Unique reconstruction of the inverse spectral problem with mixed data for AKNS operator

Abstract In this paper, the inverse spectral problem of the AKNS operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">ℋ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{H}(p,q)} defined on the interval <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> {[0,1]} with the potential pair <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(p,q)} being a priori known on the subinterval <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mfrac> <m:mi>r</m:mi> <m:mi>m</m:mi> </m:mfrac> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> {[0,\frac{r}{m}]} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mfrac> <m:mi>r</m:mi> <m:mi>m</m:mi> </m:mfrac> <m:mo>&lt;</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> </m:math> {0&lt;\frac{r}{m}&lt;\frac{1}{2}} is considered. An algorithm of reconstruction of the potential from the knowledge of a full spectrum together with a subset of the norming constants is established.