Large Coupling Constant Limit Research Articles

Wilson loop and its correlators in the limit of large coupling constant

In this paper we study Wilson loops in various representations for finite and large values of the color gauge group for supersymmetric N=4 gauge theories. We also compute correlators of Wilson loops in different representations and perform a check with the dual gravitational theory.

Trace asymptotics formula for the Schrödinger operators with constant magnetic fields

In this paper, we consider the 2D-Schrödinger operator with constant magnetic field H(V)=(Dx−By)2+Dy2+Vh(x,y), where V tends to zero at infinity and h is a small positive parameter. We will be concerned with two cases: the semi-classical limit regime Vh(x,y)=V(hx,hy), and the large coupling constant limit case Vh(x,y)=h−δV(x,y). We obtain a complete asymptotic expansion in powers of h2 of tr(Φ(H(V),h)), where Φ(⋅,h)∈C0∞(R;R). We also give a Weyl type asymptotics formula with optimal remainder estimate of the counting function of eigenvalues of H(V).

The exactly solvable spin Sutherland model of [formula omitted] type and its related spin chain

We compute the spectrum of the su(m) spin Sutherland model of BN type, including the exact degeneracy of all energy levels. By studying the large coupling constant limit of this model and of its scalar counterpart, we evaluate the partition function of their associated spin chain of Haldane–Shastry type in closed form. With the help of the formula for the partition function thus obtained we study the chainʼs spectrum, showing that it cannot be obtained as a limiting case of its BCN counterpart. The structure of the partition function also suggests that the spectrum of the Haldane–Shastry spin chain of BN type is equivalent to that of a suitable vertex model, as is the case for its AN−1 counterpart, and that the density of its eigenvalues is normally distributed when the number of sites N tends to infinity. We analyze this last conjecture numerically using again the explicit formula for the partition function, and check its validity for several values of N and m.

Discrete Spectrum of the Periodic Schrödinger Operator with a Variable Metric Perturbed by a Nonnegative Potential

We study discrete spectrum in spectral gaps of an elliptic periodic second order differential operator in L 2 (ℝd ) perturbed by a decaying potential. It is assumed that a perturbation is nonnegative and has a power-like behavior at infinity. We find asymptotics in the large coupling constant limit for the number of eigenvalues of the perturbed operator that have crossed a given point inside the gap or the edge of the gap. The corresponding asymptotics is power-like and depends on the observation point.

Nonlinear dynamics of spin and charge in spin-Calogero model

The fully nonlinear dynamics of spin and charge in spin-Calogero model is studied. The latter is an integrable one-dimensional model of quantum spin-1/2 particles interacting through inverse-square interaction and exchange. Classical hydrodynamic equations of motion are written for this model in the regime where gradient corrections to the exact hydrodynamic formulation of the theory may be neglected. In this approximation variables separate in terms of dressed Fermi momenta of the model. Hydrodynamic equations reduce to a set of decoupled Riemann-Hopf (or inviscid Burgers') equations for the dressed Fermi momenta. We study the dynamics of some nonequilibrium spin-charge configurations for times smaller than the time scale of the gradient catastrophe. We find an interesting interplay between spin and charge degrees of freedom. In the limit of large coupling constant the hydrodynamics reduces to the spin hydrodynamics of the Haldane-Shastry model.

The rotational spectrum and the attractive delta-shell potential

The spectrum of the attractive delta-shell potential is investigated in the D-dimensional space (D⩾2). A compact expression for the eigenvalues is derived in the large coupling constant limit for odd dimensions D. With respect to the ground state energy, the spectrum is rotational, being proportional to L(L+D-2), where L is the grand orbital momentum. Extension to even D is achieved on the grounds of the Hellmann–Feynman theorem. Simulating a delta-shell potential by a normalized Gaussian, we discuss finite range effects.

Spectral Shift Function in the Large Coupling Constant Limit

In the large-coupling constant limit we obtain an asymptotic expansion in powers of \( \mu ^{{ - \frac{1} {\delta }}} \) of the derivative of the spectral shift function corresponding to \( {\left( { - \Delta + \mu W(x), - \Delta } \right)} \). Here the potential W(x) is positive and \( W(x) \sim \omega _{0} (\frac{x} {{|x|}})|x|^{{ - \delta }} \) near infinity for some δ> n and ω0 ∈C∞ (Sn-1).

