The set of zeros of the Riemann zeta function as the point spectrum of an operator

A possible way of proving the Riemann hypothesis consists of constructing a selfadjoint operartor whose spectrum coincides with the set { z : | Im ⁡ z | > 1 2 ,   ζ ( 1 2 − i z ) = 0 } \{z\,: \, |\operatorname {Im}z|>\frac 12, \ \zeta \big (\frac {1}{2}-iz\big )=0\} . In the paper we construct a rank-one perturbation of a selfadjoint operator related to a certain canonical system for which a similar property is fulfilled.