On the computation of geometric features of spectra of linear operators on Hilbert spaces. (English) Zbl 07875095
Summary: Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum is not enough. Often it is vital to determine geometric features of spectra such as Lebesgue measure, capacity or fractal dimensions, different types of spectral radii and numerical ranges, or to detect gaps in essential spectra and the corresponding failure of the finite section method. Despite new results on computing spectra and the substantial interest in these geometric problems, there remain no general methods able to compute such geometric features of spectra of infinite-dimensional operators. We provide the first algorithms for the computation of many of these long-standing problems (including the above). As demonstrated with computational examples, the new algorithms yield a library of new methods. Recent progress in computational spectral problems in infinite dimensions has led to the solvability complexity index (SCI) hierarchy, which classifies the difficulty of computational problems. These results reveal that infinite-dimensional spectral problems yield an intricate infinite classification theory determining which spectral problems can be solved and with which type of algorithm. This is very much related to S. Smale’s comprehensive program on the foundations of computational mathematics initiated in the 1980s. We classify the computation of geometric features of spectra in the SCI hierarchy, allowing us to precisely determine the boundaries of what computers can achieve (in any model of computation) and prove that our algorithms are optimal. We also provide a new universal technique for establishing lower bounds in the SCI hierarchy, which both greatly simplifies previous SCI arguments and allows new, formerly unattainable, classifications.
MSC:
| 65J05 | General theory of numerical analysis in abstract spaces |
| 65F99 | Numerical linear algebra |
| 47A10 | Spectrum, resolvent |
| 46N40 | Applications of functional analysis in numerical analysis |
| 47A12 | Numerical range, numerical radius |
| 47N50 | Applications of operator theory in the physical sciences |
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
| 28A12 | Contents, measures, outer measures, capacities |
| 28A78 | Hausdorff and packing measures |
Keywords:
computational spectral problems; solvability complexity; spectral radii; spectral capacity; spectral gaps; spectral pollution; measure; fractal dimensionsReferences:
| [1] | Aiena, P.: Fredholm and local spectral theory, with applications to multipliers. Springer Science & Business Media (2007) |
| [2] | Antun, V., Colbrook, M.J., Hansen, A.C.: Proving existence is not enough: Mathematical paradoxes unravel the limits of neural networks in artificial intelligence. SIAM News 55(4), 1-4 (2022) |
| [3] | Arveson, W., Discretized CCR algebras, Journal of Operator Theory, 26, 2, 225-239, 1991 · Zbl 0797.46050 |
| [4] | Arveson, W., Improper filtrations for \(C^*\)-algebras: spectra of unilateral tridiagonal operators, Acta Sci. Math. (Szeged), 57, 1-4, 11-24, 1993 · Zbl 0819.46044 |
| [5] | Arveson, W.: Noncommutative spheres and numerical quantum mechanics. In: Operator algebras, mathematical physics, and low-dimensional topology, Res. Notes Math., vol. 5, pp. 1-10. A K Peters, Wellesley, MA (1993) · Zbl 0803.46078 |
| [6] | Arveson, W., \(C^*\)-algebras and numerical linear algebra, Journal of Functional Analysis, 122, 2, 333-360, 1994 · Zbl 0802.46069 |
| [7] | Arveson, W.: The role of \(C^\ast \)-algebras in infinite-dimensional numerical linear algebra. In: \(C^\ast \)-algebras: 1943-1993 (San Antonio, TX, 1993), Contemp. Math., vol. 167, pp. 114-129. Amer. Math. Soc., Providence, RI (1994) · Zbl 0817.46051 |
| [8] | Aubry, S.; André, G., Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Israel Phys. Soc, 3, 133, 18, 1980 · Zbl 0943.82510 |
