Quantum / CONCEPT

The twin end vertex of P'_9 is not sharply 1/9-sedentary

In the graph P'_9, the path on vertices 1 to 9 with an extra vertex 10 joined to vertex 2, the return amplitude of the quantum walk U(t) = exp(itA) at vertex 1 satisfies |U(t)_(1,1)| >= 5/18 for every real t. So vertex 1 is not sharply 1/9-sedentary, which refutes the conjecture of H. Monterde (arXiv:2401.00362) that vertex 1 of P'_n is sharply 1/n-sedentary for every odd n >= 5.