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Spectrum of Infinite Dimensional Hamiltonian Operators and Completeness of the Eigenfunction Systems

Author: WuDeYu
Tutor: ALaTanCang
School: Inner Mongolia University
Course: Applied Mathematics
Keywords: Infinite dimensional Hamiltonian operator Non- negative Hamiltonian operator Department of characteristic function Cauchy principal value Completeness Numerical range Second frequency range Reversible Krein space Greatly determine the invariant subspace Music Point spectrum Residual Spectrum Continuous spectrum Pure imaginary spectrum
CLC: O175.3
Type: PhD thesis
Year: 2008
Downloads: 202
Quote: 7
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Abstract


The existence of this dissertation to infinite dimensional Hamilton operator of eigenvalue functions (h orthogonal system) the completeness of the theme around the spectral theory of infinite dimensional Hamiltonian operators and completeness indefinite metric space greatly determine invariant subspace issues to carry out research work, thus Extensional Strum-Liouville problem and eigenfunction expansion method for solving the method of separation of variables for the Hamilton system provides a theoretical guarantee. Infinite dimensional Hamiltonian operator of eigenvalue functions (h orthogonal system) Completeness of infinite dimensional Hamilton operator theory and infinite dimensional Hamiltonian system. Strum-Liouville problem for the separation of variables-oriented partial differential equations, separation of variables is a very effective method for solving. However, the infinite-dimensional Hamiltonian operator under normal circumstances a non-self-adjoint operators, so the separation of variables in the Hamilton system under the appropriate and correct the problem becomes more important. However, the theoretical basis of the above problems is the infinite-dimensional Hamiltonian system operator characteristic function (h orthogonal system) completeness. Therefore, make full use of the infinite-dimensional Hamiltonian the operator eigenvalue functions symplectic orthogonality and a class of infinite dimensional Hamiltonian operator of the eigenvalues ??of the unique nature of the positive and negative in pairs, a class of infinite-dimensional Hamiltonian operator characteristic function sufficient condition for the Cauchy principal value of the system (h orthogonal system) complete sense, on this basis, is of such infinite dimensional Hamiltonian system of separation of variables in the Cauchy principal value sense (ie, separation of variables Cauchy principal value in the sense of the principle of superposition) to obtain the complete solution of the Cauchy principal value sense. This work for solving the infinite-dimensional Hamiltonian system for solving even the partial differential equation provides a new method, new ideas, with a high theoretical value and practical significance. To solve the more general infinite dimensional Hamiltonian system to solve the problem, it corresponds to the general infinite dimensional Hami lton operator shall take into account the characteristics of this problem belongs to the linear operator theory category. We know that the linear operator spectral analysis is an important part of the functional analysis, linear operator theory of the soul, it is the central subject of the theory of spectral decomposition. Therefore, this paper spectral theory of infinite dimensional Hamiltonian operators on a very important position, given the set of upper triangular infinite dimensional Hamiltonian operator spectrum and continuous spectrum and the main related necessary and sufficient condition, which To solve the infinite-dimensional Hamiltonian operator on triangular spectral complement and spectral perturbation provides the necessary preparations; generate a strongly continuous semigroup problems in order to solve the infinite-dimensional Hamiltonian operator, given infinite dimensional Hamiltonian operator only pure sufficient condition for continued spectrum. In addition, when the export system operator reversible, semi-analytical method to provide a strong guarantee, this time into ordinary differential equations, partial differential equations, infinite-dimensional Hamiltonian operator reversible problem also appears to be important, but the nature of this problem is zero is included in the regular point set. Thus, the use of the structural characteristics of the non-negative Hamiltonian operator, using in-house items portrait of a non-negative Hamiltonian operator reversible, to solve the problem of non-negative Hamiltonian operator when tight domain solution. It is worth noting that characterize the spectral distribution of the set value has a very important application domain, because the numerical range of bounded linear operators closure contains the spectrum set, however, recently discovered that there is a bounded linear operator two The frequency range is not only a subset of the value domain and its closure also includes Spectral portrayed Spectral two-frequency range to provide better information than the numerical domain. For these reasons, we studied a class of unbounded, infinite dimensional Hamilton operator numerical range and frequency range, and gives not only the closure of the numerical range contains the spectrum set and the closure of the two frequency range also includes spectrum The set of conclusions. This paper also studied the spectral theory of infinite-dimensional Hamilton operator in the a complete indefinite metric space. Degree of uncertainty on the regulatory space operator theory is not a Hilbert space operators and promotion on the sub-theory logic, but has a solid foundation. Its application involves physics, mathematics and mechanics. Infinite-dimensional Hamiltonian operator particularity, after the introduction of appropriate indefinite metric infinite dimensional Hamiltonian operator becomes indefinite metric the sense of anti-self-adjoint operator At this point, the nature of its complete indefinite metric space the nature of self-adjoint operators is very close, so you can get many meaningful conclusions. On this basis, this paper gives the infinite-dimensional Hamiltonian operator complete indefinite metric space greatly determine the invariant subspace sufficient condition. Secondly, since the bounded self-adjoint operator H.Weyl in 1909 isolated a limited re-set of eigenvalues ??and Weyl spectrum in the complementary set of overlapping spectral concentration (ie, the famous Weyl theorem), J.Schwartz S Berberian and many other scholars operator to meet the Weyl theorem, so to meet the Weyl type theorem operator range continues to expand. However, most results are denominated in a bounded operator as the research object, the unbounded operator Weyl theorem conclusion is very rare. This article gives a sufficient condition for unbounded self-adjoint linear operator satisfies Weyl theorem PERTURBING, only the compact operators satisfy a sufficient condition for Weyl type theorem. Paper is divided into seven chapters, the first chapter introduces the significance of the topic and the main work; given the nature of the spectrum on the triangular infinite dimensional Hamilton operator, and discuss the infinite dimensional Hamilton operator eigenvalue problem ; Chapter Department of infinite-dimensional Hamiltonian operator characteristic function of the Cauchy principal value sense complete; reversibility of the fourth chapter of the non-negative Hamiltonian operator; chapter 5, the numerical range of a class of infinite dimensional Hamiltonian operator and the nature of the frequency range; sixth chapter with perturbing unbounded self-adjoint linear operator when to meet Weyl theorem; Chapter VII of the complete spectral theory of the degree of uncertainty in the regulatory space infinite dimensional Hamilton operator and greatly determine the existence of invariant subspace problem.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Differential operator theory
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