Abstract
ohlich polaron model. It was shown that various innite sequences of non-decreasing improvable lower bound estimates for the polaron ground state energy can be derived for arbitrary values of the electron-phonon interaction constant. The proposed approach allows for explicit numerical evaluation of the thus obtained lower bound estimates at all orders and can be straightforwardly generalized for investigation of the low-lying branch of the slow-moving polaron excitation energy spectral curve adjacent to the ground state energy of the polaron at rest. In conjunction with numerous, already derived by multitudinous methods, well-known upper bound estimates for the energy spectral curve of the Fr ¤ ohlich polaron as a function of the electron-phonon interaction constant and the polaron total momentum, the aforesaid improvable lower bound estimates might provide one with virtually precise magnitude for the energy of the slow-moving polaron.
Highlights
Method of intermediate problems in the theory of linear semi-bounded self-adjoint operators on rigged Hilbert space was applied to the investigation of the ground state energy of the Frohlich polaron model
It is only natural to believe that for large enough set of the trial states (28) would yield reliable lower bounds for the polaron ground state energy. It is already seen from the formalism outlined above that the method of intermediate problems delivers improvable lower bounds and, regarding the choice of the states {|pi, i = 1, . . . , k}, seems to be flexible enough to achieve sufficiently good for all practical purposes lower bounds already for a set of trial states of moderate size k
The last term in (31) can be omitted in the limit n → ∞, En+1 → ∞, and the resulting equation provides the best lower bound available for the particular choice (29) of |p1. It is seen from the structure of (18) that, in general, neither particular choice of only one single trial state |p1 can produce lower bound to the ground state energy higher than the second eigenvalue E20 of the Hamiltonian H0 unless, 0|H |p1 = 0
Summary
It is well known that a local change in the electronic state in a crystal leads to the excitation of crystal lattice vibrations, i. e. the excitation of phonons. This combined quantum state of the moving electron and the accompanying polarization may be considered as a quasiparticle with its own particular characteristics, such as effective mass, total momentum, energy, and maybe other quantum numbers describing the internal state of the quasiparticle in the presence of an external magnetic field or in the case of a very strong lattice polarization that causes self-localization of the electron in the polarization well with the appearance of discrete energy levels Such a quasiparticle is usually called a “polaron state” or a “polaron”.
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