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Anton Zettl - One of the best experts on this subject based on the ideXlab platform.
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Multiplicity of Sturm-Liouville eigenvalues
Journal of Computational and Applied Mathematics, 2004Co-Authors: Qingkai Kong, Hongyou Wu, Anton ZettlAbstract:The geometric multiplicity of each eigenvalue of a self-adjoint Sturm-Liouville problem is equal to its algebraic multiplicity. This is true for regular problems and for singular problems with limit-circle endpoints, including the case when the Leading Coefficient changes sign.
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Sturm-Liouville Problems Whose Leading Coefficient Function Changes Sign
Canadian Journal of Mathematics, 2003Co-Authors: Xifang Cao, Qingkai Kong, Anton ZettlAbstract:Fora givenSturm-Liouville equation whoseLeading Coefficient function changessign, we es- tablish inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions. By a recent result of Binding and Volkmer, the eigenvalues(unbounded from both below and above) for a separated self-adjoint bound- ary condition can be numbered in terms of the Prangle; and our inequalities can then be used to index the eigenvalues for any coupled self-adjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such Sturm-Liouville problems, and its range as a function on the space of self-adjoint boundary conditions. We also re- late this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme.
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Semi-boundedness of Ordinary Differential Operators
Journal of Differential Equations, 1995Co-Authors: M. Moller, Anton ZettlAbstract:Abstract The boundedness from above and from below of very general symmetric quasi-differential operators S is studied. Here we show that (i) if S is regular, of even order, and has positive Leading Coefficient, then it is bounded below; (ii) if the order of S (which may be regular or singular) is odd, then S is unbounded both above and below; and (iii) if S is of even order, regular or singular, but with a Leading Coefficient which changes sign, then S is unbounded both above and below.
Guojun Zhang - One of the best experts on this subject based on the ideXlab platform.
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String-theoretic deformation of the Parke-Taylor factor
Physical Review D, 2017Co-Authors: Sebastian Mizera, Guojun ZhangAbstract:Scattering amplitudes in a range of quantum field theories can be computed using the Cachazo-He-Yuan (CHY) formalism. In theories with colour ordering, the key ingredient is the so-called Parke-Taylor factor. In this note we give a fully $\text{SL}(2,\mathbb{C})$-covariant definition and study the properties of a new integrand called the string Parke-Taylor factor. It has an $\alpha'$ expansion whose Leading Coefficient is the field-theoretic Parke-Taylor factor. Its main application is that it leads to a CHY formulation of open string tree-level amplitudes. In fact, the definition of the string Parke-Taylor factor was motivated by trying to extend the compact formula for the first $\alpha'$ correction found by He and Zhang, while the main ingredient in its definition is a determinant of a matrix introduced in the context of string theory by Stieberger and Taylor.
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a string deformation of the parke taylor factor
Physical Review D, 2017Co-Authors: Sebastian Mizera, Guojun ZhangAbstract:Scattering amplitudes in a range of quantum field theories can be computed using the Cachazo-He-Yuan (CHY) formalism. In theories with colour ordering, the key ingredient is the so-called Parke-Taylor factor. In this note we give a fully $\text{SL}(2,\mathbb{C})$-covariant definition and study the properties of a new integrand called the string Parke-Taylor factor. It has an $\alpha'$ expansion whose Leading Coefficient is the field-theoretic Parke-Taylor factor. Its main application is that it leads to a CHY formulation of open string tree-level amplitudes. In fact, the definition of the string Parke-Taylor factor was motivated by trying to extend the compact formula for the first $\alpha'$ correction found by He and Zhang, while the main ingredient in its definition is a determinant of a matrix introduced in the context of string theory by Stieberger and Taylor.
Sebastian Mizera - One of the best experts on this subject based on the ideXlab platform.
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String-theoretic deformation of the Parke-Taylor factor
Physical Review D, 2017Co-Authors: Sebastian Mizera, Guojun ZhangAbstract:Scattering amplitudes in a range of quantum field theories can be computed using the Cachazo-He-Yuan (CHY) formalism. In theories with colour ordering, the key ingredient is the so-called Parke-Taylor factor. In this note we give a fully $\text{SL}(2,\mathbb{C})$-covariant definition and study the properties of a new integrand called the string Parke-Taylor factor. It has an $\alpha'$ expansion whose Leading Coefficient is the field-theoretic Parke-Taylor factor. Its main application is that it leads to a CHY formulation of open string tree-level amplitudes. In fact, the definition of the string Parke-Taylor factor was motivated by trying to extend the compact formula for the first $\alpha'$ correction found by He and Zhang, while the main ingredient in its definition is a determinant of a matrix introduced in the context of string theory by Stieberger and Taylor.