| [9] | Avila, A., Jitomirskaya, S.: The Ten Martini Problem. Annals of Mathematics (2) 170(1), 303-342 (2009) · Zbl 1166.47031 |
| [10] | Avila, A.; Jitomirskaya, S.; Marx, C., Spectral theory of extended Harper’s model and a question by Erdős and Szekeres, Inventiones mathematicae, 210, 1, 283-339, 2017 · Zbl 1380.37019 |
| [11] | Avila, A., Krikorian, R.: Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles. Annals of Mathematics (2) 164(3), 911-940 (2006) · Zbl 1138.47033 |
| [12] | Avila, A.; Viana, M., Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture, Acta Mathematica, 198, 1, 1-56, 2007 · Zbl 1143.37001 |
| [13] | Azbel, MY, Energy spectrum of a conduction electron in a magnetic field, Sov. Phys. JETP, 19, 3, 634-645, 1964 |
| [14] | Bandres, M.A., Rechtsman, M.C., Segev, M.: Topological photonic quasicrystals: Fractal topological spectrum and protected transport. Physical Review X 6(1), 011,016 (2016) |
| [15] | Bastounis, A., Hansen, A.C., Vlačić, V.: The extended Smale’s 9th problem—On computational barriers and paradoxes in estimation, regularisation, computer-assisted proofs and learning. arXiv:2110.15734 (2021) |
| [16] | Becker, S., Hansen, A.: Computing solutions of Schrödinger equations on unbounded domains—on the brink of numerical algorithms. arXiv preprint arXiv:2010.16347 (2020) |
| [17] | Beckus, S.; Pogorzelski, F., Spectrum of Lebesgue measure zero for Jacobi matrices of quasicrystals, Mathematical Physics, Analysis and Geometry, 16, 3, 289-308, 2013 · Zbl 1273.81087 |
| [18] | Ben-Artzi, J., Colbrook, M.J., Hansen, A.C., Nevanlinna, O., Seidel, M.: Computing Spectra—On the Solvability Complexity Index hierarchy and towers of algorithms. arXiv:1508.03280v5 (2020) |
| [19] | Ben-Artzi, J.; Hansen, AC; Nevanlinna, O.; Seidel, M., New barriers in complexity theory: On the solvability complexity index and the towers of algorithms, Comptes Rendus Mathematique, 353, 10, 931-936, 2015 · Zbl 1343.68078 |
| [20] | Ben-Artzi, J., Marletta, M., Rösler, F.: Computing the sound of the sea in a seashell. Foundations of Computational Mathematics pp. 1-35 (2021) |
| [21] | Ben-Artzi, J., Marletta, M., Rösler, F.: Computing scattering resonances. J. Eur. Math. Soc. (to appear) · Zbl 07755544 |
| [22] | Ben-Artzi, J., Marletta, M., Rösler, F.: Universal algorithms for computing spectra of periodic operators. Numerische Mathematik (to appear) · Zbl 1486.35157 |
| [23] | Ben-Artzi, J., Rösler, F., Marletta, M.: Universal algorithms for solving inverse spectral problems. arXiv preprint arXiv:2203.13078 (2022) |
| [24] | Benza, V.G., Sire, C.: Band spectrum of the octagonal quasicrystal: Finite measure, gaps, and chaos. Physical Review B 44(18), 10,343 (1991) |
| [25] | Berry, M., Fractal modes of unstable lasers with polygonal and circular mirrors, Optics communications, 200, 1-6, 321-330, 2001 |
| [26] | Berry, M., Physics of nonhermitian degeneracies, Czechoslovak journal of physics, 54, 10, 1039-1047, 2004 |
| [27] | Berry, M.; Storm, C.; Van Saarloos, W., Theory of unstable laser modes: edge waves and fractality, Optics communications, 197, 4-6, 393-402, 2001 |
| [28] | Blum, L.; Cucker, F.; Shub, M.; Smale, S., Complexity and Real Computation, 1998, New York Inc, Secaucus, NJ, USA: Springer-Verlag, New York Inc, Secaucus, NJ, USA |
| [29] | Blum, L.; Shub, M.; Smale, S., On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines, American Mathematical Society. Bulletin., 21, 1, 1-46, 1989 · Zbl 0681.03020 |
| [30] | Boche, H., Pohl, V.: The solvability complexity index of sampling-based Hilbert transform approximations. In: 2019 13th International conference on Sampling Theory and Applications (SampTA), pp. 1-4. IEEE (2019) |
| [31] | Boffi, D.; Brezzi, F.; Gastaldi, L., On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form, Mathematics of Computation, 69, 229, 121-140, 2000 · Zbl 0938.65126 |
