Indefinite Hamiltonian systems whose Titchmarsh–Weyl coefficients have no finite generalized poles of non-positive type

Matthias Langer, Harald Woracek

Research output: Contribution to journalArticle

7 Citations (Scopus)
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Abstract

The two-dimensional Hamiltonian system (*)  y'(x)=zJH(x)y(x),  x∈(a,b), where the Hamiltonian H takes non-negative 2x2-matrices as values, and $J:= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, has attracted a lot of interest over the past decades.  Special emphasis has been put on operator models and direct and inverse spectral theorems.  Weyl theory plays a prominent role in the spectral theory of the equation, relating the class of all equations (*) to the class N0 of all Nevanlinna functions via the construction of Titchmarsh–Weyl coefficients.

In connection with the study of singular potentials, an indefinite (Pontryagin space) analogue of equation (*) was proposed, where the 'general Hamiltonian' is allowed to have a finite number of inner singularities. Direct and inverse spectral theorems, relating the class of all general Hamiltonians to the class <N of all generalized Nevanlinna functions, were established.

In the present paper, we investigate the spectral theory of general Hamiltonians having a particular form, namely, such which have only one singularity and the interval to the left of this singularity is a so-called indivisible interval.  Our results can comprehensively be formulated as follows.
— We prove direct and inverse spectral theorems for this class, i.e. we establish an intrinsic characterization of the totality of all Titchmarsh–Weyl coefficients corresponding to general Hamiltonians of the considered form.
—  We determine the asymptotic growth of the fundamental solution when
approaching the singularity.
—  We show that each solution of the equation has 'polynomially regularized'
boundary values at the singularity.

Besides the intrinsic interest and depth of the presented results, our motivation is drawn from forthcoming applications: the present theorems form the core for our study of Sturm–Liouville equations with two singular endpoints and our further study of the structure theory of general Hamiltonians (both to be presented elsewhere).

Original languageEnglish
Article numberOaM-07-29
Pages (from-to)477-555
Number of pages79
JournalOperators and Matrices
Volume7
Issue number3
DOIs
Publication statusPublished - Sep 2013

Fingerprint

Indefinite Systems
Hamiltonian Systems
Pole
Spectral Theorem
Inverse Theorems
Singularity
Coefficient
P-matrix
Spectral Theory
2x2 matrix
Pontryagin Space
Indivisible
Sturm-Liouville Equation
Singular Potential
Interval
Nonnegative Matrices
Two-dimensional Systems
Fundamental Solution
Boundary Value
Class

Keywords

  • Hamiltonian system with inner singularity
  • Titchmarsh–Weyl coefficient
  • inverse problem
  • asymptotics of solutions

Cite this

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title = "Indefinite Hamiltonian systems whose Titchmarsh–Weyl coefficients have no finite generalized poles of non-positive type",
abstract = "The two-dimensional Hamiltonian system (*)  y'(x)=zJH(x)y(x),  x∈(a,b), where the Hamiltonian H takes non-negative 2x2-matrices as values, and $J:= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, has attracted a lot of interest over the past decades.  Special emphasis has been put on operator models and direct and inverse spectral theorems.  Weyl theory plays a prominent role in the spectral theory of the equation, relating the class of all equations (*) to the class N0 of all Nevanlinna functions via the construction of Titchmarsh–Weyl coefficients. In connection with the study of singular potentials, an indefinite (Pontryagin space) analogue of equation (*) was proposed, where the 'general Hamiltonian' is allowed to have a finite number of inner singularities. Direct and inverse spectral theorems, relating the class of all general Hamiltonians to the class <N∞ of all generalized Nevanlinna functions, were established. In the present paper, we investigate the spectral theory of general Hamiltonians having a particular form, namely, such which have only one singularity and the interval to the left of this singularity is a so-called indivisible interval.  Our results can comprehensively be formulated as follows. — We prove direct and inverse spectral theorems for this class, i.e. we establish an intrinsic characterization of the totality of all Titchmarsh–Weyl coefficients corresponding to general Hamiltonians of the considered form. —  We determine the asymptotic growth of the fundamental solution when approaching the singularity. —  We show that each solution of the equation has 'polynomially regularized' boundary values at the singularity. Besides the intrinsic interest and depth of the presented results, our motivation is drawn from forthcoming applications: the present theorems form the core for our study of Sturm–Liouville equations with two singular endpoints and our further study of the structure theory of general Hamiltonians (both to be presented elsewhere).",
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Indefinite Hamiltonian systems whose Titchmarsh–Weyl coefficients have no finite generalized poles of non-positive type. / Langer, Matthias; Woracek, Harald.

