A Second Eigenvalue Bound for the Dirichlet Schrödinger Operator

Let λ _i (Ω,V) be the i ^th eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^nLet λ _i (Ω,V) be the i ^th eigenvalue of the Schr?dinger operator with Dirichlet boundary conditions on a bounded domain and with the positive potential V. Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V _*, we prove that λ₂(Ω,V) ≤ λ₂(S ₁,V _*). Here S ₁ denotes the ball, centered at the origin, that satisfies the condition λ₁(Ω,V)=λ₁(S ₁,V _*).Further we prove under the same convexity assumptions on a spherically symmetric potential V, that λ₂(B _R , V) / λ₁(B _R , V) decreases when the radius R of the ball B _R increases.We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.R.B. was supported by FONDECYT project # 102-0844.H.L. gratefully acknowledges financial support from DIPUC of the Pontifí cia Universidad Católica de Chile and from CONICYT.     

A lower bound for the ground state energy of a Schrödinger operator on a loop

Consider a one-dimensional quantum mechanical particle described by the Schrödinger equation on a closed curve of length . Assume that the potential is given by the square of the curve's curvature. We show that in this case the energy of the particle cannot be lower than . We also prove that it is not lower than (the conjectured optimal lower bound) for a certain class of closed curves that have an additional geometrical property.

     

Compactness properties of certain integral operators related to fractional integration

Suppose 1≤p,q≤∞ and α > (1/p−1/q)₊. Then we investigate compactness properties of the integral operator when regarded as operator from L_p[0,1] into L_q[0,1]. We prove that its Kolmogorov numbers tend to zero faster than exp(−c_αn^1/2). This extends former results of Laptev in the case p=q=2 and of the authors for p=2 and q=∞. As application we investigate compactness properties of related integral operators as, for example, of the difference between the fractional integration operators of Riemann–Liouville and Weyl type. It is shown that both types of fractional integration operators possess the same degree of compactness. In some cases this allows to determine the strong asymptotic behavior of the Kolmogorov numbers of Riemann–Liouville operators. In memoria of Eduard (University of the West Indies) who passed away in October 2004.     

High-resolution resonance measurements of the physisorption interaction

Narrow selective adsorption resonances, with line widths given by the natural lifetime of the intermediate resonance state, have been observed directly in the angular distributions of Ne nozzle beams diffracted from a 20 K Cu(110) surface. Using beams of 20Ne and 22Ne isotopes we show that this high-resolution mode of resonance detection permits, even in a case of a small isotope effect, unique assignment of bound level sequences compatible with a single gas-surface potential curve.     

Reactions of Hydrated Singly Charged First‐Row Transition‐Metal Ions M+(H2O)n (M=V,Cr, Mn,Fe, Co,Ni, Cu,and Zn) toward Nitric Oxide in the Gas Phase

Reactions of M⁺(H₂O)_n (M=V, Cr, Mn, Fe, Co, Ni, Cu, Zn; n≤40) with NO were studied by Fourier transform ion cyclotron resonance (FT‐ICR) mass spectrometry. Uptake of NO was observed for M=Cr, Fe, Co, Ni, Zn. The number of NO molecules taken up depends on the metal ion. For iron and zinc, NO uptake is followed by elimination of HNO and formation of the hydrated metal hydroxide, with strong size dependence. For manganese, only small HMnOH⁺(H₂O)_n?1 species, which are formed under the influence of room‐temperature black‐body radiation, react with NO. Here NO uptake competes with HNO formation, both being primary reactions. The results illustrate that, in the presence of water, transition‐metal ions are able to undergo quite particular and diverse reactions with NO. HNO is presumably formed through recombination of a proton and ³NO^? for M=Fe, Zn, preferentially for n=15–20. For manganese, the hydride in HMnOH⁺(H₂O)_n?1 is involved in HNO formation, preferentially for n≤4. The strong size dependence of the HNO formation efficiency illustrates that each molecule counts in the reactions of small ionic water clusters.     

