Unbounded Symmetric Operators Research Articles

Complex symmetric operators and semiclassical resonances

Unbounded complex symmetric operators on Hilbert space are discussed and various spectral results on these operators are presented in this paper. In addition, we show that the technique of complex scaling allows one to identify the quantum resonances of complex self-adjoint Schrödinger Hamiltonians.

The range and valence of a real Smirnov function

We give a complete description of the possible ranges of real Smirnov functions (quotients of two bounded analytic functions on the open unit disk where the denominator is outer and such that the radial boundary values are real almost everywhere on the unit circle). Our techniques use the theory of unbounded symmetric Toeplitz operators, some general theory of unbounded symmetric operators, classical Hardy spaces, and an application of the uniformization theorem. In addition, we completely characterize the possible valences for these real Smirnov functions when the valence is finite. To do so we construct Riemann surfaces we call disk trees by welding together copies of the unit disk and its complement in the Riemann sphere. We also make use of certain trees we call valence trees that mirror the structure of disk trees.

On local semigroups of unbounded symmetric operators

On local semigroups of unbounded symmetric operators

Self-adjointness of unbounded tridiagonal operators and spectra of their finite truncations

This paper addresses two different but related questions regarding an unbounded symmetric tridiagonal operator: its self-adjointness and the approximation of its spectrum by the eigenvalues of its finite truncations. The sufficient conditions given in both cases improve and generalize previously known results. It turns out that, not only self-adjointness helps to study limit points of eigenvalues of truncated operators, but the analysis of such limit points is a key help to prove self-adjointness. Several examples show the advantages of these new results compared with previous ones. Besides, an application to the theory of continued fractions is pointed out.

On the adjoint of a symmetric operator

In general it is a non-trivial task to determine the adjoint S . of an unbounded symmetric operator S in a Hilbert or Krein space. We propose a method to specify S . explicitly which makes use of two boundary mappings that satisfy an abstract Greeni¯s identity and a surjectivity condition, and give rise to a self-adjoint extension of S. We show for various concrete examples how convenient and easily applicable this technique is.

On the Generator of the Solution of a Quantum Stochastic Differential Equation

If the Hamiltonian of the system is a bounded selfadjoint operator, we give a simple alternative characterization of the infinitesimal generator iC of the unitary group associated with a Quantum Stochastic Differential Equation (QSDE). By using the defect indices of the symmetric operator C, we give a sufficient condition that assures that iC is the restriction of a generator of a strongly continuous semigroup of isometries in the case when the Hamiltonian of the system is an unbounded symmetric operator.

Unbounded symmetric operators in K-homology and the Baum–Connes conjecture

Using the unbounded picture of analytical K-homology, we associate a well-defined K-homology class to an unbounded symmetric operator satisfying certain mild technical conditions. We also establish an “addition formula” for the Dirac operator on the circle and for the Dolbeault operator on closed surfaces. Two proofs are provided, one using topology and the other one, surprisingly involved, sticking to analysis, on the basis of the previous result. As a second application, we construct, in a purely analytical language, various homomorphisms linking the homology of a group in low degree, the K-homology of its classifying space and the analytic K-theory of its C * -algebra, in close connection with the Baum–Connes assembly map. For groups classified by a 2-complex, this allows to reformulate the Baum–Connes conjecture.

On semibounded restrictions of self-adjoint operators

After the von Neumann's remark [10] about “pathologies” of unbounded symmetric operators and an abstract theorem about stability domain [9], we develope here a general theory allowing to construct semibounded restrictions of selfadjoint operators in explicit form. We apply this theory to quantum-mechanical momentum (position) operator to describe corresponding stability domains. Generalization to the case of measurable functions of these operators is considered. In conclusion we discuss spectral properties of self-adjoint extensions of constructed self-adjoint restrictions.

The Maslov Index: a Functional Analytical Definition and the Spectral Flow Formula

We give a functional analytical definition of the Maslov index for continuous curves in the Fredholm-Lagrangian Grassmannian. Our definition does not require assumptions either at the endpoints or at the crossings of the curve with the Maslov cycle. We demonstrate an application of our definition by developing the symplectic geometry of self-adjoint extensions of unbounded symmetric operators. We discuss continuous variations of the form $A_D+C_t$, where $A_D$ is a fixed self-adjoint unbounded Fredholm operator and $\{C_t\}$ a family of bounded self-adjoint operators. We extend the definition of the spectral flow to such families of unbounded operators in a purely functional analytical way. We then prove that the spectral flow is equal to the Maslov index of the corresponding family of abstract Cauchy data spaces.

