Direct and Inverse Sturm-Liouville Problems on Finite Intervals

Abstract

Consider Eq. ( 2.1) on the interval (0, π). If originally the equation is considered on another finite interval (a, b), then by a simple change of the independent variable \(t=\frac {x-a}{b-a}\pi \) it can always be transferred to (0, π). Let q be a real-valued function, q ∈ L2(0, π) and h, H be real numbers. Together with the equation $$\displaystyle -y^{\prime \prime }+q(x)y=\lambda y,\quad 0<x<\pi $$ consider the homogeneous boundary conditions $$\displaystyle y^{\prime }(0)-hy(0)=y^{\prime }(\pi )+Hy(\pi )=0. $$ There exists an infinite sequence of real numbers \(\left \{ \lambda _{n}\right \} _{n=0}^{\infty }\) such that λn < λm if n < m , λn → +∞ when n →∞, and for every λn the equation $$\displaystyle -y_{n}^{\prime \prime }+q(x)y_{n}=\lambda _{n}y_{n},\quad 0<x<\pi $$ admits a nontrivial solution yn satisfying the conditions (3.2).