Weyl functions of generalized dirac systems: Integral representation, the inverse problem and discrete interpolation
Tóm tắt
We study self-adjoint Dirac systems and subclasses of canonical systems (which generalize Dirac systems) and obtain explicit and global solutions for direct and inverse problems. We also derive a local Borg-Marchenko-type theorem, integral representation of the Weyl function, and results on the interpolation of Weyl functions.
Tài liệu tham khảo
citation_title=Krein systems; citation_inbook_title=
, Vol. 1; citation_publication_date=2009; citation_pages=27-63; citation_id=CR1; citation_author=D. Alpay; citation_author=I. Gohberg; citation_author=M. A. Kaashoek; citation_author=L. Lerer; citation_author=A. L. Sakhnovich; citation_publisher=Birkhäuser Verlag
citation_journal_title=Integral Equations Operator Theory; citation_title=Krein systems and canonical systems on a finite interval: accelerants with a jump discontinuity at the origin and continuous potentials; citation_author=D. Alpay, I. Gohberg, M. A. Kaashoek, L. Lerer, A. L. Sakhnovich; citation_volume=68; citation_publication_date=2010; citation_pages=115-150; citation_doi=10.1007/s00020-010-1803-x; citation_id=CR2
citation_journal_title=J. Comput. Appl. Math.; citation_title=M-operators: a generalisation of Weyl-Titchmarsh theory; citation_author=W. O. Amrein, D. B. Pearson; citation_volume=171; citation_publication_date=2004; citation_pages=1-26; citation_doi=10.1016/j.cam.2004.01.020; citation_id=CR3
citation_title=J-contractive Matrix Valued Functions and Related Topics; citation_publication_date=2008; citation_id=CR4; citation_author=D. Z. Arov; citation_author=H. Dym; citation_publisher=Cambridge University Press
citation_journal_title=Comm. Math. Phys.; citation_title=The boundary control approach to the Titchmarsh-Weyl m-function, I, The response operator and the A-amplitude; citation_author=S. Avdonin, V. Mikhaylov, A. Rybkin; citation_volume=275; citation_publication_date=2007; citation_pages=791-803; citation_doi=10.1007/s00220-007-0315-2; citation_id=CR5
citation_journal_title=J. Math. Anal. Appl.; citation_title=The interpolation of the Titchmarsh-Weyl function; citation_author=A. Boumenir, V. K. Tuan; citation_volume=335; citation_publication_date=2007; citation_pages=72-78; citation_doi=10.1016/j.jmaa.2007.01.054; citation_id=CR6
citation_journal_title=J. Funct. Anal.; citation_title=Weyl-Titchmarsh functions of vector-valued Sturm-Liouville operators on the unit interval; citation_author=D. Chelkak, E. Korotyaev; citation_volume=257; citation_publication_date=2009; citation_pages=1546-1588; citation_doi=10.1016/j.jfa.2009.05.010; citation_id=CR7
citation_journal_title=J. Phys. A; citation_title=Algebraic construction of the Darboux matrix revisited; citation_author=J. L. Ciésliński; citation_volume=42; citation_publication_date=2009; citation_pages=404003; citation_doi=10.1088/1751-8113/42/40/404003; citation_id=CR8
citation_journal_title=Trans. Amer. Math. Soc.; citation_title=Weyl-Titchmarsh M-function asymptotics, local uniqueness results, trace formulas, and Borg-type theorems for Dirac operators; citation_author=S. Clark, F. Gesztesy; citation_volume=354; citation_publication_date=2002; citation_pages=3475-3534; citation_doi=10.1090/S0002-9947-02-03025-8; citation_id=CR9
citation_title=On self-adjoint and J-self-adjoint Dirac-type operators: a case study; citation_inbook_title=Recent Advances in Differential Equations and Mathematical Physics; citation_publication_date=2006; citation_pages=103-140; citation_id=CR10; citation_author=S. Clark; citation_author=F. Gesztesy; citation_publisher=Amer. Math. Soc.
citation_journal_title=Oper. Matrices; citation_title=Weyl-Titchmarsh theory and Borg-Marchenko-type uniqueness results for CMV operators with matrix-valued Verblunsky coefficients; citation_author=S. Clark, F. Gesztesy, M. Zinchenko; citation_volume=1; citation_publication_date=2007; citation_pages=535-592; citation_id=CR11
S. Clark, F. Gesztesy and M. Zinchenko, Borg-Marchenko-type uniqueness results for CMV operators, Skr. K. Nor. Vidensk. Selsk. 2008, 1–18.