| [32] | Boffi, D.; Duran, RG; Gastaldi, L., A remark on spurious eigenvalues in a square, Appl. Math. Lett., 12, 3, 107-114, 1999 · Zbl 0941.65109 |
| [33] | Bögli, S., Brown, B.M., Marletta, M., Tretter, C., Wagenhofer, M.: Guaranteed resonance enclosures and exclosures for atoms and molecules. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470(2171), 20140,488 (2014) · Zbl 1371.81347 |
| [34] | Bögli, S., Marletta, M., Tretter, C.: The essential numerical range for unbounded linear operators. Journal of Functional Analysis p. 108509 (2020) · Zbl 1505.47007 |
| [35] | Böttcher, A., Pseudospectra and singular values of large convolution operators, Journal of Integral Equations and Applications, 6, 3, 267-301, 1994 · Zbl 0819.45002 |
| [36] | Böttcher, A.: Infinite matrices and projection methods. In: Lectures on operator theory and its applications (Waterloo, ON, 1994), Fields Inst. Monogr., vol. 3, pp. 1-72. Amer. Math. Soc., Providence, RI (1996) · Zbl 0842.65032 |
| [37] | Böttcher, A.; Brunner, H.; Iserles, A.; Nørsett, SP, On the singular values and eigenvalues of the Fox-Li and related operators, New York J. Math., 16, 539-561, 2010 · Zbl 1233.47022 |
| [38] | Böttcher, A., Grudsky, S., Iserles, A.: Spectral theory of large Wiener-Hopf operators with complex-symmetric kernels and rational symbols. Mathematical Proceedings of the Cambridge Philosophical Society 151(1), 161-191 (2011) · Zbl 1227.47015 |
| [39] | Böttcher, A.; Silbermann, B., The finite section method for Toeplitz operators on the quarter-plane with piecewise continuous symbols, Mathematische Nachrichten, 110, 279-291, 1983 · Zbl 0549.47010 |
| [40] | Böttcher, A.; Silbermann, B., Introduction to large truncated Toeplitz matrices, 1999, New York: Universitext. Springer-Verlag, New York · Zbl 0916.15012 |
| [41] | Böttcher, A.; Silbermann, B., Analysis of Toeplitz operators, 2006, Berlin: Springer Monographs in Mathematics. Springer-Verlag, Berlin · Zbl 1098.47002 |
| [42] | Brown, N.: Invariant means and finite representation theory of C*-algebras. Memoirs of the American Mathematical Society 184 (2003). doi:10.1090/memo/0865 · Zbl 1111.46030 |
| [43] | Brown, NP, AF embeddings and the numerical computation of spectra in irrational rotation algebras, Numer. Funct. Anal. Optim., 27, 5-6, 517-528, 2006 · Zbl 1105.65060 |
| [44] | Brown, NP, Quasi-diagonality and the finite section method, Mathematics of Computation, 76, 257, 339-360, 2007 · Zbl 1113.65057 |
| [45] | Brunner, H.; Iserles, A.; Nørsett, SP, The spectral problem for a class of highly oscillatory Fredholm integral operators, IMA Journal of Numerical Analysis, 30, 1, 108-130, 2008 · Zbl 1185.45002 |
| [46] | Brunner, H.; Iserles, A.; Nørsett, SP, The computation of the spectra of highly oscillatory Fredholm integral operators, J. Integral Equations Applications, 23, 4, 467-519, 2011 · Zbl 1238.65123 · doi:10.1216/JIE-2011-23-4-467 |
| [47] | Buffa, A.; Perugia, I., Discontinuous Galerkin approximation of the Maxwell eigenproblem, SIAM Journal on Numerical Analysis, 44, 5, 2198-2226, 2006 · Zbl 1344.65110 |
| [48] | Burke, JV; Greenbaum, A., Characterizations of the polynomial numerical hull of degree k, Linear algebra and its applications, 419, 1, 37-47, 2006 · Zbl 1106.15018 |
| [49] | Carmona, R.; Lacroix, J., Spectral theory of random Schrödinger operators, 1990, Boston, MA: Probability and its Applications. Birkhäuser Boston Inc, Boston, MA · Zbl 0717.60074 |
| [50] | Christiansen, SH; Winther, R., On variational eigenvalue approximation of semidefinite operators, IMA J. Numer. Anal., 33, 1, 164-189, 2013 · Zbl 1269.65121 |
| [51] | Colbrook, M.J.: The foundations of infinite-dimensional spectral computations. Ph.D. thesis, University of Cambridge (2020) |
| [52] | Colbrook, MJ, Pseudoergodic operators and periodic boundary conditions, Mathematics of Computation, 89, 322, 737-766, 2020 · Zbl 1481.47004 |
