# Pivovarchik V. N.

### On the relationship between the multiplicities of eigenvalues in finite- and infinite-dimensional problems on graphs

Boyko O. P., Martinyuk O. M., Pivovarchik V. N.

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 445-455

It is shown that some results concerning the multiplicities of eigenvalues of the spectral problem that describes small transverse vibrations of a star graph of Stieltjes strings and the multiplicities of the eigenvalues of tree-patterned matrices can be used for the description of possible multiplicities of normal eigenvalues (bound states) of the Sturm – Liouville operator on a star graph.

### Inverse Sturm-Liouville problem on a figure-eight graph

Gomilko A. M., Pivovarchik V. N.

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1168–1188

We study the inverse problem for the Strum-Liouville equation on a graph that consists of two quasione-dimensional loops of the same length having a common vertex. As spectral data, we consider the set of eigenvalues of the entire system together with the sets of eigenvalues of two Dirichlet problems for the Sturm-Liouville equations with the condition of total reflection at the vertex of the graph. We obtain conditions for three sequences of real numbers that enable one to reconstruct a pair of real potentials from L 2 corresponding to each loop. We give an algorithm for the construction of the entire set of potentials corresponding to this triple of spectra.

### On spectra of a certain class of quadratic operator pencils with one-dimensional linear part

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 702–716

We consider a class of quadratic operator pencils that occur in many problems of physics. The part of such a pencil linear with respect to the spectral parameter describes viscous friction in problems of small vibrations of strings and beams. Patterns in the location of eigenvalues of such pencils are established. If viscous friction (damping) is pointwise, then the operator in the linear part of the pencil is one-dimensional. For this case, rules in the location of purely imaginary eigenvalues are found.

### Asymptotics of Solutions of the Sturm–Liouville Equation with Respect to a Parameter

Gomilko A. M., Pivovarchik V. N.

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 742-757

On a finite segment [0, *l*], we consider the differential equation $$\left( {a\left( x \right)y\prime \left( x \right)} \right)\prime + \left[ {{\mu \rho }_{\text{1}} \left( x \right) + {\rho }_{2} \left( x \right)} \right]y\left( x \right) = 0$$ with a parameter μ ∈ *C*. In the case where *a*(*x*), ρ(*x*) ∈ *L* _{∞}[0, *l*], ρ_{ j }(*x*) ∈ *L* _{1}[0, *l*], *j* = 1, 2, *a*(*x*) ≥ *m* _{0} > 0 and ρ(*x*) ≥ *m* _{1} > 0 almost everywhere, and *a*(*x*)ρ(*x*) is a function absolutely continuous on the segment [0, *l*], we obtain exponential-type asymptotic formulas as \(\left| {\mu } \right| \to \infty\) for a fundamental system of solutions of this equation.