Self-adjoint Linear Operator Research Articles

On the supremum and infimum of bounded quantum observables

Let \documentclass[12pt]{minimal}\begin{document}$S(\mathcal {H})$\end{document}S(H) be the set of all bounded self-adjoint linear operators on a complex Hilbert space \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}.$\end{document}H. In 2006, Gudder [Math. Slovaca 56, 573 (2006)] introduced a new order ≼ on \documentclass[12pt]{minimal}\begin{document}$S(\mathcal {H}).$\end{document}S(H). Since then, the existence conditions and representations of the supremum and infimum of two elements in \documentclass[12pt]{minimal}\begin{document}$S(\mathcal {H})$\end{document}S(H) with respect to the order ≼ have been intensively studied. Specifically, Li and Sun [J. Math. Phys. 50, 122107 (2009)]10.1063/1.3272542 obtained simpler representations of A ∧ P and A ∨ P, where \documentclass[12pt]{minimal}\begin{document}$A\in S(\mathcal {H})$\end{document}A∈S(H) and P is an orthogonal projection on \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}.$\end{document}H. In this note, we present more intuitive and concise results on A ∨ P and extend the results of Li and Sun to more general cases. Moreover, some applications of our results are given to show that our results are easier to deal with.

$\mathcal{PT}$ -Symmetric Periodic Optical Potentials

In quantum theory, any Hamiltonian describing a physical system is mathematically represented by a self-adjoint linear operator to ensure the reality of the associated observables. In an attempt to extend quantum mechanics into the complex domain, it was realized few years ago that certain non-Hermitian parity-time (\(\mathcal{PT}\)) symmetric Hamiltonians can exhibit an entirely real spectrum. Much of the reported progress has been remained theoretical, and therefore hasn’t led to a viable experimental proposal for which non Hermitian quantum effects could be observed in laboratory experiments. Quite recently however, it was suggested that the concept of \(\mathcal{PT}\)-symmetry could be physically realized within the framework of classical optics. This proposal has, in turn, stimulated extensive investigations and research studies related to \(\mathcal{PT}\)-symmetric Optics and paved the way for the first experimental observation of \(\mathcal{PT}\)-symmetry breaking in any physical system. In this paper, we present recent results regarding \(\mathcal{PT}\)-symmetric Optics.

Effect Algebras of Positive Self-adjoint Operators Densely Defined on Hilbert Spaces

We show that (generalized) effect algebras may be suitable very simple and natural algebraic structures for sets of (unbounded) positive self-adjoint linear operators densely defined on an infinite-dimensional complex Hilbert space. In these cases the effect algebraic operation, as a total or partially defined binary operation, coincides with the usual addition of operators in Hilbert spaces.

Model representations for systems of selfadjoint operators satisfying commutation relations

Model representations are constructed for a system of bounded linear selfadjoint operators in a Hilbert space such that where is a linear operator from into a Hilbert space and are some selfadjoint operators in . A realization of these models in function spaces on a Riemann surface is found and a full set of invariants for is described.Bibliography: 11 titles.

TRIVIALITY OF A TRACE ON THE SPACE OF COMMUTING TRACE-CLASS SELF-ADJOINT OPERATORS

Abstract. In the present article, we investigate the possibility of a real-valued map on the space of tuples of commuting trace-class self-adjointoperators, which behaves like the usual trace map on the space of trace-class linear operators. It turns out that such maps are related with con-tinuous group homomorphisms from the Milnor’s K -group of the realnumbers into the additive group of real numbers. Using this connection,it is shown that any such trace map must be trivial, but it is proposedthat the target group of a nontrivial trace should be a linearized ver-sion of Milnor’s K -theory as with the case of universal determinant forcommuting tuples of matrices rather than just the eld of constants. 1. IntroductionCommuting self-adjoint operators are constantly encountered in various ma-thematical and physical researches. As an example of physics, let H be aHilbert space whose elements represent physical states. In quantum mechanics,observables are represented by self-adjoint linear operators on

Čebyšev's type inequalities for functions of selfadjoint operators in Hilbert spaces

Some inequalities for continuous synchronous (asynchronous) functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved operators, are given.