| [53] | Colbrook, MJ, Computing spectral measures and spectral types, Communications in Mathematical Physics, 384, 1, 433-501, 2021 · Zbl 07348142 |
| [54] | Colbrook, M.J.: Unscrambling the infinite: Can we compute spectra? IMA Mathematics Today (2021). https://ima.org.uk/16912/unscrambling-the-infinite-can-we-compute-spectra/ |
| [55] | Colbrook, MJ, Computing semigroups with error control, SIAM Journal on Numerical Analysis, 60, 1, 396-422, 2022 · Zbl 1520.65033 |
| [56] | Colbrook, M.J.: The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems. arXiv preprint arXiv:2104.09444 (2022) |
| [57] | Colbrook, M.J., Antun, V., Hansen, A.C.: The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale’s 18th problem. Proceedings of the National Academy of Sciences 119(12), e2107151,119 (2022) |
| [58] | Colbrook, M.J., Ayton, L., Szoke, M.: Residual Dynamic Mode Decomposition: Robust and verified Koopmanism. arXiv preprint (2022) |
| [59] | Colbrook, MJ; Hansen, AC, On the infinite-dimensional QR algorithm, Numerische Mathematik, 143, 1, 17-83, 2019 · Zbl 1530.65055 |
| [60] | Colbrook, M.J., Hansen, A.C.: The foundations of spectral computations via the solvability complexity index hierarchy. Journal of the European Mathematical Society (to appear) · Zbl 07774915 |
| [61] | Colbrook, M.J., Horning, A.: Specsolve: Spectral methods for spectral measures. arXiv preprint arXiv:2201.01314 (2022) |
| [62] | Colbrook, M.J., Horning, A., Thicke, K., Watson, A.B.: Computing spectral properties of topological insulators without artificial truncation or supercell approximation. arXiv preprint arXiv:2112.03942 (2021) |
| [63] | Colbrook, MJ; Horning, A.; Townsend, A., Computing spectral measures of self-adjoint operators, SIAM Review, 63, 3, 489-524, 2021 · Zbl 07379587 |
| [64] | Colbrook, M.J., Roman, B., Hansen, A.C.: How to compute spectra with error control. Physical Review Letters 122(25), 250,201 (2019) |
| [65] | Colbrook, M.J., Townsend, A.: Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. arXiv preprint arXiv:2111.14889 (2021) |
| [66] | Cucker, F., The arithmetical hierarchy over the reals, J. Logic Comput., 2, 3, 375-395, 1992 · Zbl 0765.03020 |
| [67] | Damanik, D., Embree, M., Gorodetski, A.: Spectral properties of Schrödinger operators arising in the study of quasicrystals. In: Mathematics of aperiodic order, pp. 307-370. Springer (2015) · Zbl 1378.81031 |
| [68] | Dean, CR; Wang, L.; Maher, P.; Forsythe, C.; Ghahari, F.; Gao, Y.; Katoch, J.; Ishigami, M.; Moon, P.; Koshino, M., Hofstadter’s butterfly and the fractal quantum Hall effect in Moiré superlattices, Nature, 497, 7451, 598-602, 2013 |
| [69] | Della Villa, A., Enoch, S., Tayeb, G., Pierro, V., Galdi, V., Capolino, F.: Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice. Physical Review Letters 94(18), 183,903 (2005) |
| [70] | Doyle, P.; McMullen, C., Solving the quintic by iteration, Acta Mathematica, 163, 3-4, 151-180, 1989 · Zbl 0705.65036 |
| [71] | Falconer, K., Fractal geometry, 2003, Hoboken, NJ: John Wiley & Sons Inc, Hoboken, NJ · Zbl 1060.28005 |
| [72] | Fefferman, C., Seco, L.: On the energy of a large atom. Bull. Amer. Math. Soc. (N.S.) 23(2), 525-530 (1990) · Zbl 0722.35072 |
| [73] | Fefferman, C.; Seco, L., Eigenvalues and eigenfunctions of ordinary differential operators, Adv. Math., 95, 2, 145-305, 1992 · Zbl 0797.34083 |
| [74] | Fefferman, C.; Seco, L., Aperiodicity of the Hamiltonian flow in the Thomas-Fermi potential, Rev. Mat. Iberoamericana, 9, 3, 409-551, 1993 · Zbl 0788.34004 |
| [75] | Fefferman, C.; Seco, L., The density in a one-dimensional potential, Adv. Math., 107, 2, 187-364, 1994 · Zbl 0811.34063 |
| [76] | Fefferman, C.; Seco, L., The eigenvalue sum for a one-dimensional potential, Adv. Math., 108, 2, 263-335, 1994 · Zbl 0826.34070 |
