Mathematics at Lund University

This thesis studies Sturm–Liouville operators on a compact interval with a single interior delta perturbation. More precisely, we consider eigenvalue problems of the form $$y'' + qy + \kappa y(x_0)\,\delta(x - x_0) = \lambda y, \qquad By = 0,$$ where $q \in C[a,b]$, $x_0 \in (a,b)$, $\kappa \in \mathbb{R}$ and $B$ denotes a homogeneous self-adjoint boundary operator. This problem is understood in the sense of distributions, with solutions naturally belonging to the space $$W^{2,2}_{\delta}(a,b) := W^{1,2}(a,b) \cap W^{2,2}\!\left((a,b) \setminus \{x_0\}\right).$$

It is first shown that the homogeneous equation $y'' + qy + \kappa y(x_0)\delta(x - x_0) = 0$ has a two-dimensional solution space. The thesis then studies the inhomogeneous boundary value problem $$y'' + qy + \kappa y(x_0)\,\delta(x - x_0) = f + \mu\,\delta_{x_0}, \qquad By = 0,$$ where $f \in L^2(a,b)$ and $\mu \in \mathbb{R}$. If $G$ denotes the Green's function associated with the classical Sturm–Liouville operator $L = D^2 + q$, then under the condition $1 + \kappa G(x_0,x_0) \neq 0$ the Green's function of the operator with delta interaction is of the form $$G_\delta(x,\xi) = G(x,\xi) - \frac{\kappa\, G(x,x_0)\, G(x_0,\xi)}{1 + \kappa G(x_0,x_0)}.$$ The exceptional case $1 + \kappa G(x_0,x_0) = 0$ is shown to be precisely the case in which the corresponding homogeneous boundary value problem has a non-trivial solution.

Finally, it is shown that the Green operator associated with $G_\delta$ is compact and symmetric in $L^2(a,b)$. Spectral theory for compact self-adjoint operators then yields a discrete spectral structure for the operator with delta interaction, including orthogonality of eigenfunctions.