On the solvability complexity index, the \(n\)-pseudospectrum and approximations of spectra of operators. (English) Zbl 1210.47013
Author’s abstract: We show that it is possible to compute spectra and pseudospectra of linear operators on separable Hilbert spaces given their matrix elements. The core in the theory is pseudospectral analysis and, in particular, the \(n\)-pseudospectrum and the residual pseudospectrum. We also introduce a new classification tool for spectral problems, namely, the Solvability Complexity Index. This index is an indicator of the “difficultness” of different computational spectral problems.
Reviewer: Rudolf Kodnár (Bratislava)
MSC:
| 68Q15 | Complexity classes (hierarchies, relations among complexity classes, etc.) |
| 47A10 | Spectrum, resolvent |
| 47A75 | Eigenvalue problems for linear operators |
| 46L05 | General theory of \(C^*\)-algebras |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 81Q12 | Nonselfadjoint operator theory in quantum theory including creation and destruction operators |
| 65J10 | Numerical solutions to equations with linear operators |
Software:
EigtoolReferences:
| [1] | William Arveson, Discretized CCR algebras, J. Operator Theory 26 (1991), no. 2, 225 – 239. · Zbl 0797.46050 |
| [2] | William Arveson, Improper filtrations for \?*-algebras: spectra of unilateral tridiagonal operators, Acta Sci. Math. (Szeged) 57 (1993), no. 1-4, 11 – 24. · Zbl 0819.46044 |
| [3] | William Arveson, \?*-algebras and numerical linear algebra, J. Funct. Anal. 122 (1994), no. 2, 333 – 360. · Zbl 0802.46069 · doi:10.1006/jfan.1994.1072 |
| [4] | William Arveson, The role of \?*-algebras in infinite-dimensional numerical linear algebra, \?*-algebras: 1943 – 1993 (San Antonio, TX, 1993) Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 114 – 129. · doi:10.1090/conm/167/1292012 |
| [5] | Erik Bédos, On filtrations for \?*-algebras, Houston J. Math. 20 (1994), no. 1, 63 – 74. · Zbl 0808.46081 |
| [6] | Erik Bédos, On Følner nets, Szegő’s theorem and other eigenvalue distribution theorems, Exposition. Math. 15 (1997), no. 3, 193 – 228. Erratum: ”On Følner nets, Szegő’s theorem and other eigenvalue distribution theorems” [Exposition. Math. 15 (1997), no. 3, 193 – 228; MR1458766 (98h:47065a)] by E. Bédos, Exposition. Math. 15 (1997), no. 4, 384. · Zbl 0886.46056 |
| [7] | Carl M. Bender, Properties of non-Hermitian quantum field theories, Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002), 2003, pp. 997 – 1008 (English, with English and French summaries). · Zbl 1069.81030 |
| [8] | Carl M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70 (2007), no. 6, 947 – 1018. · Zbl 1143.81312 · doi:10.1088/0034-4885/70/6/R03 |
| [9] | Albrecht Böttcher, Pseudospectra and singular values of large convolution operators, J. Integral Equations Appl. 6 (1994), no. 3, 267 – 301. · Zbl 0819.45002 · doi:10.1216/jiea/1181075815 |
| [10] | Albrecht Böttcher, \?*-algebras in numerical analysis, Irish Math. Soc. Bull. 45 (2000), 57 – 133. · Zbl 1061.46050 |
| [11] | A. Böttcher, A. V. Chithra, and M. N. N. Namboodiri, Approximation of approximation numbers by truncation, Integral Equations Operator Theory 39 (2001), no. 4, 387 – 395. · Zbl 1021.47009 · doi:10.1007/BF01203320 |
| [12] | Albrecht Böttcher and Sergei M. Grudsky, Spectral properties of banded Toeplitz matrices, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. · Zbl 1089.47001 |
| [13] | Albrecht Böttcher and Bernd Silbermann, Analysis of Toeplitz operators, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006. Prepared jointly with Alexei Karlovich. · Zbl 1098.47002 |
| [14] | Lyonell Boulton, Projection methods for discrete Schrödinger operators, Proc. London Math. Soc. (3) 88 (2004), no. 2, 526 – 544. · Zbl 1063.47023 · doi:10.1112/S0024611503014448 |
