Discrete and Embedded Eigenvalues for One-Dimensional Schrödinger Operators

I present an example of a discrete Schrödinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than exponentially. This settles a conjecture of Simon (in the negative). The potential is of von Neumann-Wigner type, with careful navigation around a previously identified borderline situation.     

Atomic Interferometry in Gravitational Fields: Influence of Gravitation on the Beam Splitter

The laser induced splitting of atomic beams inthe presence of a gravitational field is analyzed. Inthe frame of a quasiclassical approximation, the motionof the atomic beam through a laser region with rectangular profile is calculated. Beside theusual beam splitting due to the atomlaser interaction,an additional splitting occurs due to the anomalouseffective interaction with the gravitational field. In a first order approximation in thegravitational acceleration, the outcome of an atominterferometry experiment is given, which includes thevarious corrections owing to the gravitationalmodification of the beam splitting process.     

Multifrequency EPR and redox reactivity investigations of a bis(mu-thiolato)-dicopper(II,II) complex

From a new tripodal ligand [N2SS'H] with mixed N, S(thioether), and S(thiolate) donor set, the corresponding bis(mu-thiolato)dicopper(II) complex has been prepared and characterized. X-ray crystallographic analysis of the complex [Cu2(N2SS')2](ClO4)2.C4H10O (1) demonstrates that the two five-coordinated Cu atoms are bridged by two thiolates leading to a nearly planar Cu2S2 core with a Cu1...Cu1* distance of 3.418(8) A and a large bridging angle Cu1S1Cu1* of 94.92 degrees. X-band (10 GHz), Q-band (34 GHz), and F-Band (115 GHz) EPR spectra of 1 are consistent with a weakly coupled dicopper(II,II) center attributed to an S = 1 state. Simulations for the three frequencies are obtained with a unique set of electronic parameters. The mean values of the spin Hamiltonian parameters for 1 are D = 0.210(3) cm(-1), E = 0.0295(5) cm(-1), |E/D| = 0.140, gx = 2.030(2), gy = 2.032(2), gz = 2.128(2). The electrochemical one-electron reduction of 1 generates the mixed-valent CuIICuI species. EPR and UV-vis spectra are consistent with a type I localized mixed-valent species, while dinuclear CuA centers of native cytochrome c oxidase (CcO)1-3 or nitrous oxide reductase (N2OR)4 have a delocalized CuIICuI mixed-valent state. After reoxidation of the CuIICuI species, the initial complex 1 is regenerated through a reversible interconversion process.     

ESR spectra and magnetic constants of α-Al2O3:Co2+,H+ and α-Al2O3 :Co2+,X+

New anisotropic ESR spectra of Co²⁺ doped sapphire, different from hitherto known, are reported. The new spectra which are observed, beside the well-known spectra of α-Al₂O₃:Co²⁺, are shown to form two sets, each one consisting of six spectra (1–6) and (7–12). The spectra of both sets are proven to be interrelated by B_3a symmetry. g and A tensors for each set will be given. Evidence is given that the two sets are to be assigned to the defects α-Al₂O₃:Co²⁺,H⁺ and α-Al₂O₃:Co²⁺,X⁺. The former is concluded to consists of a Co²⁺ ion at the substitutional site (c) and a proton located in a potential minimum along a straight line between O^2- ions situated in O²⁺ triangles above and below the CO²⁺ ion. The potential function for the proton has been calculated by quantum-chemical calculations to clucidate the geometrical structure of the paramagnetic center. The α-Al₂O₃:Co²⁺,X⁺ could not be fully analyzed but some evidence is presented, that X⁺ might be alkali ions.     

Synthesis,Crystal Structures,and Properties of the Copper(I) Halide Coordination Polymers 2[CuX(μ‐2‐chloropyrazine‐N,N')] (X = Cl,Br), 1[CuI(2‐chloropyrazine‐N)], and [Cu2I2(2‐chloropyrazine)]

The new copper coordination polymers 2[CuX(μ‐2‐chlor‐opyrazine‐N, N')] (X = Cl ( I ), Br ( II ), 1[CuI(2‐chloropyrazine‐N)] ( III ) and [Cu₂I₂(2‐chloropyrazine)] ( IV ) has been prepared by the reaction of the copper(I) halides with 2‐chloropyrazine at roomtemperature or under hydrothermal conditions. The crystal structures of the 1:1 compounds I and II consist of zig‐zag CuX single chains running parallel to the crystallographic a‐axis which are linked by the 2‐chloropyrazine spacer molecules to sheets parallel to (010). For the iodine compound III a one‐dimensional structure is found which consists of CuX double chains running parallel to the crystallographic a‐axis. The thermic properties of all compounds were investigated in different gas atmospheres using simultaneously differential thermal analysis and thermogravimetry (DTA‐TG) as well as temperature resolved X‐ray powder diffraction. On heating, the 1:1 compounds I and II decompose directly to the corresponding copper(I) Halides, whereas the thermal decomposition of III occcur via IV as an intermediate.     

Squaric acid N-hydroxylamides: synthesis, structure, and properties of vinylogous hydroxamic acid analogues

The synthesis of squaric acid N-hydroxylamide esters 5 and amides 6 from dimethyl squarate 2a is described. These derivatives are analogues of the naturally occurring iron(III) chelator hydroxamic acid. On the basis of a comparative reactivity study, a concerted retro-Cope mechanism for the formation of the N-hydroxylamide esters 5 by reaction of dimethyl squarate with hydroxylamines is proposed. A preliminary iron(III) binding study of these hydroxamic acid analogues is presented, demonstrating binding of iron(III) to amides 6 in aqueous solutions, while the esters 5 did not show any sign of metal ion binding. 13C NMR spectroscopic data (chemical shift and spin-lattice relaxation time determination) of these and related derivatives delineate the resonance structures predominant in these molecules. The resonance structures of the derivatives rationalize their spectroscopic data, chemical reactivity, and iron(III) binding properties. Single-crystal X-ray structure analyses of squaric acid N-hydroxylamide ester 5b and squaric acid N-hydroxylamide amide 6c confirm their connectivity and provide structural evidence supporting the spectroscopically derived conclusions. The squaric acid N-hydroxylamides are potentially useful in the construction of chemosensors for iron(III).