citation_journal_title=Duke Math. J.; citation_title=Applications of a commutation formula; citation_author=P. A. Deift; citation_volume=45; citation_publication_date=1978; citation_pages=267-310; citation_doi=10.1215/S0012-7094-78-04516-7; citation_id=CR13
citation_title=Weyl’s work on singular Sturm-Liouville operators; citation_inbook_title=Groups and Analysis. The Legacy of Hermann Weyl; citation_publication_date=2008; citation_pages=63-83; citation_id=CR14; citation_author=W. N. Everitt; citation_author=H. Kalf; citation_publisher=Cambridge University Press
citation_journal_title=Oper. Matrices; citation_title=Weyl matrix-functions and inverse problems for discrete Dirac type self-adjoint system: explicit and general solutions; citation_author=B. Fritzsche, B. Kirstein, I. Ya. Roitberg, A. L. Sakhnovich; citation_volume=2; citation_publication_date=2008; citation_pages=201-231; citation_id=CR15
citation_journal_title=Integral Equations Operator Theory; citation_title=Semiseparable integral operators and explicit solution of an inverse problem for the skew-self-adjoint Dirac-type system; citation_author=B. Fritzsche, B. Kirstein, A. L. Sakhnovich; citation_volume=66; citation_publication_date=2010; citation_pages=231-251; citation_doi=10.1007/s00020-010-1739-1; citation_id=CR16
citation_journal_title=J. Funct. Anal.; citation_title=A complete spectral characterization of the double commutation method; citation_author=F. Gesztesy; citation_volume=117; citation_publication_date=1993; citation_pages=401-446; citation_doi=10.1006/jfan.1993.1132; citation_id=CR17
citation_title=
, Part 2; citation_publication_date=2007; citation_id=CR18; citation_publisher=Amer. Math. Soc
citation_journal_title=Ann. of Math. (2); citation_title=A new approach to inverse spectral theory, II, General real potentials and the connection to the spectral measure; citation_author=F. Gesztesy, B. Simon; citation_volume=152; citation_publication_date=2000; citation_pages=593-643; citation_doi=10.2307/2661393; citation_id=CR19
citation_journal_title=Proc. Amer. Math. Soc.; citation_title=On the double commutation method; citation_author=F. Gesztesy, G. Teschl; citation_volume=124; citation_publication_date=1996; citation_pages=1831-1840; citation_doi=10.1090/S0002-9939-96-03299-6; citation_id=CR20
citation_journal_title=Math. Nachr.; citation_title=On spectral theory for Schrödinger operators with strongly singular potentials; citation_author=F. Gesztesy, M. Zinchenko; citation_volume=279; citation_publication_date=2006; citation_pages=1041-1082; citation_doi=10.1002/mana.200510410; citation_id=CR21
citation_title=
, Vol. I; citation_publication_date=1990; citation_id=CR22; citation_author=I. Gohberg; citation_author=S. Goldberg; citation_author=M. A. Kaashoek; citation_publisher=Birkhäuser
citation_journal_title=Math. Nachr.; citation_title=Canonical systems with rational spectral densities: explicit formulas and applications; citation_author=I. Gohberg, M. A. Kaashoek, A. L. Sakhnovich; citation_volume=194; citation_publication_date=1998; citation_pages=93-125; citation_doi=10.1002/mana.19981940108; citation_id=CR23
citation_journal_title=Asymptot. Anal.; citation_title=Scattering problems for a canonical system with a pseudo-exponential potential; citation_author=I. Gohberg, M. A. Kaashoek, A. L. Sakhnovich; citation_volume=29; citation_publication_date=2002; citation_pages=1-38; citation_id=CR24
citation_title=Theory and Applications of Volterra Operators in Hilbert Space; citation_publication_date=1967; citation_id=CR25; citation_author=I. Gohberg; citation_author=M. G. Krein; citation_publisher=Nauka
citation_title=Darboux Transformations in Integrable Systems. Theory and Their Applications to Geometry; citation_publication_date=2005; citation_id=CR26; citation_author=C. H. Gu; citation_author=H. Hu; citation_author=Z. Zhou; citation_publisher=Springer
citation_journal_title=Integral Equations Operator Theory; citation_title=Hermitian indefinite functions and Pontryagin spaces of entire functions; citation_author=M. Kaltenbäck; citation_volume=35; citation_publication_date=1999; citation_pages=172-197; citation_doi=10.1007/BF01196382; citation_id=CR27
A. Kostenko, A. Sakhnovich, and G. Teschl, Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials, Int. Math. Res. Not. IMRN (2011) art. rnr065.
