A theory of transformation is presented for the diagonalization of a Hamiltonian that is quadratic in creation and annihilation operators or in coordinates and momenta. It is the systemization and theorization of Dirac and Bogoliubov-Valatin transformations, and thus provides us an operational procedure to answer, in a direct manner, the questions as to whether a quadratic Hamiltonian is diagonalizable, whether the diagonalization is unique, and how the transformation can be constructed if the diagonalization exists. The underlying idea is to consider the dynamic matrix. Each quadratic Hamiltonian has a dynamic matrix of its own. The eigenvalue problem of the dynamic matrix determines the diagonalizability of the quadratic Hamiltonian completely. In brief, the theory ascribes the diagonalization of a quadratic Hamiltonian to the eigenvalue problem of its dynamic matrix, which is familiar to all of us. That makes it much easy to use. Applications to various physical systems are discussed, with especial emphasis on the quantum fields, such as Klein-Gordon field, phonon field, etc..
We provide general conditions for which bosonic quadratic Hamiltonians on Fock spaces can be diagonalized by Bogoliubov transformations. Our results cover the case when quantum systems have infinite degrees of freedom and the associated one-body kinetic and paring operators are unbounded. Our sufficient conditions are optimal in the sense that they become necessary when the relevant one-body operators commute.
We provide a general method for constructing bosonic Bogoliubov transformations that diagonalize a general class of quadratic Hamiltonians. These Hamiltonians describe the pair interaction models. Bogoliubov transformations are constructed algebraically, and the resulting Hamiltonians become the second quantizations of explicit one-particle Hamiltonians. Moreover, an explicit formula for the ground state energies is given. Our method systematically diagonalizes various models of quantum field theory, including a model of a harmonic oscillator coupled to a Bose field and the Pauli-Fierz models …
In this paper the local iterative Lie-Schwinger block-diagonalization method, introduced in [FP], [DFPR1], and [DFPR2] for quantum chains, is extended to higher-dimensional quantum lattice systems with Hamiltonians that can be written as the sum of an unperturbed gapped operator, consisting of a sum of on-site terms, and a perturbation consisting of bounded interaction potentials of short range mutltiplied by a real coupling constant t. Our goal is to prove that the spectral gap above the ground-state energy of such Hamiltonians persists for sufficiently small values of |t|, independently of t…
We explain in this note how real fermionic and bosonic quadratic forms can be effectively diagonalized. Nothing like that exists for the general complex hermitian forms. Looks like this observation was missed in the Quantum Field theoretical literature. The present author observed it for the case of fermions in 1986 making some topological work dedicated to the problem: how to construct Morse-type inequalities for the generic real vector fields? This idea also is presented in the note.
We study quantum chains whose Hamiltonians are perturbations by bounded interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. We prove that, for small values of a coupling constant, the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with c…
In this paper we extend the local iterative Lie-Schwinger block-diagonalization method - introduced in [DFPR3] for quantum lattice systems with bounded interactions in arbitrary dimension- to systems with unbounded interactions, i.e., systems of bosons. We study Hamiltonians that can be written as the sum of a gapped operator consisting of a sum of on-site terms and a perturbation given by relatively bounded (but unbounded) interaction potentials of short range multiplied by a real coupling constant t. For sufficiently small values of |t| independent of the size of the lattice, we prove that t…
We study quantum chains whose Hamiltonians are perturbations by interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. For interactions that are form-bounded w.r.t. the on-site Hamiltonian terms, we prove that the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain, for small values of a coupling constant. In our proof we use a novel method introduced …
We give a partial review of what is known so far on stability of periodically driven quantum systems versus regularity of the bounded driven force. In particular we emphasize the fact that unbounded degeneracies of the unperturbed Hamiltonian are allowed. Then we give a detailed description of an extension to some unbounded driven forces. This is done by representing the Schroedinger equation in the instantaneous basis of the time dependent Hamiltonian with a method that we call progressive diagonalization.
