The Splectral Problems Of Non-symmetrlc Differential Operators

The spectral theory on self-adjoint operators in Hilbert space has been accom-plished and successfully applied in the field of differential equations, solving many significant problems in quantum mechanics and scientific technologies. In the practi-cal applications and mathematical theory, differential equations are often faced with non-symmetric problems which would be transformed into non-self-adjoint or even non-symmetric differential operator spectral problems. There has been no spectral theory on these non-symmetric problems as perfect as that of self-adjoint operators. One needs to apply existing methods of the operator theory and differential equations in order to solve corresponding boundary value problems of differential equations. What’s more, theories of non-symmetric differential equations will be more widely used in practical issues.In this dissertation, two kinds of spectral problems of non-symmetric differential operators are researched. The first kind involves the classification and characteristics of non-symmetric Sturm-Liouville (S-L for short) differential equations and systems. The other kind deals with problems related to the non-real eigenvalues of indefinite differential operators, including indefinite S-L operators, p-Laplacian and elliptic dif-ferential operators. The academic background, main conclusions and research methods will be introduced as follows.Firstly, the classification of non-symmetric S-L differential equations will be re-searched. The deficiency index classification of differential equations is the primary problem in the spectral theory of differential equations and the foundation of spectral problems of differential equations. In1910, Weyl started the research on singular S-L differential equations. He divided formally symmetric S-L differential equations, i.e., coefficient functions with real values, into two cases:the limit point case and the limit circle case. This classification was further generalized to the higher-order and symmet-ric Hamiltonian system. However, it was not until1957that Sims did research on the classification of one special kind of formally non-symmetric S-L differential equations whose potential functions are complex-valued and imaginary parts are semi-bounded. After nearly another half century, Brown, McCormack, Evans and Plum put forward the classification of S-L differential equations with coefficient functions of complex val-ues satisfying normal conditions for coefficient functions in1999. In this paper, such S-L differential equations were classified into three cases. In2003, Brown, Evans and Plum provided the classification of the non-symmetric Hamiltonian system with even dimension under strict conditions.Based on the above conclusions, the first half of this dissertation will focus on the necessary and sufficient conditions in classifying S-L differential equations with complex-valued coefficient functions. The asymptotic behavior of elements in the max-imum domain is applied to characterize the classification of non-symmetric S-L differ-ential equations given by Brown etc. in1999. Then, non-symmetric S-L differential systems are classified, whose necessary and sufficient conditions for the classification are also provided. Furthermore, the J-self-adjoint operator realizations of such non-symmetric systems are obtained.As far as the research methods are concerned, the allowable angle of rotation is applied to transform S-L differential equations with non-symmetric complex-valued coefficients into symmetric. Hamiltonian systems. Then the relation of the numbers of their linearly independent and square integrable solutions under proper weighted function spaces is set up. With this relation and the characteristics of symmetric Hamiltonian systems, one necessary and sufficient condition to classify Brown’s S-L problems with complex-valued coefficients is obtained. This method is further applied to the high dimension, which means the orthogonal transformations obtained through allowable pairs are applied to transform non-symmetric S-L systems into symmetric Hamiltonian differential systems. The theory of symmetric Hamiltonian differential systems are used to get the classification according to non-symmetric S-L systems and the numbers of their linearly independent and weighted square integrable solutions in the conjugated system. The original system and the asymptotic behavior of elements in the maximal domain of the conjugated system are applied to get the classification. Finally, the realization of J-self-adjoint operators associate with non-symmetric S-L systems is given as a practical application.Secondly, this dissertation will go on with non-real eigenvalues of differential e-quations with indefinite weights. The S-L problem with weighted functions is called right-definite or Orthogonal (given by Hilbert) if the weighted functions don’t change signs. Otherwise, the problem is called right-indefinite or Polar (by Hilbert). The spectral theory of the right definite problem with self-adjoint boundary conditions has been accomplished, but the spectral structure of right indefinite problems, especially both right and left indefinite problems, i.e., indefinite problems, is quite different from and more complicated than that of right-definite problems. For example, there is nei-ther upper nor lower bound for real eigenvalues of indefinite S-L boundary problems. What’s more, the indefinite problem may have non-real eigenvalues.Hilb in1907, Richardson in1912and1913, Bocher in1912and Haupt in1915took the lead in studying the characteristics of indefinite S-L problems. Haupt in1915, Richardson in1918first mentioned the possible existence of non-real eigenvalues. Richardson studied the indefinite S-L problem of Richardson equation with Dirichlet boundary condition in which weighted function is the sign function. In1982, Mingarelli got the number’s limitedness of non-real eigenvalues under the normal condition. He further summarized regular indefinite S-L problems in1986and some open problems were given referring to non-real