We look for best partitions of the unit interval that minimize certain functionals defined in terms of the eigenvalues of Sturm–Liouville problems. Via Γ-convergence theory, we study the asymptotic distribution of the minimizers as the number of intervals of the partition tends to infinity. Then we discuss several examples that fit in our framework, such as the sum of (positive and negative) powers of the eigenvalues and an approximation of the trace of the heat Sturm–Liouville operator.

We give an upper estimate for the order of the entire functions in the Nevanlinna parameterization of the solutions of an indeterminate Hamburger moment problem. Under a regularity condition this estimate becomes explicit and takes the form of a convergence exponent. Proofs are based on transformations of canonical systems and I.S.Kac' formula for the spectral asymptotics of a string. Combining with a lower estimate from previous work, we obtain a class of moment problems for which order can be computed. This generalizes a theorem of Yu.M.Berezanskii about spectral asymptotics of a Jacobi matrix (in the case that order is ⩽ 1/2).

We consider a class of singular Schrödinger operators H that act in L2(0,∞), each of which is constructed from a positive function ϕ on (0,∞). Our analysis is direct and elementary. In particular it does not mention the potential directly or make any assumptions about the magnitudes of the first derivatives or the existence of second derivatives of ϕ. For a large class of H that have discrete spectrum, we prove that the eigenvalue asymptotics of H does not depend on rapid oscillations of ϕ or of the potential. Similar comments apply to our treatment of the existence and completeness of the wave operators.

We consider a discontinuous Sturm–Liouville equation together with two supplementary transmission conditions at the point of discontinuity. We suggest our own approach for finding asymptotic approximation formulas for the eigenvalues of such discontinuous problems.

Under certain assumptions on $g(x)$, we obtain an asymptotic formula for computing integrals of the form

$$ F(x,\alpha)=\int_{-\infty}^\infty g(t)^\alpha\exp\biggl(-\bigg|\int_x^tg(\xi)\,\mathrm{d}\xi\bigg|\biggr)\,\mathrm{d}t,\quad\alpha\in\mathbb{R}, $$

as $|x|\to\infty$. We use this formula to study the properties (as $|x|\to\infty$) of the solutions of the correctly solvable equations in $L_p(\mathbb{R})$, $p\in[1,\infty]$,

\begin{equation} -y''(x)+q(x)y(x)=f(x),\quad x\in\mathbb{R}, \tag{1} \end{equation}

where $0\le q\in L_1^{\mathrm{loc}}(\mathbb{R})$, and $f\in L_p(\mathbb{R})$. (Equation (1) is called correctly solvable in a given space $L_p(\mathbb{R})$ if for any function $f\in L_p(\mathbb{R})$ it has a unique solution $y\in L_p(\mathbb{R})$ and if the following inequality holds with an absolute constraint $c_p\in(0,\infty)$: $\|y\|_{L_p(\mathbb{R})}\leq c(p)\|f\|_{L_p(\mathbb{R})}$, $\forall f\in L_p(\mathbb{R})$.)

We solve the inverse spectral problems for the class of Sturm–Liouville operators with singular real-valued potentials from the Sobolev space $W^{s-1}_2(0,1)$, $s\in[0,1]$. The potential is recovered from two spectra or from one spectrum and the norming constants. Necessary and sufficient conditions for the spectral data to correspond to a potential in $W^{s-1}_2(0,1)$ are established.

For a given Sturm-Liouville equation whose leading coefficient function changes sign, we establish inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions. By a recent result of Binding and Volkmer, the eigenvalues (unbounded from both below and above) for a separated self-adjoint boundary condition can be numbered in terms of the Prüfer angle; and our inequalities can then be used to index the eigenvalues for any coupled self-adjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such Sturm-Liouville problems, and its range as a function on the space of self-adjoint boundary conditions. We also relate this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme.