Quantum graph as a quantum spectral filter

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We study the transmission of a quantum particle along a straight input--output line to which a graph Γ is attached at a point. In the point of contact we impose a singularity represented by a certain properly chosen scale-invariant coupling with a coupling parameter α. We show that the probability of transmission along the line as a function of the particle energy tends to the indicator function of the energy spectrum of Γ as α→∞. This effect can be used for a spectral analysis of the given graph Γ. Its applications include a control of a transmission along the line and spectral filtering. The result is illustrated with an example where Γ is a loop exposed to a magnetic field. Two more quantum devices are designed using other special scale-invariant vertex couplings. They can serve as a band-stop filter and as a spectral separator, respectively.