Minimality of the system of root functions of Sturm–Liouville problems with decreasing affine boundary conditions

Abstract

Abstract. We consider Sturm-Liouville problems with a boundary condition linearly dependent on the eigenparameter. We study the case of decreasing dependence where non-real and multiple eigenvalues are possible. By determining the explicit form of a biorthogonal system, we prove that the system of root (i.e. eigen and associated) functions, with an arbitrary element removed, is a minimal system in L 2 (0, 1), except for some cases where this system is neither complete nor minimal.

Sturm-Liouville problems with a boundary condition depending affinely or quadratically on an eigenparameter

“…The notations and the preliminary results in this section were given in [11,12] and some of them are included here just for completeness. We define y(x, λ)…”

On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition

“…By using Rouche's theorem again, it is easy to see that there is only one root of equation (2.10) at the neighborhood O n −1 of the number n − 1 2 π (n ∈ N) for sufficiently large n.…”

“…Sturm-Liouville problems with the boundary conditions depending on the spectral parameter were studied in order to investigate their various properties in many articles (see [1,2,[6][7][8][10][11][12][13][14][15][16][17][19][20][21][22][23]25]). The problems on the basis property of system of root functions corresponding to Sturm-Liouville problems for some differential operators which contain different forms (linearly, rationally, quadratically etc.)…”

“…The problems on the basis property of system of root functions corresponding to Sturm-Liouville problems for some differential operators which contain different forms (linearly, rationally, quadratically etc.) of spectral parameter in one of the boundary conditions were considered in [1,2,8,10,12,15,16,22].…”

“…Because of the assumption b 2 ̸ = of the theorem, ψ n (1) ̸ = 0 for all n. Assume that n ̸ = m, r; m ̸ = r. The equality (ψ n , φ m ) = a n (ψn(x),ψm(1−x))+bλm(ψn(x),ψm(x)) ψm(1) − (ψn(x),ψr(1−x))+bλr(ψn(x),ψr(x))ψr(1) …”