citation_journal_title=Math. Nachr.; citation_title=Commutation methods for Schrödinger operators with strongly singular potentials; citation_author=A. Kostenko, A. Sakhnovich, G. Teschl; citation_volume=285; citation_publication_date=2012; citation_pages=392-410; citation_doi=10.1002/mana.201000108; citation_id=CR29
citation_journal_title=Dokl. Akad. Nauk SSSR; citation_title=
Continuous analogues of propositions on polynomials orthogonal on the unit circle (Russian); citation_author=M. G. Krein; citation_volume=105; citation_publication_date=1955; citation_pages=637-640; citation_id=CR30
citation_journal_title=Math. Nachr.; citation_title=
Über einige Fortsetzungsprobleme, die eng mit der Theorie Hermitescher Operatoren im Raume Πκ
zusammenhängen, I, Einige Funktionenklassen und ihre Darstellungen
; citation_author=M. G. Krein, H. Langer; citation_volume=77; citation_publication_date=1977; citation_pages=187-236; citation_doi=10.1002/mana.19770770116; citation_id=CR31
citation_journal_title=Integral Equations Operator Theory; citation_title=Direct and inverse spectral problems for generalized strings; citation_author=H. Langer, H. Winkler; citation_volume=30; citation_publication_date=1998; citation_pages=409-431; citation_doi=10.1007/BF01257875; citation_id=CR32
citation_journal_title=Inverse Problems; citation_title=A local inverse spectral theorem for Hamiltonian systems; citation_author=M. Langer, H. Woracek; citation_volume=27; citation_publication_date=2011; citation_pages=055002; citation_doi=10.1088/0266-5611/27/5/055002; citation_id=CR33
citation_title=Sturm-Liouville and Dirac Operators; citation_publication_date=1991; citation_id=CR34; citation_author=B. M. Levitan; citation_author=I. S. Sargsjan; citation_publisher=Kluwer
citation_title=Darboux Transformations and Solitons; citation_publication_date=1991; citation_id=CR35; citation_author=V. B. Matveev; citation_author=M. A. Salle; citation_publisher=Springer
citation_journal_title=Duke Math. J.; citation_title=Direct and inverse spectral problem for a system of differential equations depending rationally on the spectral parameter; citation_author=R. Mennicken, A. L. Sakhnovich, C. Tretter; citation_volume=109; citation_publication_date=2001; citation_pages=413-449; citation_doi=10.1215/S0012-7094-01-10931-9; citation_id=CR36
citation_title=Fourier Transforms in the Complex Domain; citation_publication_date=1987; citation_id=CR37; citation_author=R. E. A. C. Paley; citation_author=N. Wiener; citation_publisher=Amer. Math. Soc.
citation_journal_title=J. Operator Theory; citation_title=Spectral problems for some indefinite cases of canonical differential equations; citation_author=J. Rovnyak, L. A. Sakhnovich; citation_volume=51; citation_publication_date=2004; citation_pages=115-139; citation_id=CR38
citation_journal_title=Proc. Amer. Math. Soc.; citation_title=A new interpolation formula for the Titchmarsh-Weyl function; citation_author=A. Rybkin, V. K. Tuan; citation_volume=137; citation_publication_date=2009; citation_pages=4177-4185; citation_doi=10.1090/S0002-9939-09-09983-3; citation_id=CR39
citation_journal_title=Soviet Math. (Iz. VUZ); citation_title=Asymptotics of spectral functions of an S-node; citation_author=A. L. Sakhnovich; citation_volume=32; citation_publication_date=1988; citation_pages=92-105; citation_id=CR40
citation_journal_title=Ukrainian Math. J.; citation_title=Nonlinear Schrödinger equation on a semi-axis and an inverse problem associated with it; citation_author=A. L. Sakhnovich; citation_volume=42; citation_publication_date=1990; citation_pages=316-323; citation_doi=10.1007/BF01057016; citation_id=CR41
citation_journal_title=J. Funct. Anal.; citation_title=Iterated Bäcklund-Darboux transform for canonical systems; citation_author=A. L. Sakhnovich; citation_volume=144; citation_publication_date=1997; citation_pages=359-370; citation_doi=10.1006/jfan.1996.3003; citation_id=CR42
