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Adiabatic approximation for the evolution generated by an \(A\)-uniformly pseudo-Hermitian Hamiltonian. (English. Russian original) Zbl 1382.81101

Theor. Math. Phys. 192, No. 3, 1365-1379 (2017); translation from Teor. Mat. Fiz. 192, No. 3, 489-505 (2017).
Summary: We discuss an adiabatic approximation for the evolution generated by an \(A\)-uniformly pseudo-Hermitian Hamiltonian \(H(t)\). Such a Hamiltonian is a time-dependent operator \(H(t)\) similar to a time-dependent Hermitian Hamiltonian \(G(t)\) under a time-independent invertible operator \(A\). Using the relation between the solutions of the evolution equations \(H(t)\) and \(G(t)\), we prove that \(H(t)\) and \(H^†(t)\) have the same real eigenvalues and the corresponding eigenvectors form two biorthogonal Riesz bases for the state space. For the adiabatic approximate solution in case of the minimum eigenvalue and the ground state of the operator \(H(t)\), we prove that this solution coincides with the system state at every instant if and only if the ground eigenvector is time-independent. We also find two upper bounds for the adiabatic approximation error in terms of the norm distance and in terms of the generalized fidelity. We illustrate the obtained results with several examples.

MSC:

81Q12 Nonselfadjoint operator theory in quantum theory including creation and destruction operators
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q41 Time-dependent Schrödinger equations and Dirac equations
35A35 Theoretical approximation in context of PDEs
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