Sturm–Liouville operators

Abstract:

Let $(a,b)\subset \mathbb {R}$ be a finite or infinite interval, let $p_0(x)$, $q_0(x)$, and $p_1(x)$, $x\in (a,b)$, be real-valued measurable functions such that $p_0,p^{-1}_0$, $p^2_1p^{-1}_0$, and $q^2_0p^{-1}_0$ are locally Lebesgue integrable (i.e., lie in the space $L^1_{\operatorname {loc}}(a,b)$), and let $w(x)$, $x\in (a,b)$, be an almost everywhere positive function. This paper gives an introduction to the spectral theory of operators generated in the space $\mathcal {L}^2_w(a,b)$ by formal expressions of the form \begin{equation*} l[f]:=w^{-1}\{-(p_0f’)’+i[(q_0f)’+q_0f’]+p’_1f\}, \end{equation*} where all derivatives are understood in the sense of distributions. The construction described in the paper permits one to give a sound definition of the minimal operator $L_0$ generated by the expression $l[f]$ in $\mathcal {L}^2_w(a,b)$ and include $L_0$ in the class of operators generated by symmetric (formally self-adjoint) second-order quasi-differential expressions with locally integrable coefficients. In what follows, we refer to these operators as Sturm–Liouville operators. Thus, the well-developed spectral theory of second-order quasi-differential operators is used to study Sturm–Liouville operators with distributional coefficients. The main aim of the paper is to construct a Titchmarsh–Weyl theory for these operators. Here the problem on the deficiency indices of $L_0$, i.e., on the conditions on $p_0$, $q_0$, and $p_1$ under which Weyl’s limit point or limit circle case is realized, is a key problem. We verify the efficiency of our results for the example of a Hamiltonian with $\delta$-interactions of intensities $h_k$ centered at some points $x_k$, where \begin{equation*} l[f]=-f”+\sum _{j}h_j\delta (x-x_j)f. \end{equation*}