One-parameter Unitary Group Research Articles

Bounded perturbations of the Heisenberg commutation relation via dilation theory

We extend the notion of dilation distance to strongly continuous one-parameter unitary groups. If the dilation distance between two such groups is finite, then these groups can be represented on the same space in such a way that their generators have the same domain and are in fact a bounded perturbation of one another. This result extends to d d -tuples of one-parameter unitary groups. We apply our results to the Weyl canonical commutation relations, and as a special case we recover the result of Haagerup and Rørdam [Duke Math. J. 77 (1995), pp. 627–656 ] that the infinite ampliation of the canonical position and momentum operators satisfying the Heisenberg commutation relation are a bounded perturbation of a pair of strongly commuting selfadjoint operators. We also recover Gao’s higher-dimensional generalization of Haagerup and Rørdam’s result, and in typical cases we significantly improve control of the bound when the dimension grows.

Existence and spectral stability of multi-pulses in discrete Hamiltonian lattice systems

In the present work, we consider the existence and spectral stability of multi-pulse solutions in Hamiltonian lattice systems which are invariant under a one-parameter unitary group of symmetries. We provide a general framework for the study of such wave patterns based on a discrete analogue of Lin’s method, previously used in the continuum realm. We develop explicit conditions for the existence of multi-pulse standing wave structures and subsequently develop a reduced matrix allowing us to address their spectral stability. As a prototypical example, we consider the discrete nonlinear Schrödinger equation (DNLS). Using Lin’s method, we extend existence and linear stability results of multi-pulse solutions beyond the anti-continuum and continuum limits. Different families of 2- and 3-pulse solitary waves are discussed, and analytical expressions for the corresponding stability eigenvalues are obtained which are in very good agreement with numerical results.

Computation of the unitary group for the Rashba spin–orbit coupled operator, with application to point-interactions

We compute an explicit formula for the one-parameter unitary group of the single-particle Rashba spin–orbit coupled operator in dimension three. As an application, we derive the formula for the Green function for the two-particle operator, and then prove that the spin-dependent point-interaction is of class . The latter is thus the example of a supersingular perturbation for which no self-adjoint operator can be constructed.

Loops, projective invariants, and the realization of the Borromean topological link in quantum mechanics

All the typical global quantum mechanical observables are complex relative phases obtained by interference phenomena. They are described by means of some global geometric phase factor, which is thought of as the “memory” of a quantum system undergoing a “cyclic evolution” after coming back to its original physical state. The origin of a geometric phase factor can be traced to the local phase invariance of the transition probability assignment in quantum mechanics. Beyond this invariance, transition probabilities also remain invariant under the operation of complex conjugation. Most important, geometric phase factors distinguish between unitary and antiunitary transformations in terms of complex conjugation. These two types of invariance functions as an anchor point to investigate the role of loops and based loops in the state space of a quantum system as well as their links and interrelations. We show that arbitrary transition probabilities can be calculated using projective invariants of loops in the space of rays. The case of the double slit experiment serves as a model for this purpose. We also represent the action of one-parameter unitary groups in terms of oppositely oriented-based loops at a fixed ray. In this context, we explain the relation among observables, local Boolean frames of projectors, and one-parameter unitary groups. Next, we exploit the non-commutative group structure of oriented-based loops in 3-d space and demonstrate that it carries the topological semantics of a Borromean link. Finally, we prove that there exists a representation of this group structure in terms of one-parameter unitary groups that realizes the topological linking properties of the Borromean link.

Spectral Approximation for Quasiperiodic Jacobi Operators

Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals and in more general studies of structures exhibiting aperiodic order. The spectra of these self-adjoint operators can be quite exotic, such as Cantor sets, and their fine properties yield insight into the associated quantum dynamics, that is, the one-parameter unitary group that solves the time-dependent Schrodinger equation. Quasiperiodic operators can be approximated by periodic ones, the spectra of which can be computed via two finite dimensional eigenvalue problems. Since long periods are necessary for detailed approximations, both computational efficiency and numerical accuracy become a concern. We describe a simple method for numerically computing the spectrum of a period-K Jacobi operator in O(K 2) operations, then use the algorithm to investigate the spectra of Schrodinger operators with Fibonacci, period doubling, and Thue–Morse potentials.

