# Complex eigenvalues in Kuryshkin-Wodkiewicz quantum mechanics

**Authors:**Zorin A.V.^{1}, Malykh M.D.^{1}^{,2}, Sevastianov L.A.^{1}^{,2}-
**Affiliations:**- Peoples’ Friendship University of Russia (RUDN University)
- Joint Institute for Nuclear Research

**Issue:**Vol 30, No 2 (2022)**Pages:**139-148**Section:**Articles**URL:**https://journals.rudn.ru/miph/article/view/30952**DOI:**https://doi.org/10.22363/2658-4670-2022-30-2-139-148

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## Abstract

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 $A$ is associated with a pseudodifferential operator $\hat{O}\left(A\right)$. 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 $\psi $ the quantity $A$ is equal to $\lambda $ if $\psi $ is an eigenfunction of the operator $\hat{O}\left(A\right)$. In this case, the variance $\hat{O}\left(\right(A-\lambda \left)2\right)\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 $\hat{O}\left(\right(A-\lambda \left)2\right)$ is zero. The spectrum of the quadratic pencil $\hat{O}\left(A2\right)-2\hat{O}\left(A\right)\lambda +\lambda 2\hat{E}$ is studied by the methods of perturbation theory under the assumption of small variance $\hat{D}\left(A\right)=\hat{O}\left(A2\right)-\hat{O}\left(A\right)2$ of the observable $A$. It is shown that in the neighborhood of the real eigenvalue $\lambda $ of the operator $\hat{O}\left(A\right)$, there are two eigenvalues of the operator pencil, which differ in the first order of perturbation theory by $\pm i\sqrt{\u27e8\hat{D}\u27e9}$.

## Full Text

1. Introduction The Kuryshkin-Wodkiewicz quantum mechanics [1] is an example of a dissipative quantum system. The quantum part of the measuring device is the ‘environment of an open quantum system’. In the process of quantum measurement, an open quantum system interacts with its ‘environment’. We study the result of this interaction [2]-[12]. Therefore, wave vectors must have a finite lifetime, inversely proportional to the imaginary part of eigenvalues. In this version of quantum mechanics, the von Neumann quantization rule was abandoned and observable quantities are assigned to pseudo-differential operators, not necessarily self-adjoint. Therefore, the appearance of the imaginary part of the eigenvalues is not surprising. However, our studies of hydrogen-like atoms have shown that the operator corresponding to the Hamiltonian is essentially self-adjoint, so its discrete spectrum turned out to be real [13], [14]. This is quite surprising, since the von Neumann rule can be derived from general considerations, if we assume that the relation between the quantities## About the authors

### Alexander V. Zorin

Peoples’ Friendship University of Russia (RUDN University)
Email: zorin-av@rudn.ru

ORCID iD: 0000-0002-5721-4558

Candidate of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation### Mikhail D. Malykh

Peoples’ Friendship University of Russia (RUDN University); Joint Institute for Nuclear Research
Email: malykh-md@rudn.ru

ORCID iD: 0000-0001-6541-6603

Doctor of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation### Leonid A. Sevastianov

Peoples’ Friendship University of Russia (RUDN University); Joint Institute for Nuclear Research
**Author for correspondence.**

Email: sevastianov-la@rudn.ru

ORCID iD: 0000-0002-1856-4643

Doctor of Physical and Mathematical Sciences, Professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation## References

