Weyl-Titchmarsh M-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators

We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general Dirac-type operators on half-lines and on R. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operators and use it to prove a Borg-type theorem.

[1]  On the number of square integrable solutions and self-adjointness of symmetric first order systems of differential equations , 2000, math/0012080.

[2]  L. Dickey.  Soliton Equations and Hamiltonian Systems , 2003 .

[3]  L. Amour.  Inverse spectral theory for the AKNS system with separated boundary conditions , 1993 .

[4]  Russell Johnson,et al.  The algebraic-geometric AKNS potentials , 1987, Ergodic Theory and Dynamical Systems.

[5]  A new approach to inverse spectral theory, III. Short-range potentials , 2000 .

[6]  W. Goldberg.  On the determination of a Hill's equation from its spectrum , 1974 .

[7]  B. Dubrovin,et al.  Matrix finite-zone operators , 1985 .

[8]  J. K. Shaw,et al.  Hamiltonian systems of limit point or limit circle type with both endpoints singular , 1983 .

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

[10]  V. I. Kogan,et al.  1.—On Square-integrable Solutions of Symmetric Systems of Differential Equations of Arbitrary Order. , 1976, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[11]  A. S. Meligy,et al.  On the function , 1963, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  F. Gesztesy,et al.  A CHARACTERIZATION OF ALL ELLIPTIC SOLUTIONS OF THE AKNS HIERARCHY , 1997, solv-int/9705018.

[13]  B. Harris.  The asymptotic form of the Titchmarsh-Weyl m-function associated with a Dirac system , 1985 .

[14]  B. Simon,et al.  Absolute summability of the trace relation for certain Schrödinger operators , 1995 .

[15]  J. K. Shaw,et al.  ON THE SPECTRUM OF A SINGULAR HAMILTONIAN SYSTEM , 1982 .

[16]  C. Tretter,et al.  Direct and inverse spectral problem for a system of differential equations depending rationally on the spectral parameter , 2001 .

[17]  F. Gesztesy.  Some applications of commutation methods , 1989 .

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

[19]  J. K. Shaw,et al.  The Asymptotic Form of the Titchmarsh‐Weyl Coefficient for Dirac Systems , 1983 .

[20]  P. Deift,et al.  The Absolutely Continuous Spectrum in One Dimension , 1983 .

[21]  J. Guillot,et al.  Isospectral sets for akns systems on the unit interval with generalized periodic boundary conditions , 1996 .

[22]  F. Gesztesy,et al.  On Matrix–Valued Herglotz Functions , 1997, funct-an/9712004.

[23]  The Spectral Shift Operator , 1999, math/9901112.

[24]  F. Gesztesy,et al.  On Local Borg–Marchenko Uniqueness Results , 2000 .

[25]  L. Sakhnovich.  Evolution of spectral data and nonlinear equations , 1988 .

[26]  B. Grébert.  Inverse scattering for the Dirac operator on the real line , 1992 .

[27]  R. Carey.  A unitary invariant for pairs of self-adjoint operators. , 1976 .

[28]  I. Gohberg,et al.  Matrix and Operator Valued Functions: The Vladimir Petrovich Potapov Memorial Volume , 1994 .

[29]  M. Malamud,et al.  The Inverse Spectral Problem for First Order Systems on the Half Line , 1998, math/9805033.

[30]  Lev A. Sakhnovich,et al.  Spectral Theory of Canonical Differential Systems. Method of Operator Identities , 1999 .

[31]  Fritz Gesztesy,et al.  An Addendum to Krein's Formula , 1997 .

[32]  Y. Manin.  Matrix solitons and bundles over curves with singularities , 1978 .

[33]  F. V. Atkinson,et al.  Discrete and Continuous Boundary Problems , 1964 .

[34]  J. K. Shaw,et al.  Series representation and asymptotics for Titchmarsh-Weyl $m$-functions , 1989, Differential and Integral Equations.

[35]  H. Weyl,et al.  Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen , 1910 .

[36]  Some Applications of Operator-Valued Herglotz Functions , 1998, math/9802103.

[37]  G. Pólya,et al.  Problems and theorems in analysis , 1983 .

[38]  Helge Holden,et al.  Borg-Type Theorems for Matrix-Valued Schrödinger Operators , 1999 .

[39]  M. Kreĭn,et al.  Stability of Solutions of Differential Equations in Banach Spaces , 1974 .

[40]  The Ξ operator and its relation to Krein's spectral shift function , 1999, math/9904050.

[41]  Y. Kato,et al.  Algebraic and Spectral Methods for Nonlinear Wave Equations , 1990 .

[42]  J. Moser,et al.  The rotation number for almost periodic potentials , 1983 .

[43]  G. Pólya,et al.  Problems and Theorems in Analysis I: Series. Integral Calculus. Theory of Functions , 1976 .

[44]  L. Sakhnovich.  Inverse problems for equations systems , 1994 .

[45]  H. Dym,et al.  J-inner matrix functions, interpolation and inverse problems for canonical systems, III: More on the inverse monodromy problem , 2000 .

[46]  Levitan,et al.  Sturm―Liouville and Dirac Operators , 1990 .

[47]  Göran Borg.  Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe , 1946 .

[48]  F. Atkinson.  On the asymptotic behaviour of the Titchmarsh-Weyl m-coefficient and the spectral function for scalar second-order differential expressions , 1982 .

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

[50]  F. Atkinson.  On the order of magnitude of Titchmarsh-Weyl functions , 1988 .

[51]  F. Gesztesy,et al.  A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy , 1998 .

[52]  M. Krishna,et al.  Almost Periodicity of Some Random Potentials , 1988 .

[53]  Boris Dubrovin,et al.  Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties , 1977 .

