Computing spectral measures of self-adjoint operators

Using the resolvent operator, we develop an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators. The algorithm can achieve arbitrarily high-orders of convergence in terms of a smoothing parameter for computing spectral measures of general differential, integral, and lattice operators. Explicit pointwise and $L^p$-error bounds are derived in terms of the local regularity of the measure. We provide numerical examples, including a partial differential operator, and compute one thousand eigenvalues of a Dirac operator to near machine precision without spectral pollution. The algorithm is publicly available in $\texttt{SpecSolve}$, which is a software package written in MATLAB.

[1]  Steven Pruess,et al.  Mathematical software for Sturm-Liouville problems , 1993, TOMS.

[2]  S. Joe Discrete Collocation Methods for Second Kind Fredholm Integral Equations , 1985 .

[3]  A. Rouault,et al.  Sum rules via large deviations , 2014, 1407.1384.

[4]  B. M. Levitan,et al.  Introduction to spectral theory : selfadjoint ordinary differential operators , 1975 .

[5]  David Damanik,et al.  Singular Continuous Spectrum for a Class of Substitution Hamiltonians II , 2000 .

[6]  M. Embree,et al.  Spectral properties of Schrödinger operators arising in the study of quasicrystals , 2012, 1210.5753.

[7]  G. Kallianpur,et al.  Spectral theory of stationary H-valued processes , 1971 .

[8]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[9]  The foundations of spectral computations via the Solvability Complexity Index hierarchy: Part II , 2019, 1908.09598.

[10]  P. W. Langhoff Stieltjes-Tchebycheff Moment-Theory Approach to Photoeffect Studies in Hilbert Space , 1980 .

[11]  Lloyd N. Trefethen,et al.  An Extension of Chebfun to Two Dimensions , 2013, SIAM J. Sci. Comput..

[12]  G. Wellein,et al.  The kernel polynomial method , 2005, cond-mat/0504627.

[13]  Matthew J. Colbrook,et al.  How to Compute Spectra with Error Control. , 2019, Physical review letters.

[14]  S. P. Goldman,et al.  Application of discrete-basis-set methods to the Dirac equation , 1981 .

[15]  Jacob T. Schwartz,et al.  Linear operators. Part II. Spectral theory , 2003 .

[16]  O. Kallenberg Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[17]  E. Parzen On Consistent Estimates of the Spectrum of a Stationary Time Series , 1957 .

[18]  P. Deift Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , 2000 .

[19]  R. Silver,et al.  DENSITIES OF STATES OF MEGA-DIMENSIONAL HAMILTONIAN MATRICES , 1994 .

[20]  J. W. Silverstein,et al.  Spectral Analysis of Large Dimensional Random Matrices , 2009 .

[21]  T. Trogdon,et al.  Numerical Inverse Scattering for the Toda Lattice , 2015, 1508.01788.

[22]  Alexandre B. Tsybakov,et al.  Introduction to Nonparametric Estimation , 2008, Springer series in statistics.

[23]  F. Chatelin Spectral approximation of linear operators , 2011 .

[24]  Jon Wilkening,et al.  A Spectral Transform Method for Singular Sturm-Liouville Problems with Applications to Energy Diffusion in Plasma Physics , 2013, SIAM J. Appl. Math..

[25]  S. Olver,et al.  Spectra of Jacobi Operators via Connection Coefficient Matrices , 2017, Communications in Mathematical Physics.

[26]  P. Nevai,et al.  Orthogonal polynomials, measures and recurrence relations , 1986 .

[27]  M. Priestley Basic Considerations in the Estimation of Spectra , 1962 .

[28]  T. Trogdon,et al.  Numerical inverse scattering for the Korteweg–de Vries and modified Korteweg–de Vries equations , 2012 .

[29]  W. McLean Strongly Elliptic Systems and Boundary Integral Equations , 2000 .

[30]  B. Hall Quantum Theory for Mathematicians , 2013 .

[31]  G. Quispel,et al.  Splitting methods , 2002, Acta Numerica.

[32]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

[33]  E. Parzen Mathematical Considerations in the Estimation of Spectra , 1961 .