| [77] | Fefferman, C.; Seco, L., On the Dirac and Schwinger corrections to the ground-state energy of an atom, Adv. Math., 107, 1, 1-185, 1994 · Zbl 0820.35113 |
| [78] | Fefferman, C.; Seco, L., The density in a three-dimensional radial potential, Adv. Math., 111, 1, 88-161, 1995 · Zbl 0826.34071 |
| [79] | Fefferman, C.; Seco, L., The eigenvalue sum for a three-dimensional radial potential, Adv. Math., 119, 1, 26-116, 1996 · Zbl 0845.34086 |
| [80] | Fefferman, C., Seco, L.: Interval arithmetic in quantum mechanics. In: Applications of interval computations (El Paso, TX, 1995), Appl. Optim., vol. 3, pp. 145-167. Kluwer Acad. Publ., Dordrecht (1996) · Zbl 0841.65067 |
| [81] | Fernández-Martínez, M.; Sánchez-Granero, MA, Fractal dimension for fractal structures: a Hausdorff approach revisited, Journal of Mathematical Analysis and Applications, 409, 1, 321-330, 2014 · Zbl 1309.28003 |
| [82] | Fillmore, PA; Stampfli, JG; Williams, JP, On the essential numerical range, the essential spectrum, and a problem of Halmos, Acta Sci. Math. (Szeged), 33, 197, 179-192, 1972 · Zbl 0246.47006 |
| [83] | Geim, AK; Grigorieva, IV, Van der Waals heterostructures, Nature, 499, 7459, 419-425, 2013 |
| [84] | Gil, M.I.: Operator functions and localization of spectra. Springer (2003) · Zbl 1032.47001 |
| [85] | Gilles, MA; Townsend, A., Continuous analogues of Krylov subspace methods for differential operators, SIAM Journal on Numerical Analysis, 57, 2, 899-924, 2019 · Zbl 1411.65047 |
| [86] | Gowers, W.: Rough structure and classification. Geom. Funct. Anal. pp. 79-117 (2000) · Zbl 0989.01020 |
| [87] | Hales, T., Adams, M., Bauer, G., Dang, T.D., Harrison, J., Hoang, L.T., Kaliszyk, C., Magron, V., McLaughlin, S., Nguyen, T.T., Nguyen, Q.T., Nipkow, T., Obua, S., Pleso, J., Rute, J., Solovyev, A., Ta, T.H.A., Tran, N.T., Trieu, T.D., Urban, J., Vu, K., Zumkeller, R.: A formal proof of the Kepler conjecture. Forum Math. Pi 5, e2, 29 (2017) · Zbl 1379.52018 |
| [88] | Hales, T.C.: A proof of the Kepler conjecture. Annals of Mathematics (2) 162(3), 1065-1185 (2005) · Zbl 1096.52010 |
| [89] | Halmos, P.R.: Capacity in Banach algebras. Indiana Univ. Math. J. 20, 855-863 (1970/1971) · Zbl 0196.14803 |
| [90] | Han, J., Thouless, D., Hiramoto, H., Kohmoto, M.: Critical and bicritical properties of Harper’s equation with next-nearest-neighbor coupling. Physical Review B 50(16), 11,365 (1994) |
| [91] | Hansen, AC, On the solvability complexity index, the \(n\)-pseudospectrum and approximations of spectra of operators, Journal of the American Mathematical Society, 24, 1, 81-124, 2011 · Zbl 1210.47013 |
| [92] | Hofstadter, DR, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Physical Review B, 14, 6, 2239, 1976 |
| [93] | Horning, A.; Townsend, A., Feast for differential eigenvalue problems, SIAM Journal on Numerical Analysis, 58, 2, 1239-1262, 2020 · Zbl 1439.65088 |
| [94] | Hunt, B.; Sanchez-Yamagishi, J.; Young, A.; Yankowitz, M.; LeRoy, BJ; Watanabe, K.; Taniguchi, T.; Moon, P.; Koshino, M.; Jarillo-Herrero, P., Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure, Science, 340, 6139, 1427-1430, 2013 |
| [95] | Jitomirskaya, S.: Critical phenomena, arithmetic phase transitions, and universality: some recent results on the almost Mathieu operator. · Zbl 1518.47001 |
| [96] | Johnson, CR, Numerical determination of the field of values of a general complex matrix, SIAM Journal on Numerical Analysis, 15, 3, 595-602, 1978 · Zbl 0389.65018 |
| [97] | Johnstone, D., Colbrook, M.J., Nielsen, A.E., Öhberg, P., Duncan, C.W.: Bulk localised transport states in infinite and finite quasicrystals via magnetic aperiodicity. Physical Review B 106(4), 045,149 (2022) |
| [98] | Kechris, AS; Louveau, A., Descriptive set theory and the structure of sets of uniqueness, London Mathematical Society Lecture Note Series, 1987, Cambridge: Cambridge University Press, Cambridge · Zbl 0642.42014 |
