On recovering Sturm–Liouville operators with two delays

We study the inverse spectral problems of recovering Sturm–Liouville differential operator with two constant delays <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>a</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {a₁} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>a</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> {a₂} greater than one third of the interval. It has been proved that the operator can be recovered uniquely from four spectra under the condition <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:msub> <m:mi>a</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mfrac> <m:msub> <m:mi>a</m:mi> <m:mn>2</m:mn> </m:msub> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo>≥</m:mo> <m:mi>π</m:mi> </m:mrow> </m:math> {2a₁+{a₂}{2}≥π} , while it is not possible otherwise.