Eigenfunctions on an infinite Schrödinger network

Abstract In this article, we show that there is a one-to-one correspondence between the eigenfunctions associated with the perturbed Laplacian operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> {\Delta }_{q} on a Schrödinger infinite network <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \{X,t,q\} with weight function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> q\left(a) and the eigenfunctions associated with classical Laplacian operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>Δ</m:mi> </m:math> \Delta on the infinite network <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> <m:mo>.</m:mo> </m:math> \{X,t\}.