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Sturm-Liouville operators with measure-valued coefficients. (English) Zbl 1315.34001
The paper provides a comprehensive overview of spectral theory of Sturm-Liouville operators with measure-valued coefficients. The classical Sturm-Liouville theory deals with spectral problems $-\frac{d}{dx}p(x)\frac{dy}{dx} + q(x)y = z r(x)\, y,\quad x\in I, \tag{1}$ where the coefficients $$p$$, $$q$$ and $$r$$ are real-valued locally integrable functions on the interval $$I$$. During the last decades there was a significant progress in understanding Sturm-Liouville differential expressions with measure-valued and even with distributional coefficients. The understanding of these spectral problems is crucial in the study of numerous problems of mathematical physics. For example, they include the case of $$\delta$$ and $$\delta'$$ interactions, both are important models in solid state physics; they arise as isospectral problems for various nonlinear integrable equations (the KdV equation with non-smooth initial data; the Camassa-Holm equation; the Dym equation etc.).
In fact, spectral problems $\frac{d}{d\varrho(x)}\left( -\frac{dy}{d\varsigma(x)}(x) + \int^x y(t)d\chi(t)\right) = z\, y(x),\quad x\in I, \tag{2}$ date back to the works of M. G. Krein and W. Feller in the 1950s (for a detailed historical discussion, see the monograph [A. B. Mingarelli, Volterra-Stieltjes integral equations and generalized ordinary differential expressions. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag (1983; Zbl 0516.45012)].
Now let me briefly describe the content of the paper. Sections 1 and 2 are of introductory character. Basic properties of solutions of Sturm-Liouville spectral problems are discussed in Section 3. The next 4 sections deal with self-adjoint realizations of (2) in the Hilbert space $$L^2(I;\varrho)$$. More precisely, in Section 5, the authors discuss minimal and maximal relations associated with (2) in $$L^2(I;\varrho)$$. An extension of the classical Weyl alternative is presented in the next section. Self-adjoint realizations and the corresponding boundary conditions are discussed in Sections 6 an 7. Basic facts on spectra and resolvents of self-adjoint realizations are collected in Section 8. The final sections contain the so-called singular Weyl-Titchmarsh-Kodaira theory.

##### MSC:
 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 34B24 Sturm-Liouville theory 34B20 Weyl theory and its generalizations for ordinary differential equations 47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX) 34L05 General spectral theory of ordinary differential operators
##### Keywords:
Sturm-Liouville theory; Weyl-Titchmarsh-Kodaira theory
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