arXiv
  2. [2]

    A. A. Agrachev, On the space of symmetric operators with multiple ground states, Funct. Anal. Appl. 45 (2011), no. 4, 241–251.

  3. [3]

    A. A. Agrachev, Topology of quadratic maps and Hessians of smooth maps, J. Soviet Math. 49 (1990), no. 3, 990–1013.

  4. [4]

    A. A. Agrachev and A. V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 6, 635–690.

  5. [5]

    W. Ambrose and I. M. Singer, A theorem on holonomy, Trans. Amer. Math. Soc. 75 (1953), 428–443. DOI: 10.1090/S0002-9947-1953-0063739-1.

    DOI
  6. [6]

    V. I. Arnol’d, Modes and quasimodes, Funct. Anal. Appl. 6 (1972), 94–101.

  7. [7]

    N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957), 235–249.

  8. [8]

    A. V. Arutyunov and D. Yu. Karamzin, Regular zeros of quadratic maps and their application, Sb. Math. 202 (2011), no. 6, 783–806. DOI: 10.1070/SM2011v202n06ABEH004166.

    DOI
  9. [9]

    A. V. Arutyunov and D. Karamzin, Square-root metric regularity and related stability theorems for smooth mappings, SIAM J. Optim. 31 (2021), 1380–1409. DOI: 10.1137/20M1337697.

    DOI
  10. [10]

    A. V. Arutyunov, Investigation of the sets of real solutions of non-linear equations, Izv. Math. 83 (2019), no. 2, 199–213. DOI: 10.1070/IM8795.

    DOI
  11. [11]

    A. V. Arutyunov and S. E. Zhukovskiy, Solvability of nonlinear degenerate equations and estimates for inverse functions, Sb. Math. 216 (2025), no. 1, 1–24. DOI: 10.4213/sm10060e.

    DOI
  12. [12]

    A. V. Arutyunov and S. E. Zhukovskiy, Implicit function theorem for inclusions without a priori normality conditions, Sb. Math. 217 (2026), no. 5, 611–631. DOI: 10.4213/sm10325e.

    DOI
  13. [13]

    S. Bando and H. Urakawa, Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds, Tohoku Math. J. (2) 35 (1983), no. 2, 155–172.

  14. [14]

    G. Berkolaiko and I. Zelenko, Morse inequalities for ordered eigenvalues of generic self-adjoint families, Invent. Math. 238 (2024), 283–330.

  15. [15]

    G. Besson, Propriétés génériques des fonctions propres et multiplicité, Comment. Math. Helv. 64 (1989), no. 4, 542–588.

  16. [16]

    D. D. Bleecker and L. C. Wilson, Splitting the spectrum of a Riemannian manifold, SIAM J. Math. Anal. 11 (1980), no. 5, 813–818.

  17. [17]

    D. Cianci, C. Judge, S. Lin, and C. Sutton, Spectral multiplicity and nodal domains of torus-invariant metrics, Int. Math. Res. Not. IMRN 2024, no. 3, 2192–2218. DOI: 10.1093/imrn/rnad102.

    DOI
  18. [18]

    Y. Colin de Verdière, Sur une hypothèse de transversalité d’Arnold, Comment. Math. Helv. 63 (1988), 184–193.

  19. [19]

    Y. Colin de Verdière, Construction de laplaciens dont une partie finie du spectre est donnée, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 4, 599–615.

  20. [20]

    D. S. de Oliveira and M. A. M. Marrocos, Hidden symmetries and the generic spectral setting of generalized laplacians on homogeneous spaces, arXiv:2502.07073v1 (2025). https://arxiv.org/abs/2502.07073.

    arXiv v1
  21. [21]

    NIST Digital Library of Mathematical Functions, Sections 14.30 and 34.3, particularly the spherical-harmonic product formula. https://dlmf.nist.gov/34.3

  22. [22]

    A. N. Dranishnikov, On classifying spaces for dimension, Topology Appl. 329 (2023), 108360. DOI: 10.1016/j.topol.2022.108360.

    DOI
  23. [23]

    J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353–367. DOI: 10.2140/pjm.1951.1.353.

