Half-inverse problem for diffusion operators on the finite interval

Abstract

The potential function q(x) in the regular and singular Sturm-Liouville problem can be uniquely determined from two spectra. Inverse problem for diffusion operator given at the finite interval eigenvalues, normal numbers also on two spectra are solved. Half-inverse spectral problem for a Sturm-Liouville operator consists in reconstruction of this operator by its spectrum and half of the potential. In this study, by using the Hochstadt and Lieberman's method we show that if q(x) is prescribed on [ π 2 , π], then only one spectrum is sufficient to determine q(x) on the interval [0, π 2 ] for diffusion operator.

On the isospectrality of the scalar energy-dependent Schrödingerproblems

“…(1.1) with p(x) ∈ W 1 2 (0, 1) and q(x) ∈ L 2 (0, 1) and under (quasi)-periodic boundary conditions or interior given data were considered in [12,14,29]. This kind of problem was considered in various studies (see [1,2,9,[11][12][13][14][15][16][25][26][27]30,32,34,35]). …”

Determination of differential pencils with spectral parameter dependent boundary conditions from interior spectral data

“…Inverse spectral problems consist in recovering operators from given their spectral characteristics [2][3][4][5][6][7][8][9][10][11]. Some aspects of spectral problems for second-order differential pencils were studied in [12][13][14][15][16][17][18][19][20][21][22][23][24] and other papers.…”

Inverse spectral problems for the Sturm–Liouville operator on a d-star graph

“…Spectral and scattering properties of Schrödinger operator in such structures attracted a considerable attention during past years. There exists an extensive literature devoted to inverse spectral problems for ordinary differential operators on a finite interval; we mention only the literature [2,4,[6][7][8][9][10][11][12]16,[20][21][22]. Recently, the spectral problems of quantum graphs have become a rapidly-developing field of mathematics and mathematical physics, and spectral properties of quantum graphs and different inverse problems have been studied in both forward [13,17,24] and inverse [1,3,5,14,18,19,23,[25][26][27][28], etc.…”