We consider quantum chains whose Hamiltonians are perturbations by interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. For interactions that are form-bounded w.r.t. the on-site Hamiltonian terms, we have proven that the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain, for small values of a coupling constant; see [DFPR]. The main result of this p…
We take the Bose-Hubbard model to illustrate exact diagonalization techniques in a pedagogical way. We follow the road of first generating all the basis vectors, then setting up the Hamiltonian matrix with respect to this basis, and finally using the Lanczos algorithm to solve low lying eigenstates and eigenvalues. Emphasis is placed on how to enumerate all the basis vectors and how to use the hashing trick to set up the Hamiltonian matrix or matrices corresponding to other quantities. Although our route is not necessarily the most efficient one in practice, the techniques and ideas introduced…
The iterative methods to diagonalize matrices and many-body Hamiltonians can be reformulated as flows of Hamiltonians towards diagonalization driven by unitary transformations that preserve the spectrum. After a comparative overview of the various types of discrete flows (Jacobi, QR-algorithm) and differential flows (Toda, Wegner, White) that have been introduced in the past, we focus on the random XXZ chain with random fields in order to determine the best closed flow within a given subspace of running Hamiltonians. For the special case of the free-fermion random XX chain with random fields, …
Recently, Fehér and Kluck discovered, at the level of classical mechanics, new compactified trigonometric Ruijsenaars-Schneider $n$-particle systems, with phase space symplectomorphic to the $(n-1)$-dimensional complex projective space. In this article, we quantize the so-called type (i) instances of these systems and explicitly solve the joint eigenvalue problem for the corresponding quantum Hamiltonians by generalising previous results of van Diejen and Vinet. Specifically, the quantum Hamiltonians are realized as discrete difference operators acting in a finite-dimensional Hilbert space of …
For a gaussian interaction V(x,y)=λe^{-(x^2+y^2)/r^2} with long range r>>l_B, l_B the magnetic length, we rigorously prove that the eigenvalues of the finite volume Hamiltonian H_{N,LL}=P_{LL} H_N P_{LL}, H_N=\sum_{i=1}^N [-i\hbar \nabla_{x_i}-eA(x_i)]^2+\sum_{i,j; i\ne j} V(x_i-x_j), \rotA=(0,0,B), and P_{LL} the projection onto the lowest Landau level, are given by the following set: Let M be the number of flux quanta flowing through the sample such that ν=N/M is the filling factor. Then each eigenvalue is given by E=E(n_1,...,n_N)=\sum_{i,j=1;i\ne j}^N W(n_i-n_j). Here n_i\in {1,2,...,M}, n…
Time dependent quadratic Hamiltonians are well known as well in classical mechanics and in quantum mechanics. In particular for them the correspondance between classical and quantum mechanics is exact. But explicit formulas are non trivial (like the Mehler formula). Moreover, a good knowlege of quadratic Hamiltonians is very useful in the study of more general quantum Hamiltonians and associated Schrödinger equations in the semiclassical regime. Our goal here is to give our own presentation of this important subject. We put emphasis on computations with Gaussian coherent states. Our main motiv…
The Cauchy problem for the Gross--Pitaevsky equation with quadratic nonlocal nonlinearity is reduced to a similar problem for the correspondent linear equation. The relation between symmetry operators of the linear and nonlinear Gross--Pitaevsky equations is considered.
Different constructions for Hilbert state space for constrained systems are investigated. Properties of Gaussian states analogous to quantum mechanical Gaussian wave functions are studied. Their evolution for quadratic Hamiltonian case are discussed. A notion of Maslov complex germ is introduced for systems with linear constraints.
We consider linear and quadratic integrals of motion for general variable quadratic Hamiltonians. Fundamental relations between the eigenvalue problem for linear dynamical invariants and solutions of the corresponding Cauchy initial value problems for the time-dependent Schroedinger equation are emphasized. An eigenfunction expansionof the solution of the initial value problem is also found. A nonlinear superposition principle for the generalized Ermakov systems is established as a result of decomposition of the general quadratic invariant in terms of the linear ones.
We develop a Lie algebraic approach to systematically calculate the evolution operator of the generalized two-dimensional quadratic Hamiltonian with time-dependent coefficients. Although the development of the Lie algebraic approach presented here is mainly motivated by the two-dimensional quadratic Hamiltonian, it may be applied to investigate the evolution operators of any Hamiltonian having a dynamical algebra with a large number of elements. We illustrate the method by finding the propagator and the Heisenberg picture position and momentum operators for a two-dimensional charge subject to …
We discuss self-adjoint operators given formally by expressions quadratic in bosonic creation and annihilation operators. We give conditions when they can be defined as self-adjoint operators, possibly after an infinite renormalization. We also discuss explicit formulas for their infimum. Our main motivation comes from Local Quantum Field Theory, which furnishes interesting examples of bosonic quadratic Hamiltonians that require an infinite renormalization.
Bosonic quadratic Hamiltonians, often called Bogoliubov Hamiltonians, play an important role in the theory of many-boson systems where they arise in a natural way as an approximation to the full many-body problem. In this note we would like to give an overview of recent advances in the study of bosonic quadratic Hamiltonians. In particular, we relate the reported results to what can be called the time-dependent diagonalization problem.
Linear Poisson brackets on e(3) typical of rigid body dynamics are considered. All quadratic Hamiltonians of Kowalevski type having additional first integral of fourth degree are found. Quantum analogs of these Hamiltonians are listed.
The purpose of this article is to compute the normal form of a class of general quadratic Hamiltonian systems that generalizes naturally Euler's equations from the free rigid body dynamics.
We investigate a special class of quadratic Hamiltonians on so(4) and so(3,1) and describe Hamiltonians that have additional polynomial integrals. One of the main results is a new integrable case with an integral of sixth degree.
We point out that when a quadratic type Liénard equation is suitably interpreted shows branching due to the double valuedness of the governing Hamiltonian. Under certain approximation of the guiding coupling constant we derive its quantum counterpart that is guided by a momentum-dependent mass function.
We show that one of the five cases of a quadratic Hamiltonian, which were recently selected by Sokolov and Wolf who used the Kovalevskaya-Lyapunov test, fails to pass the Painleve test for integrability.
