Potential Approximations to δ': An Inverse Klauder Phenomenon with Norm-Resolvent Convergence

We show that there is a family Schr?dinger operators with scaled potentials which approximates the δ'-interaction Hamiltonian in the norm-resolvent sense. This approximation, based on a formal scheme proposed by Cheon and Shigehara, has nontrivial convergence properties which are in several respects opposite to those of the Klauder phenomenon. Received: 19 March 2001 / Accepted: 18 June 2001     

Polygonal-path approximations on the path spaces of quantum-mechanical systems

Properties of the subset of polygonal paths in the Hilbert space of paths referring to ad-dimensional quantum-mechanical system are examined. The results are used to discuss various types of polygonal-path approximations appearing in the functional-integration theory. The uniform approximation is applied to extend the definition of the Feynman maps from our previous paper and to prove consistency of this extension. Relations of the extendedF _−i map to the Wiener integral are given.     

Bounded energy approximation to an unstable quantum system

Any state can be approximated by the so-called bounded energy states; this gives a possibility to construct approximations with clear physical meaning to the state space, reduced evolution operator and other characteristics of an unstable system. We discuss it in detail, especially in connection with the Weisskopf-Wigner (semigroup) condition.     

Remark on the energy spectrum of a decaying system

Some time-evolution operators of a general unstable system lead to unphysical spectrum (unbounded below) of the total Hamiltonian. Various necessary conditions for boundeness of the spectrum are known. It is shown here, how this spectrum can be determined, which, in particular, gives the sufficient condition.     

Total asymmetric syntheses of β-hydroxy-δ-lactones via Umpolung with sulfur dioxide

Cyclic stereotriads and stereotetrads of the β-hydroxy-δ-lactone type, e.g. prelactones B and E, common in polyketides and polypropionates, are prepared via SO(2)-induced oxyallylations of enoxysilanes with (1E,3Z)-1-(1-phenylethoxy)penta-1,3-dien-3-yl carboxylates. Using (Z)- or (E)-enoxysilanes both 4,5-cis- or 4,5-trans-δ-lactones are obtained. Depending on the reduction method applied to the obtained aldol intermediates 5,6-trans or 5,6-cis-derivatives are formed. The δ-lactones can be prepared in both their enantiomeric forms depending on the (1R)- or (1S)-configuration of the starting 1-(1-phenylethoxy)penta-1,3-dienes.     

Essential spectrum of Schrödinger operators with δ-interactions on the union of compact Lipschitz hypersurfaces

In this note we prove that the essential spectrum of a Schrödinger operator with δ-potential supported on a finite number of compact Lipschitz hypersurfaces is given by [0, +∞). We emphasize that the union of a family of Lipschitz hypersurfaces is in general not Lipschitz. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A General Approximation of Quantum Graph Vertex Couplings by Scaled Schrödinger Operators on Thin Branched Manifolds

We demonstrate that any self-adjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schrödinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are imposed at it. The procedure involves a local change of graph topology in the vicinity of the vertex; the approximation scheme constructed on the graph is subsequently ‘lifted’ to the manifold. For the corresponding operator a norm-resolvent convergence is proved, with the natural identification map, as the tube diameters tend to zero.     

Block Matrix Operators and Weak Hyponormalities

We introduce a new model of a block matrix operator M(α, β) induced by two sequences α and β and characterize its p-hyponormality. The model may be viewed as arising from the composition operator C_T on l²₊ : = L²(\mathbb N₀)l^{2}_{+} := L^2(\mathbb {N}_0) induced by a measurable transformation T on the set of nonnegative integers \mathbb N₀\mathbb {N}_0 with point mass measure. Composition operator techniques may then be used to treat the p-hyponormality of M(α, β). Finally, we apply our results to obtain examples of these operators showing the p-hyponormal classes are distinct.     

Geometric Phase Related to Point-Interaction Transport on a Magnetic Lobachevsky Plane

We consider a charged quantum particle living in the Lobachevsky plane and interacting with a homogeneous magnetic field perpendicular to the plane and a point interaction which is transported adiabatically along a closed loop in the plane. We show that the bound-state eigenfunction acquires at that the Berry phase equal to 2 times the number of the flux quanta through the area encircled by .     

Bound States of Infinite Curved Polymer Chains

We investigate an infinite array of point interactions of the same strength in ^d , d = 2, 3, situated at vertices of a polygonal curve with a fixed edge length. We demonstrate that if the curve is not a line, but it is asymptotically straight in a suitable sense, the corresponding Hamiltonian has bound states. An example is given in which the number of these bound states can exceed any positive integer.