| [15] | Lyonell Boulton, Limiting set of second order spectra, Math. Comp. 75 (2006), no. 255, 1367 – 1382. · Zbl 1102.47019 |
| [16] | B. M. Brown and M. Marletta, Spectral inclusion and spectral exactness for singular non-self-adjoint Hamiltonian systems, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), no. 2036, 1987 – 2009. · Zbl 1078.34064 · doi:10.1098/rspa.2002.1106 |
| [17] | L. G. Brown, Lidskiĭ’s theorem in the type \?\? case, Geometric methods in operator algebras (Kyoto, 1983) Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., Harlow, 1986, pp. 1 – 35. |
| [18] | Nathanial P. Brown, AF embeddings and the numerical computation of spectra in irrational rotation algebras, Numer. Funct. Anal. Optim. 27 (2006), no. 5-6, 517 – 528. · Zbl 1105.65060 · doi:10.1080/01630560600790785 |
| [19] | Nathanial P. Brown, Quasi-diagonality and the finite section method, Math. Comp. 76 (2007), no. 257, 339 – 360. · Zbl 1113.65057 |
| [20] | H. O. Cordes and J. P. Labrousse, The invariance of the index in the metric space of closed operators, J. Math. Mech. 12 (1963), 693 – 719. · Zbl 0148.12402 |
| [21] | E. B. Davies, Semi-classical states for non-self-adjoint Schrödinger operators, Comm. Math. Phys. 200 (1999), no. 1, 35 – 41. · Zbl 0921.47060 · doi:10.1007/s002200050521 |
| [22] | E. B. Davies, Spectral properties of random non-self-adjoint matrices and operators, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2005, 191 – 206. · Zbl 1014.47002 · doi:10.1098/rspa.2000.0662 |
| [23] | E. B. Davies, Non-self-adjoint differential operators, Bull. London Math. Soc. 34 (2002), no. 5, 513 – 532. · Zbl 1052.47042 · doi:10.1112/S0024609302001248 |
| [24] | E. B. Davies and A. B. J. Kuijlaars, Spectral asymptotics of the non-self-adjoint harmonic oscillator, J. London Math. Soc. (2) 70 (2004), no. 2, 420 – 426. · Zbl 1073.34093 · doi:10.1112/S0024610704005381 |
| [25] | Michael Levitin and Eugene Shargorodsky, Spectral pollution and second-order relative spectra for self-adjoint operators, IMA J. Numer. Anal. 24 (2004), no. 3, 393 – 416. , https://doi.org/10.1093/imanum/24.3.393 E. B. Davies and M. Plum, Spectral pollution, IMA J. Numer. Anal. 24 (2004), no. 3, 417 – 438. · Zbl 1062.65056 · doi:10.1093/imanum/24.3.417 |
| [26] | P. Deift, L. C. Li, and C. Tomei, Toda flows with infinitely many variables, J. Funct. Anal. 64 (1985), no. 3, 358 – 402. · Zbl 0615.58016 · doi:10.1016/0022-1236(85)90065-5 |
| [27] | Nils Dencker, Johannes Sjöstrand, and Maciej Zworski, Pseudospectra of semiclassical (pseudo-) differential operators, Comm. Pure Appl. Math. 57 (2004), no. 3, 384 – 415. · Zbl 1054.35035 · doi:10.1002/cpa.20004 |
| [28] | Trond Digernes, V. S. Varadarajan, and S. R. S. Varadhan, Finite approximations to quantum systems, Rev. Math. Phys. 6 (1994), no. 4, 621 – 648. · Zbl 0855.47046 · doi:10.1142/S0129055X94000213 |
| [29] | Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1963 original; A Wiley-Interscience Publication. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part III, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral operators; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1971 original; A Wiley-Interscience Publication. · Zbl 0635.47002 |
| [30] | Bent Fuglede and Richard V. Kadison, Determinant theory in finite factors, Ann. of Math. (2) 55 (1952), 520 – 530. · Zbl 0046.33604 · doi:10.2307/1969645 |
| [31] | Uffe Haagerup and Hanne Schultz, Brown measures of unbounded operators affiliated with a finite von Neumann algebra, Math. Scand. 100 (2007), no. 2, 209 – 263. · Zbl 1168.46039 · doi:10.7146/math.scand.a-15023 |
