Analogues of Borg's uniqueness theorem in the case of nonseparated boundary conditions
The object of investigation is the Sturm-Liouville problem with nonseparated boundary conditions (problem L) of the form \[ -y''+q(x)y=\lambda^2y, \] \[ y'(0)+a_{11}y(0)+a_{12}y(1)=y'(1)+a_{21}y(0)+a_{22}y(1)=0, \] where \(a_{ij}\), \(j=1,2\), are real numbers and \(q(x)\in C^1[0,1]\). It is clear that the spectrum of problem \(L\) does not determine uniquely the set \(\{q(x), a_{11}, a_{12}, a_{21}, a_{22}\}\). Therefore, the authors consider the following auxiliary problems: problem \(\text{L}_1\): \[ -y''+q(x)y=\lambda^2y, \] \[ y'(0)+a_{21}y(0)=y'(1)+a_{22}y(1)=0, \] problem \(\text{L}_2\): \[ -y''+q(x)y=\lambda^2y, \] \[ y'(0)+a_{21}y(0)=y'(1)+a^*_{22}y(1)=0, \quad a_{22}\not=a^*_{22}, \] and problem R (generalized Regge problem, see \textit{G. M. Gubreev} and \textit{V. N. Pivovarchik} [Funct. Anal. 31, No. 1, 54-57 (1997); translation from Funkts. anal. Prilozh. 31, No. 1, 70-74 (1997; Zbl 0907.34069)]): \[ -y''+q(x)y=\lambda^2y, \] \[ y(0)=y'(1)+(ia_{21}\lambda+a_{22})y(1)=0. \] The authors prove that the spectra of problems \(L\), \(L_1\), \(L_2\) as well as the spectra of problems \(L\), \(R\) uniquely determine the set \(\{q(x), a_{11}, a_{12}, a_{21}, a_{22}\}\).
scientific article; zbMATH DE number 1557790
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