This book consists of two parts in which different aspects connected with spectrum problems for the Sturm-Liouville equation \(-y''+q(x)y=\lambda y\) and the one-dimensional Dirac operator \(By'+P(x)y=\lambda y\), \[ y=\left( \begin{matrix} y_1\\ y_2\end{matrix} \right),\quad B=\left( \begin{matrix} 0 & 1\\ -1 & 0\end{matrix}\right), \quad P(x)=\left( \begin{matrix} p_{11}(x) & p_{12}(x)\\ p_{21}(x) &p_{22}(x) \end{matrix}\right), \quad p_{12}(x)=p_{21}(x), \] are investigated. Starting with simple examples of boundary value problems with the above smooth operators on a bounded segment the authors give their contemporary treatment in singular cases. There are boundary value problems on an unbounded interval and those with unbounded potentials.
The authors treat such questions as asymptotics of eigenvalues of the problems on an unbounded interval and those with unbounded potentials.
The authors treat such questions as asymptotics of eigenvalues of the problems, expansions in eigenfunctions, properties of spectrum, asymptotics of distribution of eigenvalues in average, calculations of regulated traces, and some others. In particular, the inverse problems of the reconstruction of the operator for the given spectrum function or for two given spectra (for the Sturm-Liouville problem) are considered. The non-stationary Dirac problem is also studied. Part of the published material is a product of the authors and their pupils, it cannot be found in other books. The authors give new complete proofs for some well-known theorems many of which have been published in the famous book of E. C. Titchmarsh [Eigenfunction expansions associated with second order differential equations. Oxford: At the Clarendon Press (1946; Zbl 0061.13505)]. To read this book knowing of other books on the treated theory is not required. The completeness of the material and the clarity of the proofs make this book useful for both researchers and graduate students in mathematics and physics. It is desirable to translate this book into English.