Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues

In this work, we consider the inverse spectral problem for the impulsive Sturm–Liouville problem on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>π</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> {(0,π)} with the Robin boundary conditions and the jump conditions at the point <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mi>π</m:mi> <m:mn>2</m:mn> </m:mfrac> </m:math> {π/2} . We prove that the potential <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {M(x)} on the whole interval and the parameters in the boundary conditions and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potential <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {M(x)} is given on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mfrac> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>α</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>π</m:mi> </m:mrow> <m:mn>4</m:mn> </m:mfrac> <m:mo>)</m:mo> </m:mrow> </m:math> {(0,(1+α)π/4)} ; (ii) the potential <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {M(x)} is given on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mfrac> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>α</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>π</m:mi> </m:mrow> <m:mn>4</m:mn> </m:mfrac> <m:mo>,</m:mo> <m:mi>π</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> {((1+α)π/4,π)} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>α</m:mi> <m:mo><</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0<α<1} , respectively. It is also shown that the potential and all the parameters can be uniquely recovered by one spectrum and some information on the eigenfunctions at some interior point.