    DOI
  24. [24]

    S. Dyatlov, Lecture notes for 18.155: distributions, elliptic regularity, and applications to PDEs, MIT lecture notes, 2 October 2026, https://math.mit.edu/~dyatlov/18.155/155-notes.pdf.

  25. [25]

    A. El Soufi and S. Ilias, Riemannian manifolds admitting isometric immersions by their first eigenfunctions, Pacific J. Math. 195 (2000), no. 1, 91–99.

  26. [26]

    A. El Soufi and S. Ilias, Laplacian eigenvalue functionals and metric deformations on compact manifolds, J. Geom. Phys. 58 (2008), no. 1, 89–104. DOI: 10.1016/j.geomphys.2007.09.008.

    DOI
  27. [27]

    J. N. V. Gomes and M. A. M. Marrocos, On eigenvalue generic properties of the Laplace–Neumann operator, J. Geom. Phys. 135 (2019), 21–31.

  28. [28]

    J. Greilhuber and W. Kepplinger, On Arnold’s transversality conjecture for the Laplace–Beltrami operator, arXiv:2312.16939v4 (2026).

    arXiv v4
  29. [29]

    G. Grubb, Properties of normal boundary problems for elliptic even-order systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1 (1974), nos. 1–2, 1–61.

  30. [30]

    G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr. 289 (2016), no. 7, 831–844.

  31. [31]

    V. Guillemin, The Radon transform on Zoll surfaces, Adv. Math. 22 (1976), no. 1, 85–119. DOI: 10.1016/0001-8708(76)90139-0.

    DOI
  32. [32]

    X. He and Z. Wang, Riemannian metrics with prescribed volume and finite parts of Dirichlet spectrum, Calc. Var. Partial Differential Equations 63 (2024), no. 3, Paper No. 78, 24 pp.

  33. [33]

    J. Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1645–A1648.

  34. [34]

    L. Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators, Grundlehren der mathematischen Wissenschaften, vol. 274, Springer-Verlag, Berlin, 1985.

  35. [35]

    W. Hurewicz and H. Wallman, Dimension Theory, Princeton Mathematical Series 4, Princeton University Press, 1941.

  36. [36]

    A. F. Izmailov and M. V. Solodov, The theory of 2-regularity for mappings with Lipschitzian derivatives and its applications to optimality conditions, Math. Oper. Res. 27 (2002), 614–635. DOI: 10.1287/moor.27.3.614.308.

    DOI
  37. [37]

    M. Karpukhin and A. Métras, Laplace and Steklov extremal metrics via 𝑛-harmonic maps, J. Geom. Anal. 32 (2022), Article 154; arXiv:2103.15204.

    arXiv
  38. [38]

    T. Kato, Perturbation Theory for Linear Operators, second edition, Springer-Verlag, Berlin, 1976; corrected reprint, 1995.

  39. [39]

    M. Larsen and A. Lindenstrauss, The higher connectivity of intersections of real quadrics, arXiv:math/0508388.

    arXiv
  40. [40]

    J. Lohkamp, Discontinuity of geometric expansions, Comment. Math. Helv. 71 (1996), 213–228.

  41. [41]

    M. A. M. Marrocos and A. L. Pereira, Eigenvalues of the Neumann Laplacian in symmetric regions, arXiv:1310.5178v1 (2013). https://arxiv.org/abs/1310.5178.

    arXiv v1
  42. [42]

    M. A. M. Marrocos and A. L. Pereira, Eigenvalues of the Neumann Laplacian in symmetric regions, J. Math. Phys. 56 (2015), no. 11, 111502. DOI: 10.1063/1.4935300.

    DOI
  43. [43]

    A. M. Micheletti and A. Pistoia, Multiple eigenvalues of the Laplace–Beltrami operator and deformation of the Riemannian metric, Discrete Contin. Dynam. Systems 4 (1998), no. 4, 709–720.

  44. [44]

    J. W. Milnor and J. D. Stasheff, Characteristic Classes, Annals of Mathematics Studies, vol. 76, Princeton University Press, Princeton, 1974.

  45. [45]

    C. B. Morrey, Jr., and L. Nirenberg, On the analyticity of the solutions of linear elliptic systems of partial differential equations, Comm. Pure Appl. Math. 10 (1957), no. 2, 271–290. DOI: 10.1002/cpa.3160100204.