| [99] | Ketzmerick, R.; Kruse, K.; Kraut, S.; Geisel, T., What determines the spreading of a wave packet?, Physical Review Letters, 79, 11, 1959, 1997 · Zbl 0944.81002 |
| [100] | Ketzmerick, R.; Petschel, G.; Geisel, T., Slow decay of temporal correlations in quantum systems with Cantor spectra, Physical Review Letters, 69, 5, 695, 1992 |
| [101] | Killip, R.; Kiselev, A.; Last, Y., Dynamical upper bounds on wavepacket spreading, American journal of mathematics, 125, 5, 1165-1198, 2003 · Zbl 1053.81020 |
| [102] | Klaus, M.: On the point spectrum of Dirac operators. Helv. Phys. Acta 53(3), 453-462 (1981) (1980) |
| [103] | Kohmoto, M.; Sutherland, B.; Tang, C., Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model, Physical Review B, 35, 3, 1020, 1987 |
| [104] | Last, Y.: Spectral theory of Sturm-Liouville operators on infinite intervals: a review of recent developments. In: Sturm-Liouville Theory, pp. 99-120. Springer (2005) · Zbl 1098.39011 |
| [105] | Laursen, K.B., Laursen, K.B.L., Neumann, M.: An introduction to local spectral theory. 20. Oxford University Press (2000) · Zbl 0957.47004 |
| [106] | Levi, L.; Rechtsman, M.; Freedman, B.; Schwartz, T.; Manela, O.; Segev, M., Disorder-enhanced transport in photonic quasicrystals, Science, 332, 6037, 1541-1544, 2011 |
| [107] | Lewin, M., Séré, E.: Spectral pollution and how to avoid it (with applications to Dirac and periodic Schrödinger operators). Proc. Lond. Math. Soc. (3) 100(3), 864-900 (2010) · Zbl 1192.47016 |
| [108] | Lewin, M., Séré, É.: Spurious modes in Dirac calculations and how to avoid them. In: Many-Electron Approaches in Physics, Chemistry and Mathematics, pp. 31-52. Springer (2014) |
| [109] | Liesen, J.; Sète, O.; Nasser, MMS, Fast and accurate computation of the logarithmic capacity of compact sets, Computational Methods and Function Theory, 17, 4, 689-713, 2017 · Zbl 1381.65026 |
| [110] | Luitz, D.J., Lev, Y.B.: The ergodic side of the many-body localization transition. Annalen der Physik 529(7), 1600,350 (2017) · Zbl 1370.81238 |
| [111] | Malcolm Brown, B.; Langer, M.; Marletta, M.; Tretter, C.; Wagenhofer, M., Eigenvalue enclosures and exclosures for non-self-adjoint problems in hydrodynamics, LMS Journal of Computation and Mathematics, 13, 65-81, 2010 · Zbl 1301.76083 |
| [112] | Marletta, M., Neumann-Dirichlet maps and analysis of spectral pollution for non-self-adjoint elliptic PDEs with real essential spectrum, IMA J. Numer. Anal., 30, 4, 917-939, 2010 · Zbl 1206.65245 |
| [113] | Marletta, M.; Scheichl, R., Eigenvalues in spectral gaps of differential operators, Journal of Spectral Theory, 2, 3, 293-320, 2012 · Zbl 1256.65079 |
| [114] | Mattila, P., Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, 1995, Cambridge: Cambridge University Press, Cambridge · Zbl 0819.28004 |
| [115] | McMullen, C.: Families of rational maps and iterative root-finding algorithms. Annals of Mathematics (2) 125(3), 467-493 (1987) · Zbl 0634.30028 |
| [116] | McMullen, C., Braiding of the attractor and the failure of iterative algorithms, Invent. Math., 91, 2, 259-272, 1988 · Zbl 0654.58023 |
| [117] | Miekkala, U.; Nevanlinna, O., Iterative solution of systems of linear differential equations, Acta Numerica, 5, 1, 259-307, 1996 · Zbl 0870.65059 |
| [118] | Müller, V., Local behaviour of the polynomial calculus of operators, J. Reine Angew. Math., 430, 61-68, 1992 · Zbl 0749.47002 |
| [119] | Müller, V.: Spectral theory of linear operators: and spectral systems in Banach algebras, vol. 139. Springer Science & Business Media (2007) · Zbl 1208.47001 |
| [120] | Naumis, G.G., Barraza-Lopez, S., Oliva-Leyva, M., Terrones, H.: Electronic and optical properties of strained graphene and other strained 2D materials: a review. Reports on Progress in Physics 80(9), 096,501 (2017) |