| [32] | Roland Hagen, Steffen Roch, and Bernd Silbermann, \?*-algebras and numerical analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 236, Marcel Dekker, Inc., New York, 2001. · Zbl 0964.65055 |
| [33] | Anders C. Hansen, On the approximation of spectra of linear operators on Hilbert spaces, J. Funct. Anal. 254 (2008), no. 8, 2092 – 2126. · Zbl 1138.47002 · doi:10.1016/j.jfa.2008.01.006 |
| [34] | -, Infinite-dimensional numerical linear algebra: theory and applications, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science (2010). |
| [35] | N. Hatano and D. R. Nelson, Localization transitions in non-Hermitian quantum mechanics, Phys. Rev. Lett. 77 (1996), no. 3, 570-573. |
| [36] | -, Vortex pinning and non-Hermitian quantum mechanics, Phys. Rev. B 56 (1997), no. 14, 8651-8673. |
| [37] | W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. London Mathematical Society Monographs, No. 9. · Zbl 0419.31001 |
| [38] | Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. · Zbl 0836.47009 |
| [39] | Michael Levitin and Eugene Shargorodsky, Spectral pollution and second-order relative spectra for self-adjoint operators, IMA J. Numer. Anal. 24 (2004), no. 3, 393 – 416. , https://doi.org/10.1093/imanum/24.3.393 E. B. Davies and M. Plum, Spectral pollution, IMA J. Numer. Anal. 24 (2004), no. 3, 417 – 438. · Zbl 1062.65056 · doi:10.1093/imanum/24.3.417 |
| [40] | Marco Marletta, Roman Shterenberg, and Rudi Weikard, On the inverse resonance problem for Schrödinger operators, Comm. Math. Phys. 295 (2010), no. 2, 465 – 484. · Zbl 1207.34117 · doi:10.1007/s00220-009-0928-8 |
| [41] | O. Nevanlinna, Computing the spectrum and representing the resolvent, Numer. Funct. Anal. Optim. 30 (2009), no. 9-10, 1025 – 1047. · Zbl 1188.47005 · doi:10.1080/01630560903393162 |
| [42] | Eugene Shargorodsky, Geometry of higher order relative spectra and projection methods, J. Operator Theory 44 (2000), no. 1, 43 – 62. · Zbl 0994.47003 |
| [43] | E. Shargorodsky, On the level sets of the resolvent norm of a linear operator, Bull. Lond. Math. Soc. 40 (2008), no. 3, 493 – 504. · Zbl 1147.47007 · doi:10.1112/blms/bdn038 |
| [44] | Michael Shub and Steve Smale, Complexity of Bézout’s theorem. I. Geometric aspects, J. Amer. Math. Soc. 6 (1993), no. 2, 459 – 501. · Zbl 0821.65035 |
| [45] | Steve Smale, The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 1, 1 – 36. · Zbl 0456.12012 |
| [46] | Steve Smale, Complexity theory and numerical analysis, Acta numerica, 1997, Acta Numer., vol. 6, Cambridge Univ. Press, Cambridge, 1997, pp. 523 – 551. · Zbl 0883.65125 · doi:10.1017/S0962492900002774 |
| [47] | G. Szegő, Beiträge zur Theorie der Toeplitzschen Formen, Math. Z. 6 (1920), no. 3-4, 167 – 202 (German). · JFM 47.0391.04 · doi:10.1007/BF01199955 |
| [48] | Lloyd N. Trefethen, Computation of pseudospectra, Acta numerica, 1999, Acta Numer., vol. 8, Cambridge Univ. Press, Cambridge, 1999, pp. 247 – 295. · Zbl 0945.65039 · doi:10.1017/S0962492900002932 |
| [49] | Lloyd N. Trefethen and S. J. Chapman, Wave packet pseudomodes of twisted Toeplitz matrices, Comm. Pure Appl. Math. 57 (2004), no. 9, 1233 – 1264. · Zbl 1055.15014 · doi:10.1002/cpa.20034 |
| [50] | Lloyd N. Trefethen and Mark Embree, Spectra and pseudospectra, Princeton University Press, Princeton, NJ, 2005. The behavior of nonnormal matrices and operators. · Zbl 1085.15009 |
| [51] | Sergei Treil, An operator Corona theorem, Indiana Univ. Math. J. 53 (2004), no. 6, 1763 – 1780. · Zbl 1079.30049 · doi:10.1512/iumj.2004.53.2640 |
| [52] | Maciej Zworski, Resonances in physics and geometry, Notices Amer. Math. Soc. 46 (1999), no. 3, 319 – 328. · Zbl 1177.58021 |
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.