Hydrated magnesium cations Mg+(H2O)n, n ≈ 20-60, exhibit chemistry of the hydrated electron in reactions with O2 and CO2

Ion-molecule reactions of Mg(+)(H(2)O)(n), n ≈ 20-60, with O(2) and CO(2) are studied by Fourier-transform ion cyclotron resonance (FT-ICR) mass spectrometry. O(2) and CO(2) are taken up by the clusters. Both reactions correspond to the chemistry of hydrated electrons (H(2)O)(n)(-). Density functional theory calculations predicted that the solvation structures of Mg(+)(H(2)O)(16) contain a hydrated electron that is solvated remotely from a hexa-coordinated Mg(2+). Ion-molecule reactions between Mg(+)(H(2)O)(16) and O(2) or CO(2) are calculated to be highly exothermic. Initially, a solvent-separated ion pair is formed, with the hexa-coordinated Mg(2+) ionic core being well separated from the O(2)(?-) or CO(2)(?-). Rearrangements of the solvation structure are possible and produce a contact-ion pair in which one water molecule in the first solvation shell of Mg(2+) is replaced by O(2)(?-) or CO(2)(?-).     

Infrared Multiple Photon Dissociation Spectroscopy of Hydrated Cobalt Anions Doped with Carbon Dioxide CoCO2(H2O)n−, n=1–10, in the C−O Stretch Region

We investigate anionic [Co,CO₂,nH₂O]⁻ clusters as model systems for the electrochemical activation of CO₂ by infrared multiple photon dissociation (IRMPD) spectroscopy in the range of 1250–2234 cm⁻¹ using an FT-ICR mass spectrometer. We show that both CO₂ and H₂O are activated in a significant fraction of the [Co,CO₂,H₂O]⁻ clusters since it dissociates by CO loss, and the IR spectrum exhibits the characteristic C−O stretching frequency. About 25 % of the ion population can be dissociated by pumping the C−O stretching mode. With the help of quantum chemical calculations, we assign the structure of this ion as Co(CO)(OH)₂⁻. However, calculations find Co(HCOO)(OH)⁻ as the global minimum, which is stable against IRMPD under the conditions of our experiment. Weak features around 1590–1730 cm⁻¹ are most likely due to higher lying isomers of the composition Co(HOCO)(OH)⁻. Upon additional hydration, all species [Co,CO₂,nH₂O]⁻, n≥2, undergo IRMPD through loss of H₂O molecules as a relatively weakly bound messenger. The main spectral features are the C−O stretching mode of the CO ligand around 1900 cm⁻¹, the water bending mode mixed with the antisymmetric C−O stretching mode of the HCOO⁻ ligand around 1580–1730 cm⁻¹, and the symmetric C−O stretching mode of the HCOO⁻ ligand around 1300 cm⁻¹. A weak feature above 2000 cm⁻¹ is assigned to water combination bands. The spectral assignment clearly indicates the presence of at least two distinct isomers for n ≥2.     

Ultraviolet–ultraviolet dual‐cure process based on acrylate oxetane monomers

A dual‐cure process consisting of two subsequent ultraviolet‐initiated radical and cationic polymerizations was investigated. The process was studied with two acrylate oxetane monomers, one of them having a spacer between the two polymerizable moieties. It was shown by Fourier transform infrared (FTIR) that the first (radical) step was performed successfully for both systems. As for the second (cationic) step, only the monomer with the spacer was able to polymerize, allowing the crosslinking of the polyacrylic chains generated by the first step. The efficiency of the process was confirmed by differential scanning calorimetry because the glass‐transition temperatures of the cured films were ?16 and + 34 °C after the first and second steps, respectively. The dual cure of this system was further analyzed by real‐time FTIR, which showed that 86% of the acrylate and 80% of the oxetane moieties were converted after 20 and 50 s of light exposure, respectively. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 469–475, 2003     

Ultrafast structural transformations in graphite

Femtosecond laser-induced structural transitions in graphite were studied by time-resolved optical anisotropy measurements. The decay of the anisotropic reflectivity seems to indicate a loss of long-range order on a subpicosecond time scale, which is much faster than the electron-phonon relaxation time. This observation confirms the nonthermal nature of the structural transition.