arXiv subjects
Publications and source records attributed to Edwin Langmann.
We introduce variational methods for finding approximate eigenfunctions and eigenvalues of quantum Hamiltonians by constructing a set of orthogonal wave functions which approximately solve the eigenvalue equation.
We discuss interacting fermion models in two dimensions, and, in particular, such that can be solved exactly by bosonization. One solvable model of this kind was proposed by Mattis as an effective description of fermions on a square lattice. We review recent work on a specific relation between a variant of Mattis' model and such a lattice fermion system, as well as the exact solution of this model. The background for this work includes well-established results for one-dimensional systems and the high-Tc problem. We also mention exactly solvable extensions of Mattis' model.
We study an exactly solvable quantum field theory (QFT) model describing interacting fermions in 2+1 dimensions. This model is motivated by physical arguments suggesting that it provides an effective description of spinless fermions on a square lattice with local hopping and density-density interactions if, close to half filling, the system develops a partial energy gap. The necessary regularization of the QFT model is based on this proposed relation to lattice fermions. We use bosonization methods to diagonalize the Hamiltonian and to compute all correlation functions. We also discuss how, after appropriate multiplicative renormalizations, all short- and long distance cutoffs can be removed. In particular, we prove that the renormalized two-point functions have algebraic decay with non-trivial exponents depending on the interaction strengths, which is a hallmark of Luttinger-liquid behavior.
The Inozemtsev Hamiltonian is an elliptic generalization of the differential operator defining the BC_N trigonometric quantum Calogero-Sutherland model, and its eigenvalue equation is a natural many-variable generalization of the Heun differential equation. We present kernel functions for Inozemtsev Hamiltonians and Chalykh-Feigin-Veselov-Sergeev-type deformations thereof. Our main result is a solution of a heat-type equation for a generalized Inozemtsev Hamiltonian which is the source for all these kernel functions. Applications are given, including a derivation of simple exact eigenfunctions and eigenvalues for the Inozemtsev Hamiltonian.
We consider models of bosons on curved 3+1 dimensional space-time embedded in a higher dimensional flat ambient space. We propose to derive (rather than postulate) equations of motions by assuming that a standard Klein-Gordon field on ambient space is restricted to space-time by a strong confining potential. This leads to a modified Klein-Gordon equation on space-time which includes, in addition to the standard terms, a term with a so-called induced potential which depends on intrinsic- and extrinsic curvature of the embedded space-time but not on the details of the confining potential. We compute this induced potential for natural, simple embeddings of Schwarzschild- and Robertson-Walker space-times. We also discuss possible observable implications of our results and, in particular, propose and study an extension of a standard model of cosmological inflation taking into account extrinsic curvature effects. We show that the modified model allows for a solution where the scaling function vanishes like a power law with exponent 0.6830.. at some initial time.
Kernel functions related to quantum many-body systems of Calogero-Sutherland type are discussed, in particular for the elliptic case. The main result is an elliptic generalization of an identity due to Sen that is a source for many such kernel functions. Applications are given, including simple exact eigenfunctions and corresponding eigenvalues of Chalykh-Feigin-Veselov-Sergeev-type deformations of the elliptic Calogero-Sutherland model for special parameter values.
This article surveys the application of the representation theory of loop groups to simple models in quantum field theory and to certain integrable systems. The common thread in the discussion is the construction of quantum fields using vertex operators. These examples include the construction and solution of the Luttinger model and other 1+1 dimensional interacting quantum field theories, the construction of anyon field operators on the circle, the `2nd quantization' of the Calogero-Sutherland model using anyons and the geometric construction of quantum fields on Riemann surfaces. We describe some new results on the elliptic Calogero-Sutherland model. (This paper was written in 2001, and we want to make a version of it more accessible because it is a reference for us for subsequent work.)
We compute mean field phase diagrams of two closely related interacting fermion models in two spatial dimensions (2D). The first is the so-called 2D t-t'-V model describing spinless fermions on a square lattice with local hopping and density-density interactions. The second is the so-called 2D Luttinger model that provides an effective description of the 2D t-t'-V model and in which parts of the fermion degrees of freedom are treated exactly by bosonization. In mean field theory, both models have a charge-density-wave (CDW) instability making them gapped at half-filling. The 2D t-t'-V model has a significant parameter regime away from half-filling where neither the CDW nor the normal state are thermodynamically stable. We show that the 2D Luttinger model allows to obtain more detailed information about this mixed region. In particular, we find in the 2D Luttinger model a partially gapped phase that, as we argue, can be described by an exactly solvable model.