| [121] | Nevanlinna, O., Linear acceleration of Picard-Lindelöf iteration, Numerische Mathematik, 57, 1, 147-156, 1990 · Zbl 0697.65058 |
| [122] | Nevanlinna, O.: Convergence of iterations for linear equations. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1993) · Zbl 0846.47008 |
| [123] | Nevanlinna, O., Hessenberg matrices in Krylov subspaces and the computation of the spectrum, Numer. Funct. Anal. Optim., 16, 3-4, 443-473, 1995 · Zbl 0837.65056 |
| [124] | Nevanlinna, O., Computing the spectrum and representing the resolvent, Numerical Functional Analysis and Optimization, 30, 9-10, 1025-1047, 2009 · Zbl 1188.47005 |
| [125] | New, G.; Yates, M.; Woerdman, J.; McDonald, G., Diffractive origin of fractal resonator modes, Optics communications, 193, 1-6, 261-266, 2001 |
| [126] | Olver, S., GMRES for the differentiation operator, SIAM journal on numerical analysis, 47, 5, 3359-3373, 2009 · Zbl 1202.65036 |
| [127] | Olver, S.: ApproxFun.jl v0.8. github (online) https://github.com/JuliaApproximation/ApproxFun.jl (2018) |
| [128] | Olver, S.; Townsend, A., A fast and well-conditioned spectral method, SIAM Review, 55, 3, 462-489, 2013 · Zbl 1273.65182 |
| [129] | Olver, S., Townsend, A.: A Practical Framework for Infinite-dimensional Linear Algebra. In: Proceedings of the 1st First Workshop for High Performance Technical Computing in Dynamic Languages, HPTCDL ’14, pp. 57-62. IEEE Press, Piscataway, NJ, USA (2014) |
| [130] | Olver, S., Webb, M.: SpectralMeasures.jl. github (online) https://github.com/JuliaApproximation/SpectralMeasures.jl (2018) |
| [131] | Orland, GH, On a class of operators, Proc. Amer. Math. Soc., 15, 75-79, 1964 · Zbl 0123.31601 |
| [132] | Pokrzywa, A., Method of orthogonal projections and approximation of the spectrum of a bounded operator, Studia Mathematica, 65, 1, 21-29, 1979 · Zbl 0437.47004 |
| [133] | Ponomarenko, L.; Gorbachev, R.; Yu, G.; Elias, D.; Jalil, R.; Patel, A.; Mishchenko, A.; Mayorov, A.; Woods, C.; Wallbank, J., Cloning of Dirac fermions in graphene superlattices, Nature, 497, 7451, 594-597, 2013 |
| [134] | Puelz, C.; Embree, M.; Fillman, J., Spectral Approximation for Quasiperiodic Jacobi Operators, Integral Equations and Operator Theory, 82, 4, 533-554, 2015 · Zbl 1325.47069 |
| [135] | Puig, J., Cantor spectrum for the almost Mathieu operator, Communications in Mathematical Physics, 244, 2, 297-309, 2004 · Zbl 1075.39021 |
| [136] | Putnam, CR, Operators satisfying a \(G_1\) condition, Pacific Journal of Mathematics, 84, 2, 413-426, 1979 · Zbl 0437.47014 |
| [137] | Rappaz, J., Sanchez Hubert, J., Sanchez Palencia, E., Vassiliev, D.: On spectral pollution in the finite element approximation of thin elastic “membrane” shells. Numerische Mathematik 75(4), 473-500 (1997) · Zbl 0874.73066 |
| [138] | Rivera, JA; Galvin, TC; Steinforth, AW; Eden, JG, Fractal modes and multi-beam generation from hybrid microlaser resonators, Nature communications, 9, 1, 1-8, 2018 |
| [139] | Roman-Taboada, P., Naumis, G.G.: Spectral butterfly, mixed Dirac-Schrödinger fermion behavior, and topological states in armchair uniaxial strained graphene. Physical Review B 90(19), 195,435 (2014) |
| [140] | Rösler, F., On the Solvability Complexity Index for Unbounded Selfadjoint and Schrödinger Operators, Integral Equations and Operator Theory, 91, 6, 54, 2019 · Zbl 1435.35269 |
| [141] | Rösler, F., Stepanenko, A.: Computing eigenvalues of the Laplacian on rough domains. arXiv preprint arXiv:2104.09444 (2021) |
| [142] | Salinas, N., Operators with essentially disconnected spectrum, Acta Sci. Math. (Szeged), 33, 193-205, 1972 · Zbl 0248.47001 |
| [143] | Shargorodsky, E., On the level sets of the resolvent norm of a linear operator, Bull. Lond. Math. Soc., 40, 3, 493-504, 2008 · Zbl 1147.47007 |
| [144] | Shargorodsky, E., On the limit behaviour of second order relative spectra of self-adjoint operators, Journal of Spectral Theory, 3, 4, 535-552, 2013 · Zbl 1327.47003 |
