We show that all self-adjoint extensions of semibounded Sturm–Liouville operators with limit-circle endpoint(s) can be obtained via an additive singular form-bounded self-adjoint perturbation of rank equal to the deficiency indices, say $d\in \{1,2\}$. This characterization generalizes the well-known analog for semibounded Sturm–Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as

$$ \begin{align*} \boldsymbol{A}_{ {}_{\scriptstyle \Theta}}=\boldsymbol{A}_0+\mathbf{B}\Theta\mathbf{B}^*, \end{align*} $$

$\boldsymbol {A}_0$

$\Theta $

$\mathbb {C}^d$

$\boldsymbol {A}_0$

$\mathcal {H}_{-1}(\boldsymbol {A}_0)$

$\Theta $

The merging of boundary triples with perturbation theory provides a more holistic view of the operator’s matrix-valued spectral measures: identifying not just the location of the spectrum, but also certain directional information.

As an example, self-adjoint extensions of the classical Jacobi differential equation (which has two limit-circle endpoints) are obtained, and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.

In this paper, we are going to investigate the canonical property of solutions of systems of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm–Liouville equation with turning point. Using of the asymptotic estimates provided by Eberhard, Freiling, and Schneider for a special fundamental system of solutions of the Sturm–Liouville equation, we study the infinite product representation of solutions of the systems. Then we transform the Sturm–Liouville equation with turning point to the equation with singularity, then we study the asymptotic behavior of its solutions. Such representations are relevant to the inverse spectral problem.

We consider Sturm–Liouville boundary-value problems on the interval $[0,1]$ of the form $-y''+qy=\lambda y$ with boundary conditions $y'(0)\sin\alpha=y(0)\cos\alpha$ and $y'(1)=(a\lambda+b)y(1)$, where $a\lt0$. We show that via multiple Crum–Darboux transformations, this boundary-value problem can be transformed ‘almost’ isospectrally to a boundary-value problem of the same form, but with the boundary condition at $x=1$ replaced by $y'(1)\sin\beta=y(1)\cos\beta$, for some $\beta$.

AMS 2000 Mathematics subject classification: Primary 34B07; 47E05; 34L05

We consider the Sturm–Liouville equation

$$ -y''+qy=\lambda y\quad\text{on }[0,1], $$

subject to the boundary conditions

$$ y(0)\cos\alpha=y'(0)\sin\alpha,\quad\alpha\in[0,\pi), $$

and

$$\frac{y'}{y}(1)=a\lambda+b-\sum_{k=1}^N\frac{b_k}{\lambda-c_k}. $$

Topics treated include existence and asymptotics of eigenvalues, oscillation of eigenfunctions, and transformations between such problems.

AMS 2000 Mathematics subject classification: Primary 34B24; 34L20