Spectral automorphisms in CB-effect algebras

Abstract Spectral automorphisms have been introduced in [IVANOV, A.—CARAGHEORGHEOPOL, D.: Spectral automorphisms in quantum logics, Internat. J. Theoret. Phys. 49 (2010), 3146–3152]_in an attempt to construct, in the abstract framework of orthomodular lattices, an analogue of the spectral theory in Hilbert spaces. We generalize spectral automorphisms to the framework of effect algebras with compression bases and study their properties. Characterizations of spectral automorphisms as well as necessary conditions for an automorphism to be spectral are given. An example of a spectral automorphism on the standard effect algebra of a finite-dimensional Hilbert space is discussed and the consequences of spectrality of an automorphism for the unitary Hilbert space operator that generates it are shown. The last section is devoted to spectral families of automorphisms and their properties, culminating with the formulation and proof of a Stone type theorem (in the sense of Stone’s theorem on strongly continuous one-parameter unitary groups — see, e.g. [REED, M.#x2014;SIMON, B.: Methods of Modern Mathematical Physics, Vol. I, Acad. Press, New York, 1975]) for a group of spectral automorphisms.

Stability of bound states of Hamiltonian PDEs in the degenerate cases

We consider a Hamiltonian systems which is invariant under a one-parameter unitary group and give a criterion for the stability and instability of bound states for the degenerate case. We apply our theorem to the single power nonlinear Klein–Gordon equation and the double power nonlinear Schrödinger equation.

Quantum Public-Key Cryptosystem

Quantum one-way functions play a fundamental role in cryptography because of its necessity for the secure encryption schemes taking into account the quantum computer. In this paper our purpose is to establish a theoretical framework for a candidate of the quantum one-way functions and quantum trapdoor functions based on one-parameter unitary groups. The dynamics of parameterized unitary groups ensure the one-wayness and quantum undistinguishability in different levels, and the physical feasibility are derived from the simultaneous approximation of its infinitesimal generators. Moreover, these special functions are used to construct new cryptosystems-the quantum public-key cryptosystems for encrypting both the classical and quantum information.

Characterizations of Spectral Automorphisms and a Stone-Type Theorem in Orthomodular Lattices

The notion of spectral automorphism of an orthomodular lattice was introduced by Ivanov and Caragheorgheopol (Int. J. Theor. Phys. 49(12):3146–3152, 2010) to create an analogue of the Hilbert space spectral theory in the abstract framework of orthomodular lattices. We develop the theory of spectral automorphisms finding previously missing characterizations of spectral automorphisms, discussing several examples and the possibility to construct such automorphisms in direct products or horizontal sums of lattices. A factorization of the spectrum of a spectral automorphism is found. The last part of the paper addresses the problem of the unitary time evolution of a system from the point of view of the spectral automorphisms theory. An analogue of the Stone theorem concerning strongly continuous one-parameter unitary groups is given.

Feynman formulas generated by self-adjoint extensions of the Laplace operator

The Feynman formulas give a representation of a solution of the Cauchy problem for a Schrodinger-type equation (in a special case, for a heat-type equation) using the limit of integrals of finite multiplicity over Cartesian powers of the phase space (in the special case of the configuration space). The limit thus obtained, defining an explicit representation of a one-parameter unitary group eitĤ or a similar object (in our case, this concerns the semigroup etĤ, which is often referred to in the literature as the Schrodinger semigroup) by integral operators, is interpreted by using Feynman integrals, whereas the expression thus obtained is referred in turn as the Feynman formula. As a rule, the Chernoff theorem, which is a generalization of the well known Trotter formula, is used in the derivation of the Feynman formula.

SOLUTION OF THE MPC PROBLEM IN TERMS OF THE SZ.-NAGY–FOIAŞ DILATION THEORY

Motivated by physical problems, Misra, Prigogine and Courbage (MPC) studied the following problem: given a one-parameter unitary group {Ut} on a separable Hilbert space [Formula: see text], find a Hilbert space [Formula: see text], a contraction semigroup {Wt} on [Formula: see text] and an injective operator [Formula: see text] with dense range which intertwines the actions of {Ut} and {Wt} (ΛWt= UtΛ). More precisely, they studied the case where [Formula: see text] is an L2-space over a probability space and both {Ut} and {Wt} are Markovian (i.e. positivity and identity preserving). MPC gave a sufficient condition for the existence of a solution of the above problem, the existence of a time operator associated to {Ut}. In this paper we prove that, using the Sz.-Nagy–Foiaş dilation theory, it is possible to give a constructive characterization of all the solutions of the MPC problem in the general context. This criterium allows one to construct a solution of the MPC problem for which no time operator exists. When specialized to L2-spaces and Markovian {Ut} and {Wt}, the present criterium is applied to address the so-called inverse problem of Statistical Mechanics, namely to characterize the intrinsically random dynamics {Ut}.

