Spectral Theory Of Singular Continuous/Discrete Hamiltonian Systems And Matrix Difference Equations Of Mixed Order

The research content of spectral theory of linear operators is very extensive,including defect index,self-adjoint extension,spectrum,spectral radius,spectral decomposition,spectral analysis,etc.Spectral theory of linear operators plays an significant role in operator theory and quantum mechanics.It has broad applications in physics,engineering,cybernetics,signal processing,etc.The essential spectra of closed operators has been widely studied because of its stability and applications.The original idea of the essential numerical range was to give a convex enclosure of the essential spectrum,which implies that it can ensure distribution of essential spectrum.However,a key feature of the essential numerical range is that it captures spectral pollution when the operator is approximated by associated operator sequence.Defect indices of operators are very important in self-adjoint extensions and spectral properties and it has been studied for a long time.It has been found that operators generated by linear discrete Hamiltonian systems may be multi-valued or not densely defined.So it is necessary and urgent to establish theory of multi-valued operators and non-densely defined operators.Linear relations include single-valued and multi-valued operators.Perturbation theory is a classical method in studying spectral problems.This paper concerns with the stability of essential numerical ranges of singular continuous Hamiltonian systems,essential numerical ranges of linear relations and singular discrete Hamiltonian systems,defect indices and essential spectra of singular matrix difference operators of mixed order.The main results of this paper are as follows:The essential numerical ranges of singular continuous Hamiltonian systems are studied under a class of perturbations.The concept of form perturbation small at the singular endpoint is introduced.Then,the characterization of each element of the essential numerical range in terms of certain singular sequences is given since it is independent of the coefficients on finite subintervals.Hence,the stability of the essential numerical range is obtained under this perturbation.Moreover,some sufficient conditions for the invariance of the essential numerical range are given in terms of coefficients of systems and perturbations terms.The concept of essential numerical ranges of linear relations in a Hilbert space is given,and its fundamental properties are studied.Further,other various essential numerical ranges are introduced,and relationships among them are established.In addition,singular discrete Hamiltonian systems including non-symmetric cases are considered.Some sufficient and necessary conditions for the minimal relation being an operator are given,and a sufficient condition for the minimal relation being not densely defined is derived.The relevant results for essential numerical ranges of abstract linear relations are applied to the systems.Matrix differential equations of mixed order arose in magnetohydrodynamics,and the spectral analysis of the equations is important in theoretical physics.It is noted that matrix difference equations of mixed order can be regarded as a discrete analogue of matrix differential equations.In this paper,classification of singular matrix difference equations of mixed order is derived.The classification of the equations is obtained with a similar classical Weyl’s method by selecting a suitable quasi-difference.An equivalent characterization of this classification is given in terms of the number of linearly independent square summable solutions of the equation.Then,limit point and limit circle criteria are established in terms of coefficients of equations.Unlike classical differential and difference equations,the essential spectra of singular matrix differential equations of mixed order can be divided into two parts,a regular part and a singular part.The regular part is determined by the behaviour of the coefficients and it is not empty,but singular part is dependent on the singular endpoints.In this paper,essential spectra of singular matrix difference equations of mixed order are studied.Under certain conditions,characterization of singular essential spectra of the equation is derived.Moreover,an example with empty regular essential spectra is obtained,which shows that the difference between matrix differential equations and matrix difference equations.