Complex eigenvalues in Kuryshkin-Wodkiewicz quantum mechanics

One of the possible versions of quantum mechanics, known as Kuryshkin-Wodkiewicz quantum mechanics, is considered. In this version, the quantum distribution function is positive, but, as a retribution for this, the von Neumann quantization rule is replaced by a more complicated rule, in which an observed value AA is associated with a pseudodifferential operator O^(A){hat{O}(A)}. This version is an example of a dissipative quantum system and, therefore, it was expected that the eigenvalues of the Hamiltonian should have imaginary parts. However, the discrete spectrum of the Hamiltonian of a hydrogen-like atom in this theory turned out to be real-valued. In this paper, we propose the following explanation for this paradox. It is traditionally assumed that in some state ψ{psi} the quantity AA is equal to λ{lambda} if ψ{psi} is an eigenfunction of the operator O^(A){hat{O}(A)}. In this case, the variance O^((A-λ)2)ψ{hat{O}((A-lambda)2)psi} is zero in the standard version of quantum mechanics, but nonzero in Kuryshkin’s mechanics. Therefore, it is possible to consider such a range of values and states corresponding to them for which the variance O^((A-λ)2){hat{O}((A-lambda)2)} is zero. The spectrum of the quadratic pencil O^(A2)-2O^(A)λ+λ2E^{hat{O}(A2)-2hat{O}(A)lambda + lambda 2 hat{E}} is studied by the methods of perturbation theory under the assumption of small variance D^(A)=O^(A2)-O^(A)2{hat{D}(A) = hat{O}(A2) - hat{O}(A) 2} of the observable AA. It is shown that in the neighborhood of the real eigenvalue λ{lambda} of the operator  O^(A){hat{O}(A)}, there are two eigenvalues of the operator pencil, which differ in the first order of perturbation theory by  ±i⟨D^⟩{pm i sqrt{langle hat{D} rangle}}.