- V. V. Kuryshkin, “La mécanique quantique avec une fonction nonnégative de distribution dans l’espace des phases”, Annales Henri Poincaré. Physique théorique, vol. 17, no. 1, pp. 81-95, 1972.
- U. Weiss, Quantum dissipative systems, 4th ed. World Scientific, 2012. doi: 10.1142/8334.
- H.-P. Breuer and F. Petruccione, The theory of open quantum systems. Oxford: Oxford University Press, 2002.
- V. E. Tarasov, Quantum mechanics of non-Hamiltonian and dissipative systems. Elsevier, 2008.
- M. Ahmadi, D. Jennings, and T. Rudolph, “The Wigner-Araki-Yanase theorem and the quantum resource theory of asymmetry”, New Journal of Physics, vol. 15, no. 1, p. 013057, 2013. doi: 10.1088/1367-2630/15/1/013057.
- M. Ozawa, “Uncertainty relations for noise and disturbance in generalized quantum measurements”, Annals of Physics, vol. 311, no. 2, pp. 350- 416, 2004. doi: 10.1016/j.aop.2003.12.012.
- M. Ozawa, “Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement”, Physical Review A, vol. 67, no. 4, p. 042105, 2003. doi: 10.1103/PhysRevA.67. 042105.
- P. Busch and P. J. Lahti, “The standard model of quantum measurement theory: history and applications”, Foundations of Physics, vol. 26, pp. 875-893, 1996. doi: 10.1007/BF02148831.
- A. S. Holevo, Statistical structure of quantum theory. Springer, 2001.
- J. A. Wheeler and W. H. Zurek, Quantum theory and measurement. Princeton University Press, 1983.
- K. Jacobs, Quantum measurement theory and its applications. Cambridge University Press, 2014. doi: 10.1017/CBO9781139179027.
- W. H. Zurek, “Quantum theory and measurement in complexity”, in Complexity, Entropy And The Physics Of Information, W. H. Zurek, Ed., CRC Press, 1990, ch. 6.
- A. V. Zorin and L. A. Sevastianov, “Hydrogen-like atom with nonnegative quantum distribution function”, Physics of Atomic Nuclei, vol. 70, no. 4, pp. 792-799, 2007. doi: 10.1134/S1063778807040229.
- A. V. Zorin, “Kuryshkin-Wodkiewicz quantum measurement model for alkaline metal atoms”, Discrete and Continuous Models and Applied Computational Science, vol. 28, no. 3, pp. 274-288, 2020. doi: 10.22363/2658-4670-2020-28-3-274-288.
- J. P. Rybakov and J. P. Terleckij, Quantum mechanics [Kvantovaja mehanika]. Moscow: RUDN, 1991, in Russian.
- L. Cohen, “Can quantum mechanics be formulated as a classical probability theory?”, Philosophy of Science, vol. 33, no. 4, pp. 317-322, 1966. doi: 10.1086/288104.
- K.JorgensandJ.Weidmann, Spectral properties of Hamiltonian operators. Berlin, Heidelberg, New York: Springer, 1973.
- A. V. Zorin and L. A. Sevastianov, “Bottom estimation method for the eigenvalues of the Hamilton differential operator in Kuryshkin quantum mechanics [Metod ocenok snizu dlja sobstvennyh znachenij differencial’nogo operatora Gamil’tona v kvantovoj mehanike Kuryshkina]”, RUDN Journal. Seria “Prikladnaja i komp’juternaja matematika”, vol. 1, no. 1, pp. 134-144, 2002, in Russian.
- A. V. Zorin, L. A. Sevastianov, and N. P. Tretyakov, “States with MinimumDispersionofObservablesinKuryshkin-WodkiewiczQuantum Mechanics”, Lecture Notes in Computer Science,vol.11965,no.4,pp.508- 519, 2019. doi: 10.1007/978-3-030-36614-8_39.
- R. H. Nevanlinna, Uniformisierung. Berlin: Springer, 1953.
- T. Kato, Perturbation theory for linear operators. Berlin: Springer, 1966.
- A. N. Bogolyubov, M. D. Malykh, and A. G. Sveshnikov, “Instability of eigenvalues embedded in the waveguide’s continuous spectrum with respect to perturbations of its filling”, Proceedings of the USSR Academy of Sciences, vol. 385, no. 6, pp. 744-746, 2002.
- A. N. Bogolyubov and M. D. Malykh, “On the perturbation theory of spectral characteristics of waveguides”, Computational Mathematics and Mathematical Physics, vol. 43, no. 7, pp. 1004-1015, 2003.
- V. P. Shestopalov, Spectral theory and excitation of open structures [Spektral’naja teorija i vozbuzhdenie otkrytyh struktur]. Moscow: Nauka, 1987, in Russian.