Method Of Intermediate Problems Research Articles

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

Method of intermediate problems was applied to investigation of the energy spectrum of the Frohlich 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 Frohlich 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 Frohlich polaron in external magnetic field is outlined.

Method of intermediate problems in the Fröhlich polaron model

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.

Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field

Applicability of the method of intermediate problems to the investigation of the energy eigenvalues and eigen- states of a quantum dot (QD) formed by a Gaussian confining potential in the presence of an external mag- netic field is discussed. Being smooth at the QD boundaries and of finite depth and range, this potential can only confine a finite number of excess electrons thus forming a realistic model of a QD with smooth inter- face between the QD and its embedding environment. It is argued that the method of intermediate problems, which provides convergent improvable lower bound estimates for eigenvalues of linear half-bound Hermitian operators in Hilbert space, can be fused with the classical Rayleigh-Ritz variational method and stochastic variational method thus resulting in an efficient tool for an alytical and numerical studies of the energy spec- trum and eigenstates of the Gaussian quantum dots, confining small-to-medium number of excess electrons, with controllable or prescribed precision. Various theoretical and experimental aspects of the physics of nano-sized and low-dimensional systems have been under steady uninterrupted development for decades, but only in the last ten- fifteen years of research activities in the field, considerable extra momentum has been gained which can be partly ascribed to the progress in nanofabrication of these systems, partly to the development of new experimental techniques but, most of all, to a general continuous trend and desire to diminish the size of components of integrated electronic circuits down to the so-called mesoscopic scale. Among low-dimensional systems of various kinds, quantum dots are potentially fit for many practical applications and are currently thought to be promising building blocks for novel electronic, spintronic and optoelectronic devices. Reliable estimation of the energy spectrum and eigenstates of a quantum dot is a typical purpose of almost every theoretical study because their properties crucially stipulate the relevant physical characteristics of the quantum dot standing alone as a part of electric circuits or interacting with the environment through its various interfaces. As a matter of fact, nearly all the mathematical methods, developed within the domain of quan- tum mechanics so far, have already been employed in the theory of quantum dots in various specific applications though on a varying scale. The most frequently applicable methods of approximate calculation of eigenvalues and eigenstates of realistic physical models of low-dimensional quantum systems, quantum dots among them, are various numerical methods which permit either direct calculations of the magnitudes in question without proper error estimates or, at best, provide the calculations with nonincreasing or even convergent upper bounds for the eigenvalues. The widely applicable Rayleigh-Ritz method does it but, again, without error estimates. To control the error of the approximations provided by the upper bounds for some quantity it would be enough to derive the corresponding lower bounds which are highly desirable to be

ON THE STOCHASTIC VARIATIONAL APPROACH TO LOWER BOUNDS FOR ENERGY EIGENVALUES OF FEW-BODY QUANTUM SYSTEMS

It is shown that the method of intermediate problems, which provides improvable convergent lower bounds for eigenvalues of linear half-bound Hermitian operators in Hilbert space, can be linked in a natural way to the stochastic variational method, thus resulting in an effective tool for analytical and numerical investigation of the energy spectrum and eigenstates of various few-body quantum systems, among them quantum dots with relatively large numbers of electrons.

Lower and upper bounds to photonic band gap edges

A photonic band gap is determined by its edges, which are frequently calculated by the Rayleigh-Ritz method, producing a sequence of upper bounds. Since there are no error estimates available on these approximations, the level of accuracy of the computed band-gap edges, and thus of the extent of the band gap, remains undetermined. We adopt the method of intermediate problems to develop a procedure to calculate the lower bounds to the photonic band-gap edges. Lower bounds, supplemented with the Rayleigh-Ritz approximations, determine a band gap with arbitrary and known degree of accuracy, as well as provide error bounds on approximate calculations by other methods. Calculation of the lower bounds by the present method requires slightly more computational effort than the Rayleigh-Ritz method with a plane-wave basis. A parallel method to compute the alternative upper bounds is also developed in the process.

Methods for computing lower bounds to eigenvalues of self-adjoint operators

New approaches for computing tight lower bounds to the eigenvalues of a class of semibounded self-adjoint operators are presented that require comparatively little a priori spectral information and permit the effective use of (among others) finite-element trial functions. A variant of the method of intermediate problems making use of operator decompositions having the form\(T^*T\) is reviewed and then developed into a new framework based on recent inertia results in the Weinstein-Aronszajn theory. This framework provides greater flexibility in analysis and permits the formulation of a final computational task involving sparse, well-structured matrices. Although our derivation is based on an intermediate problem formulation, our results may be specialized to obtain either the Temple-Lehmann method or Weinberger's matrix method.

A rigorous justification and estimation of the rate of convergence for the partial domain method in two-dimensional eigenvalue problems for the Laplace operator

The partial domain method for problems of the type indicated in the title of this paper is treated as a version either of Weinstein's method of intermediate problems or of the Ritz method. This yields a rigorous justification of the method and enables one to estimate its rate of convergence. The justification technique is demonstrated in detail for the problem of the normal modes of an L-shaped membrane clamped at its edges.

An estimate of the rate of convergence of Weinstein's method in the eigenvalue problem for an L-shaped membrane

General estimates for the rate of convergence of Weinstein's method of intermediate problems, obtained previously by the author, are applied particularly to the case of the eigenvalue problem for the L-shaped membrane attached at its edges.

An Extension of Aronszajn’S Rule: Slicing the Spectrum for Intermediate Problems

We introduce an equivalent formulation of Aronszajn’s rule for the method of intermediate problems that provides more information of computational importance. The new formulation provides a mechani...

Corrigendum: Convergence theorems for intermediate problems

Professor R. D. Brown has brought to our attention an error in our Theorem 4.2. This would, in turn, invalidate Theorem 3.6 which depends on Theorem 4.2. The basic flaw is that the alignment hypothesis of Proposition 4.1 is not satisfied by the product space projections, Qn. Interestingly, such product space constructions are a common computational device in applications of the T*T method, but apparently seldom satisfy the alignment hypothesis of Proposition 4.1. We wish here to correct these theorems, and comment on the resulting implications for Example 6.1. In independent and as yet unpublished work, Professor Brown has obtained further new results on convergence of the method of intermediate problems.

The rate of convergence of the Bazley-Fox method of intermediate problems in the generalized eigenvalue problem of the form Au = λCu

The rate of convergence of the Bazley-Fox method of intermediate problems in the generalized eigenvalue problem of the form Au = λCu