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questions-AC

If we connect an L element and a C element in series, then there is one frequency
omega=omega(L,C)=sqrt(1/(LC))
which gives a resonance. For this frequency there is a potential which has zero boundary values but is non-zero inside the network (Note that this simple network consists of 3 vertices and 2 edges; the boundary consists of 2 outer vertices; thus the potential has just a single non-zero value.) The first question is the following. Consider a slightly larger network. For instances consider a network which Feynman used to build his ladder, but with small n (say n=1,2,3,…). Note that for n=1 we have the network described in the beginning of this paragraph. Prove or disprove the following conjecture: in the finite Feynman ladder, for each fixed n, there are at least n different frequencies for which there are non-zero potentials with zero boundary values. If this conjecture is proved or disproved, we’ll see what are the follow-up questions.