Abstract
We study general (not necessarily Hamiltonian) first-order symmetric system J y′(t)−B(t)y(t) = Δ(t) f(t) on an interval \({\mathcal{I}=[a,b) }\) with the regular endpoint a. It is assumed that the deficiency indices n ±(T min) of the minimal relation T min associated with this system in \({L^2_\Delta(\mathcal{I})}\) satisfy \({n_-(T_{\rm min})\leq n_+(T_{\rm min})}\). We are interested in boundary conditions playing the role similar to that of separated self-adjoint boundary conditions for Hamiltonian systems. Instead we define λ-depending boundary conditions with Nevanlinna type spectral parameter τ = τ(λ) at the singular endpoint b. With this boundary value problem we associate the matrix m-function m(·) of the size \({N_\Sigma = {\rm dim} {\rm ker} (iJ+I)}\). Its role is similar to that of the Titchmarsh–Weyl coefficient for the Hamiltonian system. In turn, it allows one to define the Fourier transform \({V: L^2_\Delta(\mathcal{I}) \to L^2(\Sigma)}\) where Σ (·) is a spectral matrix function of m(·). If V is an isometry, then the (exit space) self-adjoint extension \({\tilde{T}}\) of T min induced by the boundary problem is unitarily equivalent to the multiplication operator in L 2(Σ). Hence the multiplicity of spectrum of \({\tilde{T}}\) does not exceed N Σ. We also parameterize a set of spectral functions Σ(·) by means of the set of boundary parameters τ. Similar parameterizations for various classes of boundary value problems have earlier been obtained by Kac and Krein, Fulton, Hinton and Shaw, and others.
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References
Akhiezer N.I., Glazman I.M.: Theory of Linear Operators in Hilbert Space. Dover Publications, New York (1993)
Atkinson F.V.: Discrete and Continuous Boundary Problems. Academic Press, New York (1963)
Behrndt J., Hassi S., de Snoo H., Wiestma R.: Square-integrable solutions and Weyl functions for singular canonical systems. Math. Nachr. 284(11–12), 1334–1383 (2011)
Berezanskii, Yu.M.: Expansions in eigenfunctions of selfadjoint operators. American Mathematical Society, Providence (1968) (Russian edition: Naukova Dumka, Kiev, 1965)
Bruk V.M.: On a class of boundary value problems with spectral parameter in the boundary condition. Math. USSR Sbornik. 29(2), 186–192 (1976)
Derkach V.A., Hassi S., Malamud M.M., de Snoo H.S.V.: Generalized resolvents of symmetric operators and admissibility. Methods Funct. Anal. Topol. 6(3), 24–55 (2000)
Derkach V.A., Hassi S., Malamud M.M., de Snoo H.S.V.: Boundary relations and generalized resolvents of symmetric operators. Russ. J. Math. Phys. 16,–11760 (2009)
Derkach V.A., Malamud M.M.: Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal. 95, 1–95 (1991)
Derkach V.A., Malamud M.M.: The extension theory of Hermitian operators and the moment problem. J. Math. Sci. 73(2), 141–242 (1995)
Dijksma A., Langer H., de Snoo H.S.V.: Hamiltonian systems with eigenvalue depending boundary conditions. Oper. Theory Adv. Appl. 35, 37–83 (1988)
Dijksma A., Langer H., de Snoo H.S.V.: Eigenvalues and pole functions of Hamiltonian systems with eigenvalue depending boundary conditions. Math. Nachr. 161, 107–153 (1993)
Dunford N., Schwartz J.T.: Linear Operators. Part 2. Spectral Theory. Interscience Publishers, New York (1963)
Fulton Ch.T.: Parametrizations of Titchmarsh’s m(ł)-functions in the limit circle case. Trans. Am. Math. Soc. 229, 51–63 (1977)
Gohberg, I., Krein, M.G.: Theory and applications of Volterra operators in Hilbert space, Transl. Math. Monographs, vol. 24. American Mathematical Society, Providence (1970)
Gorbachuk M.L.: On spectral functios of a differential equation of the second order with operator-valued coefficients. Ukrain. Mat. Zh. 18(2), 3–21 (1966)
Gorbachuk, V.I., Gorbachuk, M.L.: Boundary problems for differential-operator equations. Kluver Academic Publishers, Dordrecht (1991) (Russian edition: Naukova Dumka, Kiev, 1984)
Hassi S., de Snoo H.S.V., Winkler H.: Boundary-value problems for two-dimensional canonical systems. Integr. Equ. Oper. Theory 36, 445–479 (2000)
Hinton, D.B., Shaw, J.K.: Parameterization of the M(ł) function for a Hamiltonian system of limit circle type. Proc. R Soc. Edinb. Sect. A 93(3–4), 349–360 (1982/1983)
Hinton D.B., Schneider A.: On the Titchmarsh–Weyl coefficients for singular S-Hermitian systems I. Math. Nachr. 163, 323–342 (1993)
Hinton D.B., Schneider A.: Titchmarsh–Weyl coefficients for odd order linear Hamiltonian systems. J. Spectr. Math. 1, 1–36 (2006)
Hinton D.B., Schneider A.: On the Spectral Representation for Singular Selfadjoint Boundary Eigenfunction Problems. Oper. Theory: Advances and Applications, vol. 106. Birkhauser, Basel (1998)
