**1. Introduction**

The spectral theory of operators finds numerous uses in various fields of mathematics and their applications.

An important part of the spectral theory of differential operators is the distribution of their eigenvalues. This classical question was studied for a second-order operator on a finite interval by Liouville and Sturm. Later, G.D. Birkhoff [1–3] studied the distribution of eigenvalues for an ordinary differential operator of arbitrary order on a finite interval with regular boundary conditions.

For quantum mechanics, it is especially interesting to distribute the eigenvalues of operators defined throughout the space and having a discrete spectrum. E.C. Titchmarsh [4–9] was the first to rigorously establish the formula for the distribution of the number of eigenvalues for a one-dimensional Sturm-Liouville operator on the whole axis with potential growing at infinity. He also first strictly established the distribution formula for the Schrödinger operator. B.M. Levitan [10–12] deserves much credit for the improvement of E.C. Titchmarsh's method.

In solving many mathematical physics problems, the need arises for the expansion of an arbitrary function in a Fourier series with respect to Sturm-Liouville eigenvalues. The so-called regular case of the Sturm-Liouville problem corresponding to a finite interval and a continuous coefficient of the equation has been studied for a relatively long time and is usually described in detail in the manuals on the equations of mathematical physics and integral equations.

The Sturm-Liouville problem for the so-called singular case, as well as with nonlocal boundary conditions, is much less known.

As it is known, so-called fractal media are studied in solid-state physics and, in particular, diffusion phenomena in them. In one of the models studied in [13], diffusion in a strongly porous (fractal) medium is described by an equation of the type of heat-conduction equation, but with a fractional derivative with respect to time coordinate

$$D\_t^{(a)}u(\mathbf{x},t) = \frac{\partial^2(u(\mathbf{x},t)}{\partial \mathbf{x}^2} \; , \; 0 < a < 1. \tag{1}$$

The formulation of initial-boundary value problems for Equation (1), similar to the problems for parabolic differential equations, makes sense if by a regularized fractional derivative:

$$D^{(a)}\varphi(t) = \frac{1}{\Gamma(1-a)} \left[ \frac{d}{dt} \int\_0^1 (t-\tau)^{-a} \varphi(\tau) d\tau - t^{-a} \varphi(0) \right], \ t \ge 0 \tag{2}$$

Study of the form equations

$$D\_t^{(a)}u = Au$$

where *A* is an elliptic operator (in [14–16]). In recent years, many authors studied fractional differential equations in [17–34].
