Ordinary Differential Operators with Complex Coefficients

Abstract

<p> The object of this dissertation is to investigate the properties, associated boundary conditions and generalized resolvents of symmetric ordinary differential operators associated with formally self-adjoint nth order ordinary differential expressions with complex coefficients. </p> <p> While symmetric differential operators with equal deficiency indices have been studied in some detail, expecially the particular case when the underlying differential expression has real coefficients, little research has been done on the properties of symmetric differential operators with unequal deficiency indices which are associated with a differential expression with complex coefficients. </p> <p> By extending the symmetric differential operators with unequal deficiency indices to suitable operators with equal deficiency indices in larger Hilbert spaces and introducing a new type of boundary conditions to these extensions, we obtain important information about the original symmetric differential operators with unequal deficiency indices. We are able to generate some well-known theorems of I. M. Glazman (1950) and E. A. Coddington (1954) dealing with the characterization of self-adjoint extensions of symmetric operators in terms of boundary conditions, the relation between the deficiency indices of operators on the whole real line and on the half-line, and the resolvent of self-adjoint extensions, from the theory of symmetric, in particular real, differential operators with equal deficiency indices. We also generalize the result of W. N. Everitt (1959) concerning the number of integrable-square solutions of differential equations with one particular and one singular end-point to the case in which both end-points are singular. Finally, under certain assumptions, we extend some of the fundamental results of K. Kodaira (1950) based upon the methods of algebraic geometry, concerning Green's functions and the minimal symmetric differential operator associated with an even-order formally self-adjoint ordinary differential expansion with real coefficients to the case of Green's functions and the minimal symmetric differential operator associated with an even-order formally self-adjoint ordinary differential expression with complex coefficients. </p>