Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Condensed Matter Physics

Method of intermediate problems in the Fröhlich polaron model

Soldatov

Published Jan 1, 2009• 2 citations
Publisher

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 Fr öhlich 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 Fr öhlich 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.

DOI
10.5488/cmp.12.4.665

Related papers

On lower bound estimates for the energy spectrum of the Fröhlich polaron model

Andrey Vladimirovich Soldatov

Method of intermediate problems was applied to investigation of the energy spectrum of the Fröhlich polaron model. It was shown that various infinite sequences of non-decreasing improvable lower bound estimates for the low-lying branch of the slow-moving polaron excitation energy spectral curve adjacent to the ground state energy can be derived for arbitrary values of the electron-phonon interaction constant. These bound estimates allow for explicit numerical evaluation at all orders. In conjunction with numerous well-known upper bound estimates for the energy spectral curve of the Fröhlich polaron as a function of the 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. Possible generalization to the case of the Fröhlich polaron in external magnetic field is outlined.

Physics of Particles and Nuclei•Nov 25, 2010•0 citations

Infinitely improvable upper bounds in the theory of polarons

А. В. Солдатов, V. A. Steklov

An infinite convergent sequence of improving non-increasing upper bounds to the low-lying branch of the slow-moving "physical" Fröhlich polaron ground-state energy spectral curve, adjacent to the ground state energy of the polaron at rest, was obtained by means of generalized variational method. The proposed approach is especially well-suited for massive analytical and numerical computations of experimentally measurable properties of realistic polarons, such as inverse effective mass tensor and excitation gap, and can be elaborated even further, without major alterations, to allow for treatment of multitudinous polaron-like models, those describing polarons of various sorts placed in external magnetic and electric fields among them.

•Jan 1, 2010•0 citations

Infinitely improvable upper bounds in the theory of polarons

Soldatov

An infinite convergent sequence of improving non-increasing upper bounds to the low-lying branch of the slowmoving "physical" Fr öhlich polaron ground-state energy spectral curve, adjacent to the ground state energy of the polaron at rest, was obtained by means of generalized variational method.The proposed approach is especially well-suited for massive analytical and numerical computations of experimentally measurable properties of realistic polarons, such as inverse effective mass tensor and excitation gap, and can be elaborated even further, without major alterations, to allow for treatment of multitudinous polaron-like models, those describing polarons of various sorts placed in external magnetic and electric fields among them.

Condensed Matter Physics•Jan 1, 2010•0 citations

Infinitely Improvable Upper Bound Estimates for Acoustical Polaron Ground State Energy

Nikolay N. Bogolubov, Soldatov, A. V.

It was shown that an infinite convergent sequence of improving non-increasing upper bounds to the ground state energy of a slow-moving acoustical polaron can be obtained by means of generalized variational method. The proposed approach is especially well-suited for massive analytical and numerical computations of experimentally measurable properties of realistic polarons and can be elaborated even further, without major alterations, to allow for treatment of various polaron-type models.

arXiv:1311.6278•Nov 25, 2013•0 citations

Weak and Intermediate Coupling Solutions of the Fröhlich Hamiltonian

Ashok Chatterjee, Soma Mukhopadhyay

After deriving the quantum mechanical model polaron Hamiltonian, Frohlich, Pelzar and Zineau presented a perturbative solution to the problem. This chapter examines the perturbation theory from Frohlich et al. and presents other important weak and intermediate coupling polaron theories. It discusses several polaronic properties like average number of virtual phonons in the polaron cloud, electron life-time, lattice polarization potential using the first-order perturbative wave function and obtained the polaron energy and the polaron mass using the second-order Rayleigh-Schrodinger perturbation theory (RSPT). The second-order RSPT is known to have a limited range of validity. In the one-phonon Tamm-Dancoff approximation, the exact ground state energy appears in the denominator instead of the unperturbed energy that appears in RSPT. The chapter also discusses the work of Lee and Pines who have used the Tomonaga's approximation made in the formulation of the meson-theoretic problem to the case of polaron.