SYMMETRIC DIFFERENTIATION AND HAMILTONIAN OF A QUANTUM STOCHASTIC PROCESS

The main object of this paper is to establish the Hamiltonian of the Hudson–Parthasarathy quantum stochastic differential equation [Formula: see text] As its solution forms a cocycle with respect to the time shift, its product with the time shift establishes a strongly continuous one-parameter unitary group W(t)= exp (-itH) and H is called its Hamiltonian. Our main result is that H is the closure of the restriction of the singular operator [Formula: see text] on an appropriate domain that shall be explicitly described. The symmetric differentiation [Formula: see text] is a generalization of the generator of the time shift, [Formula: see text] and [Formula: see text] are annihilation and creation operators and [Formula: see text] is the symmetric Dirac δ-function. In one dimension the symmetric differentiation might be used to establish a quantum stochastic calculus without Ito term.

Rotating Singular Perturbations of the Laplacian

We study a system of a quantum particle interacting with a singular time-dependent uniformly rotating potential in 2 and 3 dimensions: in particular we consider an interaction with support on a point (rotating point interaction) and on a set of codimension 1 (rotating blade). We prove the existence of the Hamiltonians of such systems as suitable self-adjoint operators and we give an explicit expression for their unitary semigroups. Moreover we analyze the asymptotic limit of large angular velocity and we prove strong convergence of the time-dependent propagator to some one-parameter unitary group as (\omega \to \infty).

A Converse Hawking-Unruh Effect and dS2/CFT Correspondence

Given a local quantum field theory net A on the de Sitter spacetime dS^d, where geodesic observers are thermalized at Gibbons-Hawking temperature, we look for observers that feel to be in a ground state, i.e. particle evolutions with positive generator, providing a sort of converse to the Hawking-Unruh effect. Such positive energy evolutions always exist as noncommutative flows, but have only a partial geometric meaning, yet they map localized observables into localized observables. We characterize the local conformal nets on dS^d. Only in this case our positive energy evolutions have a complete geometrical meaning. We show that each net has a unique maximal expected conformal subnet, where our evolutions are thus geometrical. In the two-dimensional case, we construct a holographic one-to-one correspondence between local nets A on dS^2 and local conformal non-isotonic families (pseudonets) B on S^1. The pseudonet B gives rise to two local conformal nets B(+/-) on S^1, that correspond to the H(+/-)-horizon components of A, and to the chiral components of the maximal conformal subnet of A. In particular, A is holographically reconstructed by a single horizon component, namely the pseudonet is a net, iff the translations on H(+/-) have positive energy and the translations on H(-/+) are trivial. This is the case iff the one-parameter unitary group implementing rotations on dS^2 has positive/negative generator.

Representation-theoretic aspects of two-dimensional quantum systems in singular vector potentials: Canonical commutation relations, quantum algebras, and reduction to lattice quantum systems

Some representation-theoretic aspects of a two-dimensional quantum system of a charged particle in a vector potential A, which may be singular on an infinite discrete subset D of R2 are investigated. For each vector v in a set V(D)⊂R2\{0}, the projection Pv of the physical momentum operator P≔p−αA to the direction of v is defined by Pv≔v⋅P as an operator acting in L2(R2), where p=(−iDx,−iDy)[(x,y)∈R2] with Dx (resp., Dy) being the generalized partial differential operator in the variable x (resp., y) and α∈R is a parameter denoting the charge of the particle. It is proven that Pv is essentially self-adjoint and an explicit formula is derived for the strongly continuous one-parameter unitary group {eitP̄v}t∈R generated by the self-adjoint operator P̄v (the closure of Pv), i.e., the magnetic translation to the direction of the vector v. The magnetic translations along curves in R2\D are also considered. Conjugately to Pv and Pw [w∈V(D)], a self-adjoint multiplication operator Qv,w is introduced, which is a linear combination of the position operators x and y, such that, if A is flat on R2\D, then πv,wA≔{Qv,w,Qw,v,Pv,Pw} gives a representation of the canonical commutation relations (CCR) with two degrees of freedom. Properties of the representation πv,wA are analyzed. In particular, a necessary and sufficient condition for πv,wA to be unitarily equivalent (or inequivalent) to the Schrödinger representation of CCR is established. The case where πv,wA is inequivalent to the Schrödinger representation corresponds to the Aharonov–Bohm effect. Quantum algebraic structures [quantum plane and the quantum group Uq(sl2)] associated with the pair {P̄v,P̄w} are also discussed. Moreover, for every A in a class of vector potentials having singularities on the infinite lattice L(ω1,ω2)≔{mω1+nω2|m,n∈Z} [the case D=L(ω1,ω2)], where ω1∈R2 and ω2∈R2 are linearly independent, it is shown that the magnetic translations eiP̄ωj, j=1,2, with A replaced by a modified vector potential are reduced by the Hilbert space l2(L(ω1,ω2)) identified with a closed subspace of L2(R2). This result, which may be regarded as one of the most important novel results of the present paper, establishes a connection of continuous quantum systems in vector potentials to lattice ones.

