Method of intermediate problems in the Fröhlich polaron model

Abstract

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 Fröhlich polaron model. It was shown that various infinite 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 Fröhlich 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. The polaron concept and the Fröhlich polaron modelIt 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. And vice versa, any local change in the state of the lattice ions alters the local electronic state. This situation is commonly referred to as an "electron-phonon interaction". This interaction manifests itself even at the absolute zero of temperature, and results in a number of specific microscopic and macroscopic phenomena such as, for example, lattice polarization. When a conduction electron with band mass m moves through the crystal, this state of polarization can move together with it. 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 simply a "polaron". Hence, polaron formation is a consequence of dynamic electron-lattice interaction.The concept of the polaron was introduced first by L.D. Landau in a very short paper [1], followed by much more detailed work by S.I. Pekar [2] who investigated the most essential properties of stationary polaron in the limiting case of very intense electron-phonon interaction, so that the polaron behavior could be analyzed in the so-called adiabatic approximation. Subsequently, Landau and Pekar [3] investigated the self-energy and the effective mass of the polaron for the adiabatic or strong-coupling regime. Many other famous researchers, amon...