citation_journal_title=J. Math. Anal. Appl.; citation_title=Generalized Bäcklund-Darboux transformation: spectral properties and nonlinear equations; citation_author=A. L. Sakhnovich; citation_volume=262; citation_publication_date=2001; citation_pages=274-306; citation_doi=10.1006/jmaa.2001.7577; citation_id=CR43
citation_journal_title=Inverse Problems; citation_title=Dirac type and canonical systems: spectral and Weyl-Titchmarsh matrix functions, direct and inverse problems; citation_author=A. L. Sakhnovich; citation_volume=18; citation_publication_date=2002; citation_pages=331-348; citation_doi=10.1088/0266-5611/18/2/303; citation_id=CR44
citation_journal_title=Inverse Problems; citation_title=Weyl functions, the inverse problem and special solutions for the system auxiliary to the nonlinear optics equation; citation_author=A. L. Sakhnovich; citation_volume=24; citation_publication_date=2008; citation_pages=025026; citation_doi=10.1088/0266-5611/24/2/025026; citation_id=CR45
citation_journal_title=Integral Equations Operator Theory; citation_title=Construction of the solution of the inverse spectral problem for a system depending rationally on the spectral parameter, Borg-Marchenko-type theorem, and sine-Gordon equation; citation_author=A. L. Sakhnovich; citation_volume=69; citation_publication_date=2011; citation_pages=567-600; citation_doi=10.1007/s00020-010-1843-2; citation_id=CR46
citation_journal_title=Positivity; citation_title=On explicit inversion of a subclass of operators with D-difference kernels and Weyl theory of the corresponding canonical systems; citation_author=A. L. Sakhnovich, A. A. Karelin, J. Seck-Tuoh-Mora, G. Perez-Lechuga, M. Gonzalez-Hernandez; citation_volume=14; citation_publication_date=2010; citation_pages=547-564; citation_doi=10.1007/s11117-009-0036-1; citation_id=CR47
citation_journal_title=Ukr. Mat. J.; citation_title=
Spectral analysis of Volterra’s operators defined in the space of vectorfunctions L
(0, l); citation_author=L. A. Sakhnovich; citation_volume=16; citation_publication_date=1964; citation_pages=259-268; citation_id=CR48
citation_journal_title=Russian Math. Surveys; citation_title=Factorisation problems and operator identities; citation_author=L. A. Sakhnovich; citation_volume=41; citation_publication_date=1986; citation_pages=1-64; citation_doi=10.1070/RM1986v041n01ABEH003200; citation_id=CR49
citation_title=Integral Equations with Difference Kernels on Finite Intervals; citation_publication_date=1996; citation_id=CR50; citation_author=L. A. Sakhnovich; citation_publisher=Birkhäuser Verlag
citation_title=Interpolation Theory and its Applications; citation_publication_date=1997; citation_id=CR51; citation_author=L. A. Sakhnovich; citation_publisher=Kluwer Academic Publishers
citation_journal_title=Integral Equations Operator Theory; citation_title=On a class of canonical systems on half-axis; citation_author=L. A. Sakhnovich; citation_volume=31; citation_publication_date=1998; citation_pages=92-112; citation_doi=10.1007/BF01203458; citation_id=CR52
L. A. Sakhnovich, Spectral Theory of Canonical Differential Systems, Method of Operator Identities, Birkhäuser Verlag, 1999.
citation_journal_title=Ann. of Math. (2); citation_title=A new approach to inverse spectral theory, I, Fundamental formalism; citation_author=B. Simon; citation_volume=150; citation_publication_date=1999; citation_pages=1029-1057; citation_doi=10.2307/121061; citation_id=CR54
citation_title=Mathematical Methods in Quantum Mechanics With Applications to Schrödinger Operators; citation_publication_date=2009; citation_id=CR55; citation_author=G. Teschl; citation_publisher=Amer. Math. Soc.
citation_journal_title=Comm. Math. Phys.; citation_title=On the integrability of classical spinor models in two-dimensional space-time; citation_author=V. E. Zakharov, A. V. Mikhailov; citation_volume=74; citation_publication_date=1980; citation_pages=21-40; citation_doi=10.1007/BF01197576; citation_id=CR56