V. N. Pivovarchik
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Methods Funct. Anal. Topology 22 (2016), no. 1, 74-80
We consider a generalization of the three spectral inverse problem, that is, for given spectrum of the Dirichlet-Dirichlet problem (the Sturm-Liouville problem with Dirichlet conditions at both ends) on the whole interval $[0,a]$, parts of spectra of the Dirichlet-Neumann and Dirichlet-Dirichlet problems on $[0,a/2]$ and parts of spectra of the Dirichlet-Newman and Dirichlet-Dirichlet problems on $[a/2,a]$, we find the potential of the Sturm-Liouville equation.
Methods Funct. Anal. Topology 20 (2014), no. 2, 164-174
A spectral problem generated by the Stieltjes string recurrence relations with a finite number of point masses on a connected graph is considered with Neumann conditions at pendant vertices and continuity and Kirchhoff conditions at interior vertices. The strings on the edges are supposed to be the same and symmetric with respect to the midpoint of the string. The characteristic function of such a problem is expressed via characteristic functions of two spectral problems on an edge: one with Dirichlet conditions at the both ends and the other one with the Neumann condition at one end and the Dirichlet condition at the other end. This permits to find values of the point masses and the lengths of the subintervals into which the masses divide the string from knowing the spectrum of the problem on the graph and the length of an edge. If the number of vertices is less than five then the spectrum uniquely determines the form of the graph.
Methods Funct. Anal. Topology 18 (2012), no. 2, 189-197
Known connection between discrete and continuous Laplacians in case of same symmetric potential on the edges of a quantum graph is used to construct characteristic functions of quantum graphs and to find some parameters of graphs using spectra of boundary value problems.
Methods Funct. Anal. Topology 16 (2010), no. 4, 349-358
We describe the spectrum of the problem generated by the Stieltjes string recurrence relations on a figure-of-eight graph. The continuity and the force balance conditions are imposed at the vertex of the graph. It is shown that the eigenvalues of such (main) problem are interlaced with the elements of the union of sets of eigenvalues of the Dirichlet problems generated by the parts of the string which correspond to the loops of the figure-of-eight graph. Also the eigenvalues of the main problem are interlaced with the elements of the union of sets of eigenvalues of the periodic problems generated by the same parts of the string.
Methods Funct. Anal. Topology 14 (2008), no. 2, 159-167
We solve the inverse spectral problem for a star graph of Stieltjes strings (these are threads bearing a finite number of point masses) with the pendant ends fixed, i.e., we recover the masses and lengths of the intervals between them from the spectra of small transverse vibrations of the graph together with the spectra of the Dirichlet problems on the edges and the total lengths of the edges.
Methods Funct. Anal. Topology 14 (2008), no. 1, 10-19
Small transversal vibrations of the Stieltjes string, i.e., an elastic thread bearing point masses is considered for the case of one end being fixed and the other end moving with viscous friction in the direction orthogonal to the equilibrium position of the string. The inverse problem of recovering the masses, the lengths of subintervals and the coefficient of damping by the spectrum of vibrations of such a string and its total length is solved.
Methods Funct. Anal. Topology 13 (2007), no. 2, 187-200
We investigate the subclass of symmetric indefinite Hermite-Biehler functions which is obtained from positive definite Hermite-Biehler functions by means of the square-transform. It is known that functions of this class can be characterized in terms of location of their zeros. We give another, more elementary and geometric, proof of this result. The present proof employs a `shifting-of-zeros' perturbation method. We apply our results to obtain information on the eigenvalues of a concrete boundary value problems.
Methods Funct. Anal. Topology 11 (2005), no. 3, 270-274
Methods Funct. Anal. Topology 6 (2000), no. 4, 26-42
Methods Funct. Anal. Topology 4 (1998), no. 3, 61-71