eigenvalues. Two of them are listed as follows.Open problem1. Can one find an a priori bound on the modulus, real and imag-inary parts of the "largest" non-real eigenvalue which might appear?Open problem2. Find a sufficient condition which will guarantee the existence of a non-real eigenvalue for an indefinite problem.Kong, Muller, Wu and Zettl in2003and Zettl again in2005, raised similar prob-lems about the upper and lower bounds of indefinite problems. In1996, Binding and Volkmer again mentioned the necessary and sufficient conditions for the existence of a non-real eigenvalue for an indefinite problem. In2009, Behrndt, Katatbeh and Trunk gave sufficient conditions for the existence of non-real eigenvalues of the singular indef-inite S-L problems with the weighted function being the sign function. Further more, Behrndt, Philipp and Trunk in2013, obtained explicit bounds on non-real eigenvalues for singular indefinite problems under conditions that the weighted function is the sign function and potential functions are essentially bounded. But their methods require the whole real-axis as its essential spectrum, therefore the methods can not be applied to the regular indefinite S-L problems.Until2013, Qi and Chen evaluated the upper bound of non-real eigenvalues with both real and imaginary parts, under Dirichlet boundary condition and the condition that weighted functions change signs for only once or are absolutely continuous. They also got the sufficient conditions for the existence and non-existence of non-real eigen-values under the conditions that there is only one non-negative eigenvalue for corre- sponding right definite problems as well as that both potential functions and weighted functions satisfy symmetry conditions, respectively. Behrndt, Chen. Philipp and Qi evaluated the upper bound of non-real eigenvalues for indefinite S-L problems whose weighted functions can change signs any times. However, all these conclusions needed additional conditions to weighted functions other than their integrability.Based on the achievement and limitedness above, the second half of this disserta-tion will deal with following issues:to evaluate the upper bound of non-real eigenvalues, with both real and imaginary parts, of indefinite S-L problems, under the conditions that the coefficient functions and weighted functions only satisfy the standard condi-tion. Therefore, Open problem1is completely solved which was raised by Mingarelli in1986about the upper bound of non-real eigenvalues. Then, under the condition that the corresponding right definite problems have negative eigenvalues, which is the necessary condition for the existence of non-real eigenvalues, the lower bound is given. Therefore non-real eigenvalues’lower bound evaluation of Open problem1is solved. Besides, the sufficient condition for the existence and non-existence of non-real eigen-values is obtained under standard conditions, which solves Open problem2. As to the boundary problem of the indefinite p-Laplace, the upper bound estimation and the sufficient condition for the non-existence of non-real eigenvalues are obtained. As to indefinite elliptic operators, the evaluation of the upper and lower bounds of non-real eigenvalues is given, as well as the sufficient condition for their existence and non-existence.As far as researching methods are concerned, pure analysis and measure theory are used as tools to evaluate the upper bound of non-real eigenvalues. Then, the operator theories in Krein space are applied to evaluate the lower bound of the non-real eigenvalues of general indefinite operators. The volume which can show the oscillation of weighted function is introduced. This depiction of oscillation is exploited to give more accurate evaluation on the lower bound of non-real eigenvalue and examples are given to illustrate the accuracy of upper and lower bounds. In the research of the existence of non-real eigenvalues, the eigencurve method and the study on the relationship between real and imaginary spectral curves are used. Further, theorems of real spectral curves’monotonicity and the local extreme point are obtained. Finally, the connection problem between real and imaginary spectral curves in three dimensions is solved and therefore the sufficient condition for the existence of non-real eigenvalues is obtained. The methods dealing with indefinite problems are applied to one dimensional reg-ular p-Laplacian problems. The evaluation of the upper bound of non-real eigenvalues is obtained as well as the sufficient condition for the non-existence. Furthermore, the above methods and conclusions are further extended to indefinite elliptic operators. With the increase of dimensions, the methods above are difficult to work. Therefore, Sobolev space theory is applied to evaluate non-real eigenvalues ’upper bound of indef-inite elliptic operators. With operator theory in Krein Space, the sufficient condition for the existence of non-real eigenvalues is obtained under the condition provided that both potential and weight functions are symmetry in the symmetric area under con-sideration.There are six chapters in this dissertation. The first chapter is the introduction part, where the background, main methods and conclusions are presented. The first half part of this dissertation includes the second and third chapters. Chapter2gives the necessary and sufficient condition for the classification of S-L problems with complex-valued coefficients. In Chapter3, one classification of non-symmetric S-L systems is got and described and the J-self-adjoint operator realizations are given. The second half consists of the fourth, fifth and sixth chapters. In Chapter4, there are evaluations on the upper and lower bounds of non-real eigenvalues of indefinite S-L problems. Chapter5provides the upper bound’s evaluation and sufficient conditions for the non-existence of non-real eigenvalues of indefinite p-Laplacian. In the last chapter, the upper and lower bounds evaluation of indefinite elliptic differential operators as well as the sufficient condition for the existence and non-existence of non-real eigenvalues are presented.