Absence of absolutely continuous spectrum of Floquet operators

The spectrum of the Floquet operator associated with time-periodic perturbations of discrete Hamiltonians is considered. If the gap between successive eigenvalues _j of the unperturbed Hamiltonian grows as _j - _j-1 j and the multiplicity of _j grows asj with >0 asj tends to infinity, then the corresponding Floquet operator possesses no absolutely continuous spectrum provided the perturbation is smooth enough.     

Stability of a singular continuous spectrum of Sturm-Liouville operators

We prove that there exists a continuous potentialq such that the operator generated by $$(lu)(x) = - u^n (x) + \{ q(x) + \upsilon (x)\} u(x), 0 \leqslant (x)< \infty$$ and boundary conditionsu(0) cos α+u′(0) sin α=0 has a singular continuous spectrum in [0, 1] for every locally integrable functionv with compact support and every α ∈ [0, 2π].     

Operators with singular continuous spectrum: II. Rank one operators

For an operator,A, with cyclic vector , we studyA+P, whereP is the rank one projection onto multiples of . If [,] spec (A) andA has no a.c. spectrum, we prove thatA+P has purely singular continuous spectrum on (,) for a denseG of 's.Research partially supported by DGAPA-UNAM and CONACYT.This material is based upon work suported by the National Science Foundation under Grant No. DMS-9207071. The Government has certain rights in this material.This material is based upon work supported by the National Science Foundation under Grant No. DMS-9101715. The Government has certain rights in this material.     

Purely absolutely continuous spectrum for almost Mathieu operators

Using a recent result of Sinai, we prove that the almost Mathieu operators acting onl ²(), (l _Y, )(n) = (l+1)+(l–)+ cos(n+) (n) have a purely absolutely continuous spectrum for almost all a provided that is a good irrational and is sufficiently small. Furthermore, the generalized eigen-functions are quasiperiodic.     

Some quantum operators with discrete spectrum but classically continuous spectrum

We consider a number of simple quantum Hamiltonians H(?i?,x) with the following property: H(?i?,x) has discrete spectrum even though {(p,q) | H(p,q) <E} has infinite volume.     

Generalized uncertainty principle and self-adjoint operators

In this work we explore the self-adjointness of the GUP-modified momentum and Hamiltonian operators over different domains. In particular, we utilize the theorem by von-Neumann for symmetric operators in order to determine whether the momentum and Hamiltonian operators are self-adjoint or not, or they have self-adjoint extensions over the given domain. In addition, a simple example of the Hamiltonian operator describing a particle in a box is given. The solutions of the boundary conditions that describe the self-adjoint extensions of the specific Hamiltonian operator are obtained.     

The minimaximin and maximinimax theorems for eigenvalues of semibounded self-adjoint operators

A new result, namely the minimaximin theorem for eigenvalues of a semibounded self-adjoint operator whose spectrum at the bounded end consists entirely of isolated point eigenvalues of finite multiplicities, is established. A particularly simple proof of the maximinimax theorem is also given. The relevance of these results for establishing bounds on the eigenvalues of atomic hamiltonians is explained.     

On the inequivalence of renormalization and self-adjoint extensions for quantum singular interactions

A unified S-matrix framework of quantum singular interactions is presented for the comparison of self-adjoint extensions and physical renormalization. For the long-range conformal interaction the two methods are not equivalent, with renormalization acting as selector of a preferred extension and regulator of the unbounded Hamiltonian.     

Local nets and self-adjoint extensions of quantum field operators

We consider Wightman fields having the property that some closed extensions of the field operators generate locally commuting von Neumann algebras. We show that for such fields the hermitian field operators have self-adjoint extensions, possibly in an enlarged Hilbert space, such that bounded functions of the self-adjoint operators commute locally.     

Constructing quantum observables and self-adjoint extensions of symmetric operators. II. Differential operators

We discuss a problem of constructing self-adjoint ordinary differential operators starting from self-adjoint differential expressions based on the general theory of self-adjoint extensions of symmetric operators outlined in [1]. We describe one of the possible ways of constructing in terms of the closure of an initial symmetric operator associated with a given differential expression and deficient spaces. Particular attention is focused on the features peculiar to differential operators, among them on the notion of natural domain and the representation of asymmetry forms generated by adjoint operators in terms of boundary forms. Main assertions are illustrated in detail by simple examples of quantum-mechanical operators like the momentum or Hamiltonian. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 3–36, August, 2007.     

Absence of wave operators for one-dimensional quantum walks

Letters in Mathematical Physics - We show that there exist pairs of two time evolution operators which do not have wave operators in a context of one-dimensional discrete time quantum walks. As a...     

Characterization and uniqueness of distinguished self-adjoint extensions of Dirac operators

Distinguished self-adjoint extensions of Dirac operators are characterized by Nenciu and constructed by means of cut-off potentials by Wüst. In this paper it is shown that the existence and a more explicit characterization of Nenciu's self-adjoint extensions can be obtained as a consequence from results of the cut-off method, that these extensions are the same as the extensions constructed with cut-off potentials and that they are unique in some sense.On leave from Universität Zürich, Schöneberggasse 9, CH-8001 Zürich. Supported by Swiss National Science FoundationOn leave from Technische Universität Berlin, Straße des 17. Juni 135, D-1000 Berlin     

Operators with singular continuous spectrum,VII. Examples with borderline time decay

We construct one-dimensional potentialsV(x) so that if onL ²(), thenH has purely singular spectrum; but for a dense setD, D implies that |,e ^-itH |C |t|^-1/2 ln(|t|) for t>2. This implies the spectral measures have Hausdorff dimension one and also, following an idea of Malozemov-Molchanov, provides counterexamples to the direct extension of the theorem of Simon-Spencer on one-dimensional infinity high barriers.This material is based upon work supported by the National Science Foundation under Grant No. DMS-9401491. The Government has certain rights in this material.     

The conjugate operator method for strongly singular operators

In this Letter, we adapt the version of the conjugate operator method for Hamiltonians defined as quadratic forms developed by Boutet de Monvel-Berthier and Georgescu, to study a class of self-adjoint operators of the form , whereH is conjugate to a self-adjoint operatorA but itself is not. The spectral theory for such operators is considered and applications to strongly singular second-order operators as the wave propagators in inhomogeneous and stratified media are given.     

Møller operators for scattering on singular potentials

The existence of Møller operators is proved for singular potentials which decrease more rapidly at infinity than the Coulomb potential. The question of their uniqueness is discussed.     

Gauge-invariant operators for singular knots in Chern-Simons gauge theory

We construct gauge-invariant operators for singular knots in the context of Chern-Simons gauge theory. These new operators provide polynomial invariants and Vassiliev invariants for singular knots. As an application we present the form of the Kontsevich integral for the case of singular knots.     

Feynman method for disentangling operators and a singular oscillator

The problem of the evolution of a singular quantum oscillator with a frequency exhibiting an arbitrary time dependence has been solved. The probabilities w _mn of transitions in the oscillator spectrum and generating functions have been calculated, and the sum rules for w _mn have been derived. The calculations are based on the Feynman disentangling method and the theory of representations of the SU(1, 1) group.     

Strongly elliptic operators with singular coefficients

In this paper, we deal with operators of the form