In: Operators and Matrices, Vol. 7, No. 3, OaM-07-29, 09.2013, p. 477-555.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Indefinite Hamiltonian systems whose Titchmarsh–Weyl coefficients have no finite generalized poles of non-positive type

AU - Langer, Matthias

AU - Woracek, Harald

PY - 2013/9

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N2 - The two-dimensional Hamiltonian system (*)  y'(x)=zJH(x)y(x),  x∈(a,b), where the Hamiltonian H takes non-negative 2x2-matrices as values, and $J:= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, has attracted a lot of interest over the past decades.  Special emphasis has been put on operator models and direct and inverse spectral theorems.  Weyl theory plays a prominent role in the spectral theory of the equation, relating the class of all equations (*) to the class N0 of all Nevanlinna functions via the construction of Titchmarsh–Weyl coefficients. In connection with the study of singular potentials, an indefinite (Pontryagin space) analogue of equation (*) was proposed, where the 'general Hamiltonian' is allowed to have a finite number of inner singularities. Direct and inverse spectral theorems, relating the class of all general Hamiltonians to the class <N∞ of all generalized Nevanlinna functions, were established. In the present paper, we investigate the spectral theory of general Hamiltonians having a particular form, namely, such which have only one singularity and the interval to the left of this singularity is a so-called indivisible interval.  Our results can comprehensively be formulated as follows. — We prove direct and inverse spectral theorems for this class, i.e. we establish an intrinsic characterization of the totality of all Titchmarsh–Weyl coefficients corresponding to general Hamiltonians of the considered form. —  We determine the asymptotic growth of the fundamental solution when approaching the singularity. —  We show that each solution of the equation has 'polynomially regularized' boundary values at the singularity. Besides the intrinsic interest and depth of the presented results, our motivation is drawn from forthcoming applications: the present theorems form the core for our study of Sturm–Liouville equations with two singular endpoints and our further study of the structure theory of general Hamiltonians (both to be presented elsewhere).

AB - The two-dimensional Hamiltonian system (*)  y'(x)=zJH(x)y(x),  x∈(a,b), where the Hamiltonian H takes non-negative 2x2-matrices as values, and $J:= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, has attracted a lot of interest over the past decades.  Special emphasis has been put on operator models and direct and inverse spectral theorems.  Weyl theory plays a prominent role in the spectral theory of the equation, relating the class of all equations (*) to the class N0 of all Nevanlinna functions via the construction of Titchmarsh–Weyl coefficients. In connection with the study of singular potentials, an indefinite (Pontryagin space) analogue of equation (*) was proposed, where the 'general Hamiltonian' is allowed to have a finite number of inner singularities. Direct and inverse spectral theorems, relating the class of all general Hamiltonians to the class <N∞ of all generalized Nevanlinna functions, were established. In the present paper, we investigate the spectral theory of general Hamiltonians having a particular form, namely, such which have only one singularity and the interval to the left of this singularity is a so-called indivisible interval.  Our results can comprehensively be formulated as follows. — We prove direct and inverse spectral theorems for this class, i.e. we establish an intrinsic characterization of the totality of all Titchmarsh–Weyl coefficients corresponding to general Hamiltonians of the considered form. —  We determine the asymptotic growth of the fundamental solution when approaching the singularity. —  We show that each solution of the equation has 'polynomially regularized' boundary values at the singularity. Besides the intrinsic interest and depth of the presented results, our motivation is drawn from forthcoming applications: the present theorems form the core for our study of Sturm–Liouville equations with two singular endpoints and our further study of the structure theory of general Hamiltonians (both to be presented elsewhere).

KW - Hamiltonian system with inner singularity

KW - Titchmarsh–Weyl coefficient

KW - inverse problem

KW - asymptotics of solutions

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