Abstract
In this paper, regular and singular fourth order differential operators with distributional potentials are investigated. In particular, existence and uniqueness of solutions of the fourth order differential equations are proved, deficiency indices theory of the corresponding minimal symmetric operators are studied. These symmetric operators are considered as acting on the single and direct sum Hilbert spaces. The latter one consists of three Hilbert spaces such that a squarely integrable space and two spaces of complex numbers. Moreover all maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal symmetric operators including direct sum operators are given in the single and direct sum Hilbert spaces.
Highlights
Weyl theory is an important tool to understand the nature of the solutions of an differential equation on an unbounded domain
In 1910, Weyl proved that following second order differential equation
Two linearly indpendent solutions of (1.1) and any combinations of them may be squarely integrable. These results are based on the nested property of the corresponding circles which are related with the regular boundary conditions
Summary
Weyl theory is an important tool to understand the nature of the solutions of an differential equation on an unbounded domain. We investigate the number of the squarely integrable solutions of the following fourth order differential equation q2 y(2) − s1y(1) + s2y (1) + q2s1 y(2) − s1y(1) − q1(y(1) + s4y) + s3y (1) +q2s2y(2) − s3y(1) + q1s4 y(1) + s4y + q0y = λwy (1.5). (1.7) is the well-known fourth order Sturm-Liouville equation and it should be noted that the number of the squarely integrable solutions of (1.7) with w ≡ 1 was investigated by Everitt in 1963 [4]. This investigation was done with the help of the nested property of the corresponding surfaces.
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