| [145] | Shechtman, D.; Blech, I.; Gratias, D.; Cahn, JW, Metallic phase with long-range orientational order and no translational symmetry, Physical Review Letters, 53, 1951-1953, 1984 |
| [146] | Simon, B., Schrödinger operators in the twenty-first century, Mathematical physics, 2000, 283-288, 2000 · Zbl 1074.81521 |
| [147] | Sire, C., Electronic spectrum of a 2D quasi-crystal related to the octagonal quasi-periodic tiling, EPL (Europhysics Letters), 10, 5, 483, 1989 |
| [148] | Smale, S., The fundamental theorem of algebra and complexity theory, American Mathematical Society. Bulletin., 4, 1, 1-36, 1981 · Zbl 0456.12012 |
| [149] | Smale, S.: On the efficiency of algorithms of analysis. Bull. Amer. Math. Soc. (N.S.) 13(2), 87-121 (1985) · Zbl 0592.65032 |
| [150] | Smale, S.: Complexity theory and numerical analysis. In: Acta numerica, 1997, Acta Numer., vol. 6, pp. 523-551. Cambridge Univ. Press, Cambridge (1997) · Zbl 0883.65125 |
| [151] | Stadnik, Z.M.: Physical properties of quasicrystals, vol. 126. Springer Science & Business Media (2012) |
| [152] | Stampfli, JG; Williams, JP, Growth conditions and the numerical range in a Banach algebra, Tohoku Mathematical Journal, Second Series, 20, 4, 417-424, 1968 · Zbl 0175.43902 |
| [153] | Stewart, D., Towards numerically estimating Hausdorff dimensions, The ANZIAM Journal, 42, 4, 451-461, 2001 · Zbl 0992.65005 |
| [154] | Sütő, A., Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, Journal of Statistical Physics, 56, 3-4, 525-531, 1989 · Zbl 0712.58046 |
| [155] | Szegő, G., Beiträge zur Theorie der Toeplitzschen Formen, Mathematische Zeitschrift, 6, 3-4, 167-202, 1920 · JFM 47.0391.04 |
| [156] | Tanese, D., Gurevich, E., Baboux, F., Jacqmin, T., Lemaître, A., Galopin, E., Sagnes, I., Amo, A., Bloch, J., Akkermans, E.: Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential. Physical Review Letters 112(14), 146,404 (2014) |
| [157] | Thouless, D., Bandwidths for a quasiperiodic tight-binding model, Physical Review B, 28, 8, 4272, 1983 |
| [158] | Thouless, D., Scaling for the discrete Mathieu equation, Communications in mathematical physics, 127, 1, 187-193, 1990 · Zbl 0692.34021 |
| [159] | Thouless, D., Tan, Y.: Total bandwidth for the Harper equation. III. corrections to scaling. Journal of Physics A: Mathematical and General 24(17), 4055 (1991) |
| [160] | Thouless, DJ; Kohmoto, M.; Nightingale, MP; den Nijs, M., Quantized Hall conductance in a two-dimensional periodic potential, Physical Review Letters, 49, 6, 405, 1982 |
| [161] | Torres-Herrera, E., Santos, L.F.: Dynamics at the many-body localization transition. Physical Review B 92(1), 014,208 (2015) |
| [162] | Trefethen, LN; Embree, M., Spectra and pseudospectra, 2005, Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 1085.15009 |
| [163] | Tucker, W.: Validated numerics: a short introduction to rigorous computations. Princeton University Press (2011) · Zbl 1231.65077 |
| [164] | Turing, A.M.: On Computable Numbers, with an Application to the Entscheidungsproblem. Proc. London Math. Soc. (2) 42(3), 230-265 (1936) · Zbl 0016.09701 |
| [165] | Vardeny, ZV; Nahata, A.; Agrawal, A., Optics of photonic quasicrystals. Nature Phot., 7, 3, 177-187, 2013 |
| [166] | Webb, M.; Olver, S., Spectra of Jacobi operators via connection coefficient matrices, Communications in Mathematical Physics, 382, 2, 657-707, 2021 · Zbl 1518.47055 |
| [167] | Weinberger, S., Computers, Rigidity, and Moduli: The Large-Scale Fractal Geometry of Riemannian Moduli Space, 2004, USA: Princeton University Press, USA |
| [168] | Zhao, S., On the spurious solutions in the high-order finite difference methods for eigenvalue problems, Computer methods in applied mechanics and engineering, 196, 49-52, 5031-5046, 2007 · Zbl 1173.74462 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