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a string deformation of the parke taylor factor
Physical Review D, 2017Co-Authors: Sebastian Mizera, Guojun ZhangAbstract:Scattering amplitudes in a range of quantum field theories can be computed using the Cachazo-He-Yuan (CHY) formalism. In theories with colour ordering, the key ingredient is the so-called Parke-Taylor factor. In this note we give a fully $\text{SL}(2,\mathbb{C})$-covariant definition and study the properties of a new integrand called the string Parke-Taylor factor. It has an $\alpha'$ expansion whose Leading Coefficient is the field-theoretic Parke-Taylor factor. Its main application is that it leads to a CHY formulation of open string tree-level amplitudes. In fact, the definition of the string Parke-Taylor factor was motivated by trying to extend the compact formula for the first $\alpha'$ correction found by He and Zhang, while the main ingredient in its definition is a determinant of a matrix introduced in the context of string theory by Stieberger and Taylor.
Thomas Berger - One of the best experts on this subject based on the ideXlab platform.
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On perturbations in the Leading Coefficient matrix of time‐varying index‐1 DAEs
PAMM, 2012Co-Authors: Thomas BergerAbstract:Time-varying index-1 DAEs and the effect of perturbations in the Leading Coefficient matrix is investigated. An appropriate class of allowable perturbations is introduced. Robustness of exponential stability with respect to a certain class of perturbations is shown in terms of the Bohl exponent and perturbation operator. Finally, a stability radius involving these perturbations is introduced and investigated. In particular, a lower bound for the stability radius is derived.
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on perturbations in the Leading Coefficient matrix of time varying index 1 daes
Pamm, 2012Co-Authors: Thomas BergerAbstract:Time-varying index-1 DAEs and the effect of perturbations in the Leading Coefficient matrix is investigated. An appropriate class of allowable perturbations is introduced. Robustness of exponential stability with respect to a certain class of perturbations is shown in terms of the Bohl exponent and perturbation operator. Finally, a stability radius involving these perturbations is introduced and investigated. In particular, a lower bound for the stability radius is derived.
Roman Šimon Hilscher - One of the best experts on this subject based on the ideXlab platform.
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sturm liouville matrix differential systems with singular Leading Coefficient
Annali di Matematica Pura ed Applicata, 2017Co-Authors: Iva Dřímalová, Werner Kratz, Roman Šimon HilscherAbstract:In this paper we study a general even-order symmetric Sturm–Liouville matrix differential equation, whose Leading Coefficient may be singular on the whole interval under consideration. Such an equation is new in the current literature, as it is equivalent with a system of Sturm–Liouville equations with different orders. We identify the so-called normal form of this equation, which allows to transform this equation into a standard (controllable) linear Hamiltonian system. Based on this new transformation we prove that the associated eigenvalue problem with Dirichlet boundary conditions possesses all the traditional spectral properties, such as the equality of the geometric and algebraic multiplicities of the eigenvalues, orthogonality of the eigenfunctions, the oscillation theorem and Rayleigh’s principle, and the Fourier expansion theorem. We also discuss sufficient conditions, which allow to reduce a general even-order symmetric Sturm–Liouville matrix differential equation into the normal form. Throughout the paper we provide several examples, which illustrate our new theory.
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Sturm–Liouville matrix differential systems with singular Leading Coefficient
Annali di Matematica Pura ed Applicata (1923 -), 2016Co-Authors: Iva Dřímalová, Werner Kratz, Roman Šimon HilscherAbstract:In this paper we study a general even-order symmetric Sturm–Liouville matrix differential equation, whose Leading Coefficient may be singular on the whole interval under consideration. Such an equation is new in the current literature, as it is equivalent with a system of Sturm–Liouville equations with different orders. We identify the so-called normal form of this equation, which allows to transform this equation into a standard (controllable) linear Hamiltonian system. Based on this new transformation we prove that the associated eigenvalue problem with Dirichlet boundary conditions possesses all the traditional spectral properties, such as the equality of the geometric and algebraic multiplicities of the eigenvalues, orthogonality of the eigenfunctions, the oscillation theorem and Rayleigh’s principle, and the Fourier expansion theorem. We also discuss sufficient conditions, which allow to reduce a general even-order symmetric Sturm–Liouville matrix differential equation into the normal form. Throughout the paper we provide several examples, which illustrate our new theory.
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Spectral and oscillation theory for general second order Sturm-Liouville difference equations
Advances in Difference Equations, 2012Co-Authors: Roman Šimon HilscherAbstract:In this article we establish an oscillation theorem for second order Sturm-Liouville difference equations with general nonlinear dependence on the spectral parameter λ. This nonlinear dependence on λ is allowed both in the Leading Coefficient and in the potential. We extend the traditional notions of eigenvalues and eigenfunctions to this more general setting. Our main result generalizes the recently obtained oscillation theorem for second order Sturm-Liouville difference equations, in which the Leading Coefficient is constant in λ. Problems with Dirichlet boundary conditions as well as with variable endpoints are considered.