Nonstandard hulls of unbounded symmetric operators

Nonstandard hulls of unbounded symmetric operators

On position and momentum operators in the q-oscillator algebra

The aim of this paper is to study the position and momentum operators in q-deformed oscillator algebras. The natural form of the position operator is Xp=qpN(a++a)qpN, where p is a real number. This operator is an operator representable by a Jacobi matrix. Using the theory of Jacobi matrices, the theory of classical moment problem and the theory of basic hypergeometric functions, it is shown that, depending on values of q and p, Xp can be unbounded symmetric operator [which has the deficiency indices (1,1) and, hence, is not self-adjoint, but has self-adjoint extensions], bounded self-adjoint operator with continuous simple spectrum or self-adjoint operator of trace class (therefore, with discrete spectrum with zero as the point of accumulation of eigenvalues). The connection of the q-deformed Heisenberg relation PX−qXP=1 for the position and momentum operators with a q-deformation of the quantum harmonic oscillator is also considered.

Spectra, eigenvectors and overlap functions for representation operators ofq-deformed algebras

Operators of representations corresponding to symmetric elements of theq-deformed algebrasUq(su1,1),Uq(so2,1),Uq(so3,1),Uq(son) and representable by Jacobi matrices are studied. Closures of unbounded symmetric operators of representations of the algebrasUq(su1,1) andUq(so2,1) are not selfadjoint operators. For representations of the discrete series their deficiency indices are (1,1). Bounded symmetric operators of these representations are trace class operators or have continuous simple spectra. Eigenvectors of some operators of representations are evaluated explicitly. Coefficients of transition to eigenvectors (overlap coefficients) are given in terms ofq-orthogonal polynomials. It is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq-deformed algebras.

Computation of expectation values with Dirichlet one‐dimensional wave functions

AbstractRecently, it was proven that the wave functions ψ of one‐dimensional Schrödinger operators can be approximated by those ψn corresponding to the Dirichlet eigenproblem in a finite box sufficiently large. In this article, sufficient conditions to compute expectation values are used to prove that the expectation value 〈ψ, Sψ〉 can be approximated by 〈ψn, Sψn〉 as n → ∞, for a wide class of unbounded symmetric operators S. © 1995 John Wiley & Sons, Inc.

THE PROBLEM OF STEADY-STATE OSCILLATIONS OF A TRANSVERSALLY ISOTROPIC HALF-CYLINDER

The authors study the problem of solvability on the semiaxis of the equation where , and , , , and are unbounded symmetric operators in a Hilbert space. Models of this equation are problems of steady-state oscillations of an elastic half-cylinder with various boundary conditions. The main results are theorems on factorization of a pencil related to this problem and solvability theorems.

On tuples of commuting symmetric, non-selfadjoint operators

Generalizing the Cowen-Douglas-Theory to certain tuples of unbounded symmetric operators we obtain canonical models for such tuples, which are realized in holomorphic functional Hilbert spaces. The results are applied to multidimensional moment problems.

Unbounded commutants and intertwining spaces of unbounded symmetric operators and ∗-representations

This paper investigates the strong commutant, the weak commutant and the form commutant of an unbounded symmetric (nonself-adjoint) operator and of an unbounded ∗-representation on a Hilbert space. For two examples of unbounded symmetric operators these commutants are described in terms of singular integral operators and of Toeplitz operators, respectively.

A Theory of Quantum Measurement Based on the CCR Algebra L$^+(\mathcal W)$

Starting from L ^+(\mathcal W) as the algebra of observables, \mathcal W a space of type \mathcal S , a theory of quantum measurement is devised. It is based on the theory of Davies and Lewis, but adapted to unbounded symmetric operators defined on a common dense domain and no restriction on the spectra.

Extensions of Quasi-Free Derivations on the CAR algebra

We classify all of the infinitesimal generator extensions of a particular quasi-free derivation δ\_A\_ on the CAR (canonical anticommutation relations) algebra, where A is an unbounded symmetric operator on a Hilbert space having deficiency indices (1,1). We show that each of these extensions generates a strongly continuous one-parameter group of Bogoliubov transformations of the CAR algebra.