Inverse nodal problems for Sturm–Liouville operators with many frozen arguments

In this paper, we study inverse nodal problems for Sturm–Liouville operators with many frozen arguments. We obtain the asymptotics of eigenvalues and nodal points (zeros) of the corresponding eigenfunction. Moreover, we establish a uniqueness theorem concerning the potential under condition <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msubsup> <m:mi>W</m:mi> <m:mn>2</m:mn> <m:mn>1</m:mn> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>π</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {q∈ W₂¹[0,π]} . Numerical examples are presented to show the validity and effectiveness of the numerical scheme. Since the potential <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {q(x)} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>π</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {q∈ L₂[0,π]} cannot be recovered by the classical method, we use an alternative method to reconstruct the potential as an application of numerical methods.