[34]  David Ruelle,et al.  A remark on bound states in potential-scattering theory , 1969 .

[35]  N. Barnea,et al.  TOPICAL REVIEW: The Lorentz integral transform (LIT) method and its applications to perturbation-induced reactions , 2007, 0708.2803.

[36]  Alex Townsend,et al.  FEAST for Differential Eigenvalue Problems , 2019, SIAM J. Numer. Anal..

[37]  Mathieu Lewin,et al.  Spectral pollution and how to avoid it , 2008, 0812.2153.

[38]  B. Simon,et al.  Sum rules for Jacobi matrices and their applications to spectral theory , 2001, math-ph/0112008.

[39]  V M Shabaev,et al.  Dual kinetic balance approach to basis-set expansions for the dirac equation. , 2004, Physical review letters.

[40]  Werner Kutzelnigg,et al.  RELATIVISTIC ONE-ELECTRON HAMILTONIANS 'FOR ELECTRONS ONLY' AND THE VARIATIONAL TREATMENT OF THE DIRAC EQUATION , 1997 .

[41]  H. Hellmann,et al.  A New Approximation Method in the Problem of Many Electrons , 1935 .

[42]  Steven Pruess,et al.  Computing the spectral function for singular Sturm-Liouville problems , 2005 .

[43]  William D. Shoaff,et al.  Parallel Computation of Sturm-Liouville Spectral Density Functions , 1994, Parallel Algorithms Appl..

[44]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[45]  András Sütő,et al.  Singular continuous spectrum on a cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian , 1989 .

[46]  K. F. Wojciechowski,et al.  Metal Surface Electron Physics , 1996 .

[47]  The Recursion Method and the Estimation of Local Densities of States , 1984 .

[48]  Bernd Thaller,et al.  The Dirac Equation , 1992 .

[49]  E. Stein,et al.  Real Analysis: Measure Theory, Integration, and Hilbert Spaces , 2005 .

[50]  W. Kutzelnigg Basis set expansion of the dirac operator without variational collapse , 1984 .

[51]  Yousef Saad,et al.  Approximating Spectral Densities of Large Matrices , 2013, SIAM Rev..

[52]  M. Embree,et al.  Spectral Approximation for Quasiperiodic Jacobi Operators , 2014, Integral Equations and Operator Theory.

[53]  L. Trefethen,et al.  Numerical linear algebra , 1997 .

[54]  Matthew J. Colbrook,et al.  Computing Spectral Measures and Spectral Types , 2019, Communications in Mathematical Physics.

[55]  V. Enss Asymptotic completeness for quantum mechanical potential scattering , 1978 .

[56]  A. Böttcher,et al.  Introduction to Large Truncated Toeplitz Matrices , 1998 .

[57]  B. Simon,et al.  Duality and singular continuous spectrum in the almost Mathieu equation , 1997 .

[58]  Lloyd N. Trefethen,et al.  The Exponentially Convergent Trapezoidal Rule , 2014, SIAM Rev..

[59]  New characterizations of spectral density functions for singular Sturm-Liouville problems , 2008 .

[60]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[61]  K. Gustafson Operator spectral states , 1997 .

[62]  Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics , 2005, math/0502486.

[63]  The Colored Hofstadter Butterfly for the Honeycomb Lattice , 2014, 1403.1270.

[64]  K. Novoselov Nobel Lecture: Graphene: Materials in the Flatland , 2011 .

[65]  Austin R. Benson,et al.  Network Density of States , 2019, KDD.

[66]  C. Lubich From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis , 2008 .

[67]  V. Heine,et al.  Electronic structure based on the local atomic environment for tight-binding bands. II , 1972 .

[68]  B. Simon Szegő's Theorem and Its Descendants: Spectral Theory for L 2 Perturbations of Orthogonal Polynomials , 2010 .

[69]  M. Hochbruck,et al.  Exponential integrators , 2010, Acta Numerica.

[70]  A scattering problem by means of the spectral representation of Green's function for a layered acoustic half-space , 2000 .

[71]  Murray Rosenblatt,et al.  Stochastic Curve Estimation , 1991 .