    DOI
  46. [46]

    M. S. Narasimhan and S. Ramanan, Existence of universal connections, Amer. J. Math. 83 (1961), no. 3, 563–572. DOI: 10.2307/2372896.

    DOI
  47. [47]

    N. Nowaczyk, Existence of Dirac eigenvalues of higher multiplicity, Math. Z. 284 (2016), 285–307; arXiv:1504.00961v1, with statement numbering from that arXiv version.

  48. [48]

    K. Okikiolu, Hessians of spectral zeta functions, arXiv:math/0211100v3 (2003). https://arxiv.org/abs/math/0211100.

    arXiv v3
  49. [49]

    A. L. Pereira, Eigenvalues of the Laplacian on symmetric regions, Nonlinear Differential Equations Appl. 2 (1995), no. 1, 63–109. DOI: 10.1007/BF01194014.

    DOI
  50. [50]

    R. Ponge, Connes’ integration and Weyl’s laws, J. Noncommut. Geom. 17 (2023), no. 2, 719–767. DOI: 10.4171/JNCG/509; arXiv:2107.01242, Theorem 6.1 and Remark 6.2.

    DOI · arXiv
  51. [51]

    T. R. Ramadas, On the space of maps inducing isomorphic connections, Ann. Inst. Fourier 32 (1982), no. 1, 263–276. DOI: 10.5802/aif.868.

    DOI
  52. [52]

    A. Ramos, Second-order optimality conditions for degenerate optimization problems and higher-order tangents sets, Appl. Math. Optim. 94 (2026), Art. 5, 40 pp. DOI: 10.1007/s00245-026-10466-4.

    DOI
  53. [53]

    E. Shelukhin, Generic irreducibility of Laplace eigenspaces with finite symmetry, Oxford Mathematical Institute, seminar announcement, 29 May 2026, joint work in progress with E. Morozov. https://www.maths.ox.ac.uk/node/81234.

  54. [54]

    B. Simon, Holonomy, the quantum adiabatic theorem, and Berry’s phase, Phys. Rev. Lett. 51 (1983), 2167–2170. DOI: 10.1103/PhysRevLett.51.2167.

    DOI
  55. [55]

    O. Sönnerborn, Isoholonomic inequality and tight implementations of holonomic quantum gates, Phys. Rev. A 111 (2025), 052430. DOI: 10.1103/PhysRevA.111.052430.

    DOI
  56. [56]

    S. Tanimura, M. Nakahara, and D. Hayashi, Exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation, J. Math. Phys. 46 (2005), 022101; arXiv:quant-ph/0406038. DOI: 10.1063/1.1835545.

    DOI · arXiv
  57. [57]

    M. Teytel, How rare are multiple eigenvalues?, Comm. Pure Appl. Math. 52 (1999), no. 8, 917–934.

  58. [58]

    K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math. 98 (1976), no. 4, 1059–1078.

  59. [59]

    F. Wilczek and A. Zee, Appearance of gauge structure in simple dynamical systems, Phys. Rev. Lett. 52 (1984), 2111–2114. DOI: 10.1103/PhysRevLett. 52.2111.

    DOI
  60. [60]

    S. Wu, Quantum adiabatic theorem and universal holonomy, Lett. Math. Phys. 16 (1988), 339–345. DOI: 10.1007/BF00402042.

    DOI
  61. [61]

    P. Zanardi and M. Rasetti, Holonomic quantum computation, Phys. Lett. A 264 (1999), 94–99; arXiv:quant-ph/9904011. DOI: 10.1016/S0375-9601(99)00803-8.

    DOI · arXiv
  62. [62]

    S. Zelditch, Gaussian beams on Zoll manifolds and maximally degenerate Laplacians, Contemp. Math. 640 (2015), 169–198; arXiv:1404.2906. https://arxiv.org/abs/1404.2906.

  63. [63]

    S. Zelditch, Maximally degenerate laplacians, Ann. Inst. Fourier 46 (1996), no. 2, 547–587. https://www.numdam.org/item/AIF_1996__46_2_547_0.pdf.

  64. [64]

    S. Zelditch, On the generic spectrum of a Riemannian cover, Ann. Inst. Fourier 40 (1990), no. 2, 407–442. DOI: 10.5802/aif.1219.

    DOI