On the infimum of bounded quantum observables

The set of all bounded observables corresponding to a quantum system is usually represented by S(H), the set of all bounded linear self-adjoint operators on a Hilbert space H. In 2006, Gudder introduced a logic order on S(H). Given A, B∊S(H), let A∧B be the greatest lower bound (infimum) of A and B with respect to the Gudder order. In this paper, we characterize A∧B using the invariant subspaces of A and B.

Multidimensional parametrization and numerical solution of systems of nonlinear equations

The numerical solution of a system of nonlinear algebraic or transcendental equations is examined within the framework of the parameter continuation method. An earlier result of the author according to which the best parameters should be sought in the tangent space of the solution set of this system is now refined to show that the directions of the eigenvectors of a certain linear self-adjoint operator should be used for finding these parameters. These directions correspond to the extremal values of the quadratic form associated with the above operator. The parametric approximation of curves and surfaces is considered.

New bounds for the Cebysev functional of two functions of selfadjoint operators in Hilbert spaces

Some new inequalities for the Cebysev functional of two functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved functions and operators, are given. 2010 Mathematics Subject Classifications. 47A63; 47A99. .

General preservers of quasi-commutativity on self-adjoint operators

Let H be a separable Hilbert space and B sa ( H ) the set of all bounded linear self-adjoint operators. We say that A , B ∈ B sa ( H ) quasi-commute if there exists a nonzero ξ ∈ C such that A B = ξ B A . Bijective maps on B sa ( H ) which preserve quasi-commutativity in both directions are classified.

Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations

We develop index theories for linear selfadjoint operator equations and investigate multiple solutions for asymptotically linear operator equations. The operator equations consist of two kinds: the first has finite Morse index and can be used to investigate second order Hamiltonian systems and elliptic partial differential equations; the second may have infinite Morse index and can be used to investigate first order Hamiltonian systems.

On the existence of waves guided by a cavity in an elastic film

In the regime where linearised elasticity is a suitable approximation to the behaviour of an elastic body, the existence of a wave guided along the cavity in a film of a homogeneous and isotropic elastic material is proved, at least for a certain range of frequencies. Using the theory of linear self-adjoint operators, it is shown that the associated eigenvalue problem has a nontrivial solution in an appropriate Sobolev class, with the corresponding eigenvalue lying below the continuous spectrum. We study the existence of localised modes of this kind in two particular cases: under the assumption that the cavity is sufficiently narrow in the direction transverse to the film and for a rectangular cavity of arbitrary size.

An explicit expression of supremum of bounded quantum observables

Bounded observables corresponding to a quantum system are usually represented by S(H), the set of all bounded linear self-adjoint operators on a Hilbert space H. In 2006, Gudder introduced a logic order, ≼, on S(H). For A,B∊S(H), A≼B if and only if there exists C∊S(H) such that AC=0 and A+C=B. Given A,B∊S(H), let A∨B be the least upper bound (supremum) for A and B with respect to the Gudder order. In 2007, Pulmannová and Vincenková proved that A∨B exists if and only if A and B have an upper bound for the Gudder order. In this paper, we present some new necessary and sufficient conditions for which A∨B exists and give an explicit representation of A∨B (if A∨B exists).

A-priori bounds and asymptotics on the eigenvalues in bifurcation problems for perturbed self-adjoint operators

We prove upper and lower bounds on the eigenvalues and discuss their asymptotic behaviour (as the norm of the eigenvector tends to zero) in bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lyapounov–Schmidt reduction. The results are applied to a class of semilinear elliptic operators in bounded domains of R N and in particular to Sturm–Liouville operators.

Unbounded Fredholm modules and double operator integrals

In noncommutative geometry one is interested in invariants such as the Fredholm index or spectral flow and their calculation using cyclic cocycles. A variety of formulae have been established under side conditions called summability constraints. These can be formulated in two ways, either for spectral triples or for bounded Fredholm modules. We study the relationship between these by proving various properties of the map on unbounded self adjoint operators D given by ƒ( D ) = D (1 + D 2 ) −1/2 . In particular we prove commutator estimates which are needed for the bounded case. In fact our methods work in the setting of semifinite noncommutative geometry where one has D as an unbounded self adjoint linear operator affiliated with a semi-finite von Neumann algebra ℳ. More precisely we show that for a pair D , D 0 of such operators with D – D 0 a bounded self-adjoint linear operator from ℳ and , where 𝓔 is a noncommutative symmetric space associated with ℳ, then . This result is further used to show continuous differentiability of the mapping between an odd 𝓔-summable spectral triple and its bounded counterpart.