Superconductivity due to massless boson exchange in the strong-coupling limit

We solve the problem of fermionic pairing mediated by a massless boson in the limit of large coupling constant. At weak coupling, the transition temperature is exponentially small and superconductivity is robust against phase fluctuation. In the strong coupling limit, the pair formation occurs at a temperature of the order of the Fermi energy, however, the actual transition temperature is much smaller due to phase and amplitude fluctuations of the pairing gap. Our model calculations describe superconductivity due to color magnetic interactions in quark matter and in systems close to a ferromagnetic quantum critical point with Ising symmetry. Our strong-coupling results are, however, more general and can be applied to other systems as well, including the antiferromagnetic exchange in 2D used for description of the cuprates.

Spectral shift function of the Schrodinger operator in the large coupling constant limit

The spectral shift function of a Schrodinger operator with a perturbation of definite sign is considered. The asymptotics of the spectral shift function for large coupling constant is studied, and results concerning positive and negative perturbations are compared. A more general case of unperturbed operator given by a function of the Laplacian is discussed. This case explains the dependence of the asymptotics of the spectral shift function on the perturbation potential on the one hand and on the order of the unperturbed operator on the other hand.

Spectral Shift Function in the Large Coupling Constant Limit

Given two selfadjoint operators H0 and V=V+−V−, we study the spectrum of the operator H(α)=H0+αV, α>0. We consider the quantity ξ(λ, H(α), H0), λ∈R, which coincides with Krein's spectral shift function of the pair H(α), H0 if V is of trace class and study its asymptotic behavior as α→∞. Applications to differential operators are given.

Spectral shift function of the schrödinger operator in the large coupling constant limit

We consider the spectral shift function where H 0 is a Schrödinger operator with a variable Riemannian metric and an electromagnetic field and V is a perturbation by a multiplication operator. We prove the Weyl type asymptotic formula for in the large coupling constant limit

Variational Approach to Uniform Rigid Rotation of Ginzburg—Landau Vortices

Time periodic solutions for the hyperbolic gauged Ginzburg–Landau system, with spatial domain the unit disc, are shown to exist. Time periodic solutions representing bound states of vortices rotating about one another have been previously obtained in the near self-dual limit, using perturbative techniques. In contrast, we here take a variational approach, the solutions being obtained as critical points of an indefinite functional. We consider a special class of solutions which map out, uniformly in time, an orbit of the rotation group SO(2). It is shown that in the limit of large coupling constant the solutions have nontrivial time dependence or, as is shown to be equivalent, are not radially symmetric in any gauge.

Exact Ground State Energy of the Strong-Coupling Polaron

The polaron has been of interest in condensed matter theory and field theory for about half a century, especially the limit of large coupling constant. It was not until 1983, however, that a proof of the asymptotic formula for the ground state energy was finally given by using difficult arguments involving the large deviation theory of path integrals. Here we derive the same asymptotic result and with explicit error bounds, by simple, rigorous methods applied directly to the Hamiltonian. Our method is easily generalizable to other settings, e.g., the excitonic and magnetic polarons. This version of the article includes an erratum concerning the ultra-violet bound given in the published article. The erratum is to appear in the same journal.

Exact Ground State Energy of the Strong-Coupling Polaron

The polaron has been of interest in condensed matter theory and field theory for about half a century, especially the limit of large coupling constant, α. It was not until 1983, however, that a proof of the asymptotic formula for the ground state energy was finally given by using difficult arguments involving the large deviation theory of path integrals. Here we derive the same asymptotic result, , and with explicit error bounds, by simple, rigorous methods applied directly to the Hamiltonian. Our method is easily generalizable to other settings, e.g., the excitonic and magnetic polarons.

Delta expansion applied to quantum electrodynamics.

A recently proposed technique known as the \ensuremath{\delta} expansion provides a nonperturbative treatment of a quantum field theory. The \ensuremath{\delta}-expansion approach can be applied to electrodynamics in such a way that local gauge invariance is preserved. In this paper it is shown that for electrodynamic processes involving only external photon lines and no external electron lines the \ensuremath{\delta} expansion is equivalent to a fermion loop expansion. That is, the coefficient of ${\mathrm{\ensuremath{\delta}}}^{\mathit{n}}$ in the \ensuremath{\delta} expansion is precisely the sum of all n-electron-loop Feynman diagrams in a conventional weak-coupling approximation. This equivalence does not extend to processes having external electron lines. When external electron lines are present, the \ensuremath{\delta} expansion is truly nonperturbative and does not have a simple interpretation as a resummation of conventional Feynman diagrams. To illustrate the nonperturbative character of the \ensuremath{\delta} expansion we perform a speculative calculation of the fermion condensate in the massive Schwinger model in the limit of large coupling constant.