Abstract
The problem of construction a quantum mechanical evolution for the Schrödinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have selfadjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of these terms are corresponded to the unitary and shift components of Wold’s decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.
Similar content being viewed by others
References
G. Fichera, “On a unified theory of boundary value problems for elliptic-parabolic equations of second order,” Boundary Problems in Differential Equations, 97–120 (Univ. of Wisconsin Press, Madison, 1960).
O. A. Oleinik and E. V. Radkevich, “Second order equations with nonnegative characteristic form,” Itogi Nauki, Ser. Mat.,Mat. Anal. 1969, 7–252 (1971) [in Russian].
A. D. Ventcel’ and M. I. Freidlin, “Some problems concerning stability under small random perturbations,” Theory Probab. Appl. 17 (2), 269–283 (1973).
I. V. Volovich and V. Zh. Sakbaev, “Universal boundary value problem for equations ofmathematical physics,” Proc. Steklov Inst. Math. 285 (1), 56–80 (2014).
V. Zh. Sakbaev and O.G. Smolyanov, “Diffusion and quantum dynamics of particles with position-dependent mass,” DokladyMath. 86 (1), 460–463 (2012).
V. Zh. Sakbaev, “Averaging of quantum dynamical semigroups,” Theor. Math. Phys. 164 (3), 1215–1221 (2010).
V. Zh. Sakbaev, “On dynamics of quantum states generated by the Cauchy problem for the Schroedinger equation with degeneration on the half-line, J. Math. Sci. 151 (1), 2741–2753 (2008).
V. Zh. Sakbaev, “On the Cauchy problem for linear differential equation with degeneration and the averaging of its regularization,” Sovrem. Mat. Fundam.Napravl. 43, 3–174 (2012).
M. Gadella, S. Kuru and J. Negro, “Self-adjoint Hamiltonians with a mass jump: General matching conditions,” Phys. Lett. A 362, 265–268 (2007).
M. Gadella and O. G. Smolyanov, “Feynman formulas for particles with position-dependent mass,” Doklady Math. 77 (1), 120–123 (2007).
V. L. Chernyshev, A. A. Tolchennikov and A. I. Shafarevich, “Behavior of quasi-particles on hybrid spaces. Relations to the geometry of geodesics and to the problems of analytic number theory,” Regul. Chaotic Dyn. 21 (5), 531–537 (2016).
Di Perna and P. Lions, “Ordinary differential equation, transport theory and Sobolev spaces,” Invent. Math. V 98), 511–547 (1989).
M. Ohya and I. V. Volovich, Mathematical Foundations of Quantum Information and Computation and its Applications to Nano- and Bio-Systems (Springer, Dordrecht, 2011).
L. Accardi, Yu. G. Lu and I. V. Volovich, Quantum Theory and its Stochastic Limit (Springer, 2002).
A. S. Trushechkin and I. V. Volovich, “Perturbative treatment of inter-site couplings in the local description of open quantum networks,” EPL 113 (3), 30005 (2016).
I. Ya. Aref’eva, I. V. Volovich and S. V. Kozyrev, “Stochastic limit method and interference in quantum manyparticle systems,” Theor. Math. Phys. 183 (3), 782–799 (2015).
I. V. Volovich and S. V. Kozyrev, “Manipulation of states of a degenerate quantum system,” Proc. Steklov Inst.Math. 294, 241–251 (2016).
O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics (Springer-Verlag, 1979).
M. Takesaki, “On the conjugate space of operator algebra,” TohikuMath. J. 10, 194–203 (1958).
W. I. M. Wils, “Stone-Cech compactification and representations of operator algebras,” http://hdl.handle.net/2066/107571.
V. Zh. Sakbaev, “On the variational description of the trajectories of averaging quantum dynamical maps,” p-Adic Numbers Ultrametric Anal. Appl. 4 (2), 120–134 (2012).
G. F. Dell’ Antonio, “On the limits of sequences of normal states,” Comm. Pure Appl. Math. 20, 413–429 (1967).
B. Sz.-Nagy and C. Foias, Analyse harmonique des operateurs de l’espace de Hilbert (1967).
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the authors in English.
Rights and permissions
About this article
Cite this article
Sakbaev, V.Z., Volovich, I.V. Self-adjoint approximations of the degenerate Schrödinger operator. P-Adic Num Ultrametr Anal Appl 9, 39–52 (2017). https://doi.org/10.1134/S2070046617010046
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046617010046