Empirical quantum mechanics

Machida and Namiki developed a many-Hilbert-spaces formalism for dealing with the interaction between a quantum object and a measuring apparatus. Their mathematically rugged formalism was polished first by Araki from an operator-algebraic standpoint and then by Ozawa for Boolean quantum mechanics, which approaches a quantum system with a compatible family of continuous superselection rules from a notable and perspicacious viewpoint. On the other hand, Foulis and Randall set up a formal theory for the empirical foundation of all sciences, at the hub of which lies the notion of a manual of operations. They deem an operation as the set of possible outcomes and put down a manual of operations at a family of partially overlapping operations. Their notion of a manual of operations was incorporated into a category-theoretic standpoint into that of a manual of Boolean locales by Nishimura, who looked upon an operation as the complete Boolean algebra of observable events. Considering a family of Hilbert spaces not over a single Boolean locale but over a manual of Boolean locales as a whole, Ozawa's Boolean quantum mechanics is elevated into empirical quantum mechanics, which is, roughly speaking, the study of quantum systems with incompatible families of continuous superselection rules. To this end, we are obliged to develop empirical Hilbert space theory. In particular, empirical versions of the square root lemma for bounded positive operators, the spectral theorem for (possibly unbounded) self-adjoint operators, and Stone's theorem for one-parameter unitary groups are established.

On ground states for CCR algebras and Bogoliubov automorphism groups

AbstractLet ACCR(L) be the CCR algebra over a Hubert space L, and let {σt}t∈R be a Bogoliubov automorphism group of ACCR(L) induced by a strongly continuous one-parameter unitary group on L. In this paper, we introduce some continuity for linear functionals on ACCR(L) and, under this continuity, we study existence, uniqueness, and non-uniqueness of ground states for {ACCR(L), {σt}t∈R}.

A geometric criterion for adiabatic chaos

Chaos in adiabatic Hamiltonian systems is a recent discovery and a pervasive phenomenon in physics. In this work, a geometric criterion is discussed based on the theory of action from classical mechanics to detect the existence of Smale horseshoe chaos in adiabatic systems. It is used to show that generic adiabatic planar Hamiltonian systems exhibit stochastic dynamics in large regions of phase space. To illustrate the method, results are obtained for three problems concerning relativistic particle dynamics, fluid mechanics, and passage through resonance, results which either could not be obtained with existing methods, or which were difficult and analytically impractical to obtain with them.

On KMS states for self-dual CCR algebras and Bogoliubov automorphism groups

AbstractLet be the self-dual CCR algebra over a triplet (K, r, Γ), where K is assumed to be a Hilbert space, and let {τt}t ∈ R be a Bogoliubov automorphism group of induced by a strongly continuous one-parameter unitary group on K. In this paper, we introduce some continuity for linear functionals on and, under this continuity, we study existence, uniqueness, and non-uniqueness of KMS states for .

Stability theory of solitary waves in the presence of symmetry, I

Consider an abstract Hamiltonian system which is invariant under a one-parameter unitary group of operators. By a “solitary wave” we mean a solution the time development of which is given exactly by the one-parameter group. We find sharp conditions for the stability and instability of solitary waves. Applications are given to bound states and traveling waves of nonlinear PDEs such Klein-Gordon and Schrödinger equations.