[72]  Jost functions and Jost solutions for Jacobi matrices, III. Asymptotic series for decay and meromorphicity , 2005, math/0503392.

[73]  G. Weiss,et al.  EIGENFUNCTION EXPANSIONS. Associated with Second-order Differential Equations. Part I. , 1962 .

[74]  K. Friedrichs On the perturbation of continuous spectra , 1948 .

[75]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[76]  K. Thylwe,et al.  Approximate Bound States Solution of the Hellmann Potential , 2013 .

[77]  Kenji Watanabe,et al.  Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene , 2019, Nature.

[78]  G. Teschl Jacobi Operators and Completely Integrable Nonlinear Lattices , 1999 .

[79]  Talman Minimax principle for the Dirac equation. , 1986, Physical review letters.

[80]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[81]  B. Silverman Density estimation for statistics and data analysis , 1986 .

[82]  A. Avila,et al.  The Ten Martini Problem , 2009 .

[83]  R. Senoussi,et al.  Semigroup stationary processes and spectral representation , 2003 .

[84]  K. Hoffman Banach Spaces of Analytic Functions , 1962 .

[85]  B. Simon,et al.  Schrödinger Semigroups , 2007 .

[86]  On Calculating Response Functions Via Their Lorentz Integral Transforms , 2019, Few-Body Systems.

[87]  Tosio Kato Perturbation theory for linear operators , 1966 .

[88]  V. Marchenko Sturm-Liouville Operators and Applications , 1986 .

[89]  Steven Pruess,et al.  The computation of spectral density functions for singular Sturm-Liouville problems involving simple continuous spectra , 1998, TOMS.

[90]  Paul M. Goldbart,et al.  Mathematics for Physics: A Guided Tour for Graduate Students , 2009 .

[91]  Z. Strakos,et al.  Krylov Subspace Methods: Principles and Analysis , 2012 .

[92]  Bernard W. Silverman,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[93]  D. Hofstadter Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields , 1976 .

[94]  E. Stein,et al.  Functional Analysis: Introduction to Further Topics in Analysis , 2011 .

[95]  H. Cramér On Some Classes of Nonstationary Stochastic Processes , 1961 .

[96]  F. Guinea,et al.  The electronic properties of graphene , 2007, Reviews of Modern Physics.

[97]  W. Amrein,et al.  Characterization of bound states and scattering states in quantum mechanics , 1973 .

[98]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[99]  R. Carmona,et al.  Spectral Theory of Random Schrödinger Operators , 1990 .

[100]  Petre Stoica,et al.  Spectral Analysis of Signals , 2009 .

[101]  M. Potemski,et al.  Cloning of Dirac fermions in graphene superlattices , 2013, Nature.

[102]  K. Dyall,et al.  Kinetic balance and variational bounds failure in the solution of the Dirac equation in a finite Gaussian basis set , 1990 .

[103]  Matthew P. Wand,et al.  Kernel Smoothing , 1995 .

[104]  M. Rosenblatt Remarks on Some Nonparametric Estimates of a Density Function , 1956 .

[105]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[106]  Second Order Linear Evolution Equations with General Dissipation , 2018, Applied Mathematics & Optimization.

[107]  D. Tannor,et al.  Introduction to Quantum Mechanics: A Time-Dependent Perspective , 2006 .

[108]  E. Parzen On Estimation of a Probability Density Function and Mode , 1962 .

[109]  T. Chihara,et al.  An Introduction to Orthogonal Polynomials , 1979 .

[110]  Sheehan Olver,et al.  A Fast and Well-Conditioned Spectral Method , 2012, SIAM Rev..

[111]  G. Orlandini,et al.  Response functions from integral transforms with a Lorentz kernel , 1994, nucl-th/9409004.

[112]  ON THE SPECTRAL THEORY OF SINGULAR INTEGRAL OPERATORS. , 1958, Proceedings of the National Academy of Sciences of the United States of America.

[113]  Richard E. Stanton,et al.  Kinetic balance: A partial solution to the problem of variational safety in Dirac calculations , 1984 .

[114]  I. M. Glazman Direct methods of qualitative spectral analysis of singular differential operators , 1965 .