A detailed derivation of a two dimensional (2D) low energy effective model for spinless fermions on a square lattice with local interactions is given. This derivation utilizes a particular continuum limit that is justified by physical arguments. It is shown that the effective model thus obtained can be treated by exact bosonization methods. It is also discussed how this effective model can be used to obtain physical information about the corresponding lattice fermion system.
We present a fermion model that is, as we suggest, a natural 2D analogue of the Luttinger model. We derive this model as a partial continuum limit of a 2D spinless lattice fermion system with local interactions and away from half filling. In this derivation, we use certain approximations that we motivate by physical arguments. We also present mathematical results that allow an exact treatment of parts of the degrees of freedom of this model by bosonization, and we propose to treat the remaining degrees of freedom by mean field theory.
In this paper we consider a large class of many-variable polynomials which contains generalisations of the classical Hermite, Laguerre, Jacobi and Bessel polynomials as special cases, and which occur as the polynomial part in the eigenfunctions of Calogero-Sutherland type operators and their deformations recently found and studied by Chalykh, Feigin, Sergeev, and Veselov. We present a unified and explicit construction of all these polynomials.
We derive explicit formulas for the eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland model as infinite series, to all orders and for arbitrary particle numbers and coupling parameters. The eigenfunctions obtained provide an elliptic deformation of the Jack polynomials. We prove in certain special cases that these series have a finite radius of convergence in the nome $q$ of the elliptic functions, including the two particle (= Lamé) case for non-integer coupling parameters.
I present the exact energy eigenstates and eigenvalues of a quantum many-body system of bosons on non-commutative space and in a harmonic oszillator confining potential at the selfdual point. I also argue that this exactly solvable system is a prototype model which provides a generalization of mean field theory taking into account non-trivial correlations which are peculiar to boson systems in two space dimensions and relevant in condensed matter physics. The prologue and epilogue contain a few remarks to relate my main story to recent developments in non-commutative quantum field theory and an addendum to our previous work together with Szabo and Zarembo on this latter subject.
There exists a large class of quantum many-body systems of Calogero-Sutherland type where all particles can have different masses and coupling constants and which nevertheless are such that one can construct a complete (in a certain sense) set of exact eigenfunctions and corresponding eigenvalues, explicitly. Of course there is a catch to this result: if one insists on these eigenfunctions to be square integrable then the corresponding Hamiltonian is necessarily non-hermitean (and thus provides an example of an exactly solvable PT-symmetric quantum-many body system), and if one insists on the Hamiltonian to be hermitean then the eigenfunctions are singular and thus not acceptable as quantum mechanical eigenfunctions. The standard Calogero-Sutherland Hamiltonian is special due to the existence of an integral operator which allows to transform these singular eigenfunctions into regular ones.
We consider the quantum Calogero model, which describes N non-distinguishable quantum particles on the real line confined by a harmonic oscillator potential and interacting via two-body interactions proportional to the inverse square of the inter-particle distance. We elaborate a novel solution algorithm which allows us to obtain fully explicit formulas for its eigenfunctions, for arbitrary coupling parameter and particle number. We also show that our method applies, with minor changes, to all Calogero models associated with classical root systems.
We review a method providing explicit formulas for the Jack polynomials. Our method is based on the relation of the Jack polynomials to the eigenfunctions of a well-known exactly solvable quantum many-body system of Calogero-Sutherland type. We also sketch a generalization of our method allowing to find the exact solution of the elliptic generalization of the Calogero-Sutherland model. We present the resulting explicit formulas for certain symmetric functions generalizing the Jack polynomials to the elliptic case.
We present further remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers $N$ and $M$, and a particular function of $N+M$ variables arising as anyon correlation function of $N$ particles and $M$ anti-particles. In addition to identities obtained from anyons with the same statistics parameter $λ$, we also obtain ``dual'' relations involving ``mixed'' correlation functions of anyons with two different statistics parameters $λ$ and $1/λ$. We also give alternative, elementary proofs of these identities by direct computations.
We study self-adjoint operators defined by factorizing second order differential operators in first order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum mechanical models like the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero-Sutherland type.