CRC Press eBooks•Oct 9, 2018•0 citations

Upper and Lower Bounds for the Intermediate-Coupling Polaron Ground-State Energy

David M. Larsen

Using a variational ansatz in which pair correlations between wave vectors of virtually emitted phonons are taken into account, we have obtained better upper bounds to the polaron ground-state energy than have been available heretofore for $\ensuremath{\alpha}\ensuremath{\le}3.5$. The same variational trial functions are also used to obtain lower bounds for $\ensuremath{\alpha}\ensuremath{\le}2.5$. These lower bounds, although not completely rigorous, represent a considerable improvement over the Lieb-Yamazaki values, which are, to the author's knowledge, the only other lower bounds in the literature. The variational ansatz chosen is also suitable for practical calculations of the polaron effective mass.

Physical Review•Aug 15, 1968•50 citations
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 Fr öhlich 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 Fr öhlich 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.
Method of intermediate problems was applied to investigation of the energy spectrum of the Fröhlich polaron model. It was shown that various infinite sequences of non-decreasing improvable lower bound estimates for the low-lying branch of the slow-moving polaron excitation energy spectral curve adjacent to the ground state energy can be derived for arbitrary values of the electron-phonon interaction constant. These bound estimates allow for explicit numerical evaluation at all orders. In conjunction with numerous well-known upper bound estimates for the energy spectral curve of the Fröhlich polaron as a function of the 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. Possible generalization to the case of the Fröhlich polaron in external magnetic field is outlined.
An infinite convergent sequence of improving non-increasing upper bounds to the low-lying branch of the slow-moving "physical" Fröhlich polaron ground-state energy spectral curve, adjacent to the ground state energy of the polaron at rest, was obtained by means of generalized variational method. The proposed approach is especially well-suited for massive analytical and numerical computations of experimentally measurable properties of realistic polarons, such as inverse effective mass tensor and excitation gap, and can be elaborated even further, without major alterations, to allow for treatment of multitudinous polaron-like models, those describing polarons of various sorts placed in external magnetic and electric fields among them.
An infinite convergent sequence of improving non-increasing upper bounds to the low-lying branch of the slowmoving "physical" Fr öhlich polaron ground-state energy spectral curve, adjacent to the ground state energy of the polaron at rest, was obtained by means of generalized variational method.The proposed approach is especially well-suited for massive analytical and numerical computations of experimentally measurable properties of realistic polarons, such as inverse effective mass tensor and excitation gap, and can be elaborated even further, without major alterations, to allow for treatment of multitudinous polaron-like models, those describing polarons of various sorts placed in external magnetic and electric fields among them.
After deriving the quantum mechanical model polaron Hamiltonian, Frohlich, Pelzar and Zineau presented a perturbative solution to the problem. This chapter examines the perturbation theory from Frohlich et al. and presents other important weak and intermediate coupling polaron theories. It discusses several polaronic properties like average number of virtual phonons in the polaron cloud, electron life-time, lattice polarization potential using the first-order perturbative wave function and obtained the polaron energy and the polaron mass using the second-order Rayleigh-Schrodinger perturbation theory (RSPT). The second-order RSPT is known to have a limited range of validity. In the one-phonon Tamm-Dancoff approximation, the exact ground state energy appears in the denominator instead of the unperturbed energy that appears in RSPT. The chapter also discusses the work of Lee and Pines who have used the Tomonaga's approximation made in the formulation of the meson-theoretic problem to the case of polaron.
Using a variational ansatz in which pair correlations between wave vectors of virtually emitted phonons are taken into account, we have obtained better upper bounds to the polaron ground-state energy than have been available heretofore for $\ensuremath{\alpha}\ensuremath{\le}3.5$. The same variational trial functions are also used to obtain lower bounds for $\ensuremath{\alpha}\ensuremath{\le}2.5$. These lower bounds, although not completely rigorous, represent a considerable improvement over the Lieb-Yamazaki values, which are, to the author's knowledge, the only other lower bounds in the literature. The variational ansatz chosen is also suitable for practical calculations of the polaron effective mass.