Khol’kin A.M.: Description of selfadjoint extensions of differential operators of an arbitrary order on the infinite interval in the absolutely indefinite case. Teor. Funkcii Funkcional. Anal. Prilozhen. 44, 112–122 (1985)
Kats, I.S.: On Hilbert spaces generated by monotone Hermitian matrix-functions. Khar’kov. Gos. Univ. Uchen. Zap. 34, 95–113 (1950) [Zap. Mat. Otdel. Fiz. -Mat. Fak. i Khar’kov. Mat. Obshch. 22(4), 95–113 (1950)]
Kats I.S.: Linear relations generated by the canonical differential equation of phase dimension 2, and eigenfunction expansion. St. Petersburg Math. J. 14, 429–452 (2003)
Kac, I.S., Krein, M.G.: On Spectral Functions of a String. Supplement to the Russian edition of F.V. Atkinson. Discrete and continuous boundary problems, Mir, Moscow (1968)
Khrabustovsky V.I.: On the characteristic operators and projections and on the solutions of Weyl type of dissipative and accumulative operator systems. 3. Separated boundary conditions. J. Math. Phys. Anal. Geom. 2(4), 449–473 (2006)
Kogan, V.I., Rofe-Beketov, F.S.: On square-integrable solutions of symmetric systems of differential equations of arbitrary order. Proc. R Soc. Edinb. Sect. A 74, 5–40 (1974/1975)
Kovalishina I.V.: Analytic theory of a class of interpolation problems. Izv. Akad. Nauk SSSR Ser. Mat. 47(3), 455–497 (1983)
Krall A.M.: M(ł)-theory for singular Hamiltonian systems with one singular endpoint. SIAM J. Math. Anal. 20, 664–700 (1989)
Langer H., Textorious B.: On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space. Pac. J. Math. 72(1), 135–165 (1977)
Langer H., Textorius B.: L-resolvent matrices of symmetric linear relations with equal defect numbers; appliccations to canonical differential relations. Integr. Equ. Oper. Theory 5, 208–243 (1982)
Lesch M., Malamud M.M.: On the deficiency indices and self-adjointness of symmetric Hamiltonian systems. J. Differ. Equ. 189, 556–615 (2003)
Malamud M.M.: On the formula of generalized resolvents of a nondensely defined Hermitian operator. Ukr. Math. Zh. 44(12), 1658–1688 (1992)
Malamud M.M., Malamud S.M.: Spectral theory of operator measures in Hilbert space. St. Petersburg Math. J. 15(3), 323–373 (2003)
Malamud M.M., Mogilevskii V.I.: Krein type formula for canonical resolvents of dual pairs of linear relations. Methods Funct. Anal. Topol. 8(4), 72–100 (2002)
Malamud M., Neidhardt H.: Sturm–Liouville boundary value problems with operator potentials and unitary equivalence. J. Differ. Equ. 252, 5875–5922 (2012)
Mogilevskii V.I.: Nevanlinna type families of linear relations and the dilation theorem. Methods Funct. Anal. Topol. 12(1), 38–56 (2006)
Mogilevskii V.I.: Boundary triplets and Krein type resolvent formula for symmetric operators with unequal defect numbers. Methods Funct. Anal. Topol. 12(3), 258–280 (2006)
Mogilevskii V.I.: Description of spectral functions of differential operators with arbitrary deficiency indices. Math. Notes 81(4), 553–559 (2007)
Mogilevskii V.I.: Boundary triplets and Titchmarsh–Weyl functions of differential operators with arbitrary deficiency indices. Methods Funct. Anal. Topol. 15(3), 280–300 (2009)
Mogilevskii V.I.: Boundary pairs and boundary conditions for general (not necessarily definite) first-order symmetric systems with arbitrary deficiency indices. Math. Nachr. 285(14–15), 1895–1931 (2012)
Naimark M.A.: Linear Differential Operators, vols. 1, 2. Harrap, London (1967)
Orcutt, B.C.: Canonical differential equations. Dissertation, University of Virginia (1969)
Rofe-Beketov F.S.: Self-adjoint extensions of differential operators in the space of vector-valued functions. Teor. Funkcii Funkcional. Anal. Prilozhen. 8, 3–23 (1969)
S̆traus A.V.: On generalized resolvents and spectral functions of differential operators of an even order. Izv. Akad. Nauk. SSSR Ser. Mat. 21, 785–808 (1957)
Qi J.: Non-limit-circle criteria for singular Hamiltonian differential systems. Math. Anal. Appl. 305, 599–616 (2005)
Qi J.: Limit-point criteria for semi-degenerate singular Hamiltonian differential systems with perturbation terms. Math. Anal. Appl. 334, 983–997 (2007)
Weidmann J.: Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics, vol. 1258. Springer, Berlin (1987)
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Albeverio, S., Malamud, M. & Mogilevskii, V. On Titchmarsh–Weyl Functions and Eigenfunction Expansions of First-Order Symmetric Systems. Integr. Equ. Oper. Theory 77, 303–354 (2013). https://doi.org/10.1007/s00020-013-2090-0
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DOI: https://doi.org/10.1007/s00020-013-2090-0