[54]  W. N. Everitt.  On a Property of the M -Coefficient of a Secondorder Linear Differential Equation , 1972 .

[55]  N. S. Barnett,et al.  Private communication , 1969 .

[56]  B. Simon,et al.  HIGHER ORDER TRACE RELATIONS FOR SCHRÖDINGER OPERATORS , 1995 .

[57]  A. Sakhnovich.  Canonical systems and transfer matrix-functions , 1997 .

[58]  Christer Bennewitz,et al.  A Proof of the Local Borg–Marchenko Theorem , 2001 .

[59]  H. Dym,et al.  J-inner matrix functions, interpolation and inverse problems for canonical systems, I: Foundations , 1997 .

[60]  M. Malamud.  Borg type theorems for first-order systems on a finite interval , 1999 .

[61]  H. Holden,et al.  ON TRACE FORMULAS FOR SCHRODINGER-TYPE OPERATORS , 1997 .

[62]  T. Misyura.  An asymptotic formula for Weyl solutions of the Dirac equations , 1995 .

[63]  Russell Johnson.  m-functions and Floquet exponents for linear differential systems , 1987 .

[64]  A Trace Formula for One-Dimensional Dirac Operators , 2001 .

[65]  Alexei Rybkin,et al.  On the Trace Approach to the Inverse Scattering Problem in Dimension One , 2001, SIAM J. Math. Anal..

[66]  J. K. Shaw,et al.  Asymptotic phase, asymptotic modulus, and Titchmarsh-Weyl coefficient for a Dirac system , 1989 .

[67]  A. Sakhnovich,et al.  SPECTRAL FUNCTIONS OF A CANONICAL SYSTEM OF ORDER $ 2n$ , 1992 .

[68]  B. Simon,et al.  Stochastic Schrödinger operators and Jacobi matrices on the strip , 1988 .

[69]  A. Offord.  Introduction to the Theory of Fourier Integrals , 1938, Nature.

[70]  N. Aronszajn,et al.  On exponential representations of analytic functions in the upper half-plane with positive imaginary part , 1956 .

[71]  B. Grébert,et al.  Gaps of One-Dimensional Periodic AKNS Systems , 1993 .

[72]  Steve Clark,et al.  Weyl-titchmarsh M -function Asymptotics for Matrix-valued Schr¨odinger Operators , 2022 .

[73]  W. Craig.  The trace formula for Schrödinger operators on the line , 1989 .

[74]  V. Marchenko.  Nonlinear Equations and Operator Algebras , 1987 .

[75]  F. Gesztesy,et al.  Uniqueness Results for Matrix-Valued Schrodinger, Jacobi, and Dirac-Type Operators , 2000 .

[76]  Allan M. Krall,et al.  M (λ) theory for singular Hamiltonian systems with two singular points , 1989 .

[77]  J. K. Shaw,et al.  Embedded Half‐Bound States for Potentials of Wigner‐Von Neumann Type , 1991 .

[78]  B. Simon,et al.  Commutation methods applied to the mKdV-equation , 1991 .

[79]  G. Teschl,et al.  On isospectral sets of Jacobi operators , 1996 .

[80]  P. Yuditskii,et al.  Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum , 1995 .

[81]  J. K. Shaw,et al.  Inverse scattering on the line for a Dirac system , 1991 .

[82]  S. Clark.  On the absolutely continuous spectrum of a vector-matrix Dirac system , 1994, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[83]  Some Applications of the Spectral Shift Operator , 1999, math/9903186.

[84]  B. Després.  The Borg theorem for the vectorial Hill's equation , 1995 .

[85]  P. Deift,et al.  Almost periodic Schrödinger operators , 1983 .

[86]  Russell Johnson.  The recurrent Hill's equation , 1982 .

[87]  R. Giachetti,et al.  Spectral theory of second-order almost periodic differential operators and its relation to classes of nonlinear evolution equations , 1984 .

[88]  Israel Michael Sigal,et al.  Introduction to Spectral Theory , 1996 .

[89]  H. Hochstadt.  On the determination of a Hill's equation from its spectrum II , 1965 .

[90]  Barry Simon,et al.  A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure , 1998, math/9809182.

[91]  Barry Simon,et al.  A new approach to inverse spectral theory, I. Fundamental formalism , 1999, math/9906118.

[92]  Rafael Obaya,et al.  Ergodic properties and Weyl M-functions for random linear Hamiltonian systems , 2000, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[93]  S. Kotani.  Ljapunov Indices Determine Absolutely Continuous Spectra of Stationary Random One-dimensional Schrödinger Operators , 1984 .

[94]  F. Atkinson.  On the location of the Weyl circles , 1981, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[95]  J. Weidmann.  Oszillationsmethoden für Systeme gewöhnlicher Differentialgleichungen , 1971 .

[96]  Albert Schneider,et al.  On the Titchmarsh‐Weyl Coefficients for Singular S‐Hermitian Systems II , 1993 .

[97]  加藤 祐輔,et al.  Algebraic and spectral methods for nonlinear wave equations , 1990 .

[98]  K. R. Prasad,et al.  Upper and lower bounds for the solution of the general matrix Riccati differential equation , 1990 .

[99]  H. Dym,et al.  J-Inner matrix functions, interpolation and inverse problems for canonical systems, II: The inverse monodromy problem , 2000 .

[100]  W. N. Everitt,et al.  On the asymptotic form of the Titchmarsh-Weyl m-coefficient† , 1978 .

[101]  J. K. Shaw,et al.  On Boundary Value Problems for Hamiltonian Systems with Two Singular Points , 1984 .

[102]  J. K. Shaw,et al.  On Titchmarsh-Weyl M(λ)-functions for linear Hamiltonian systems , 1981 .

[103]  B. Simon,et al.  The xi function , 1996 .