Spectral Asymptotics of Self-Adjoint Fourth Order Differential Operators with Eigenvalue Parameter Dependent Boundary Conditions

Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications

“…Theorem 2.7.2 was proved in [195]. Some other spectral problems generated by fourth-order ordinary differential equation with dissipative terms can be found in [197], [198], [199].…”

“…For asymptotics of eigenvalues of boundary value problems generated by a fourth-order differential equations and spectral parameter dependent boundary conditions see [195], [197], [198], [199].…”

Asymptotics of the Eigenvalues of a Self-Adjoint Fourth Order Boundary Value Problem with Four Eigenvalue Parameter Dependent Boundary Conditions

“…In the remainder of the section, we establish more precise asymptotic expansions of the eigenvalues. For this, it is more convenient to replace ( ) 4 =1 with the asymptotic fundamental system ( ] ) 4 ]=1 obtained in [10, Theorem 8.2.1], (1), which can be written as ; ] = 1, . .…”

“…Necessary and sufficient conditions such that the associated operator pencil consists of self-adjoint operators have been obtained. In [4,5], we have derived eigenvalue asymptotics associated with particular boundary conditions. In this paper, we are considering the case of separable boundary conditions where all four of these boundary conditions depend on the eigenvalue parameter.…”

Sixth order differential operators with eigenvalue dependent boundary conditions

“…For the same differential operator as in [19], we have investigated a more general class of eigenvalue parameter dependent boundary conditions. Necessary and sufficient conditions for the associated operator pencil to consist of selfadjoint operators have been obtained in [21], while in [22,23] we have continued the work in the direction of [19] to find the asymptotic distribution of eigenvalues for boundary conditions which lead to self-adjoint operator representations. In this paper we start to extend this investigation to a corresponding problem for a sixth order differential equation.…”