Quantum graph as a quantum spectral filter

We study the transmission of a quantum particle along a straight input-output line to which a graph Gamma 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 alpha. 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 Gamma as alpha -> infinity. This effect can be used for a spectral analysis of the given graph Gamma. Its applications include a control of a transmission along the line and spectral filtering. The result is illustrated with an example where Gamma 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. (C) 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4795404]