Completeness of Eigenfunctions of Sturm-Liouville Problems with Transmission Conditions

In this paper, we investigate a class of Sturm-Liouville problems with eigenparameter- dependent boundary conditions and transmission conditions at an interior point. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of such a problem coincide with those of A. We show that the operator A has only point spectrum, the eigenvalues of the problem are algebraically simple, and the eigenfunctions of A are complete in H.

Tight frames of exponentially decaying Wannier functions

Let L be a Schrödinger operator with periodic magnetic and electric potentials A, V, a Maxwell operator in a periodic medium or an arbitrary self-adjoint elliptic linear partial differential operator in with coefficients periodic with respect to a lattice Γ. Also let S be a finite part of its spectrum separated by gaps from the rest of the spectrum. We consider the old question of the existence of a finite set of exponentially decaying Wannier functions wj(x) such that their Γ-shifts wj,γ(x) = wj(x − γ) for γ ∊ Γ span the whole spectral subspace corresponding to S in some ‘nice’ manner. It is known that a topological obstruction might exist to finding exponentially decaying wj,γ that form an orthonormal basis of the spectral subspace. This obstruction has the form of non-triviality of a certain finite-dimensional (with the dimension of the fiber equal to the number m of spectral bands in S) analytic vector bundle ΛS over the n-dimensional torus. It was shown by G Nenciu in 1983 that in the presence of time reversal symmetry (which implies the absence of magnetic fields), and if S is a single band, the bundle is trivial and thus the desired Wannier functions do exist. In 2007, G Panati proved that in dimensions n ⩽ 3, even if S consists of several spectral bands, the time reversal symmetry removes the obstruction as well, if one uses the so-called composite Wannier functions. It has not been known what could be achieved when the bundle is non-trivial (which can occur, for instance, in the presence of magnetic fields or for Chern insulators). Let τ be the type of the bundle ΛS, i.e. the number of open sub-domains over which it is trivial (for the trivial bundle τ = 1, and τ never exceeds 2n, where n is the dimension of the coordinate space). We show that it is always possible to find a finite number l ⩽ τm (and thus m ⩽ l ⩽ 2nm) of exponentially decaying composite Wannier functions wj such that their Γ-shifts form a 1-tight frame in the spectral subspace. Here the 1-tight frame is a redundant analog of an orthogonal basis, which appears in many applications, e.g. to signal processing and communication theory. This appears to be the best one can do when the topological obstruction is present. The number l is the smallest dimension of a trivial bundle containing an equivalent copy of ΛS. In particular, l = m if and only if ΛS is trivial, in which case an orthonormal basis of exponentially decaying composite Wannier functions is known to exist.

Exact Values of Bernstein -Widths for Some Classes of Periodic Functions with Formal Self-Adjoint Linear Differential Operators

We consider the classes of periodic functions with formal self-adjoint linear differential operators , which include the classical Sobolev class as its special case. With the help of the spectral of linear differential equations, we find the exact values of Bernstein -width of the classes in the for .

Positive partial transpose from spectra

In this contribution we solve the following problem. Let H_{nm} be a Hilbert space of dimension nm, and let A be a positive semidefinite self-adjoint linear operator on H_{nm}. Under which conditions on the spectrum has A a positive partial transpose (is PPT) with respect to any partition H_n \otimes H_m of the space H_{nm} as a tensor product of an n-dimensional and an m-dimensional Hilbert space? We show that the necessary and sufficient conditions can be expressed as a set of linear matrix inequalities on the eigenvalues of A.

A spectral approach to ill-posed problems for wave equations

In this article, we consider the problem of finding a solution to ill-posed problems for abstract wave equations in a Hilbert space, of the form $$\frac{{\rm d}^{2}u}{{\rm d}t^{2}}\left( t\right) + Au \left(t\right) = 0,\quad t \in \left(0, T\right),\quad u \left(0\right) = 0,\quad u \left(T\right) = u_{0},$$ when A is a general linear selfadjoint operator. We study issues like existence, uniqueness and continuance dependance of data and stability for this problem. Under precise constraint conditions on T, we make such problems well posed and in effect, generalize known results about these equations.