Spectral shift function for a discretized continuum

A spectral shift function (SSF) is an important object in the scattering theory which is related both to the spectral density and to the scattering matrix. In the paper, it is shown how to employ the SSF formalism to solve scattering problems when the continuum is discretized, e.g. when solving a scattering problem in a finite volume or in the representation of some finite square-integrable basis. A new algorithm is proposed for reconstructing integrated densities of states and the SSF using a union of discretized spectra corresponding to a set of Gaussian bases with the shifted scale parameters. The examples given show that knowledge of the discretized spectra of the total and asymptotic Hamiltonians is sufficient to find the scattering partial phase shifts at any required energy, as well as the resonances parameters.

Spectral asymptotics for the Schrödinger operator with a non-decaying potential

We study Schrödinger operators H ( h ) = − h 2 Δ + V ( x ) acting in L 2 ( R n ) for non-decaying potentials V. We give a full asymptotic expansion of the spectral shift function for a pair of such operators in the high energy limit. In particular for asymptotically homogeneous potentials W at infinity of degree zero, we also study the semiclassical asymptotics to give a Weyl formula of the spectral shift function above the threshold max W and Mourre estimates in the range of W except at its critical values.

Evaluation of Stereolithography-Based Additive Manufacturing Technology for BaTiO3 Ceramics at 465 nm

A piezoceramic BaTiO3 material that is difficult for 3D printing was tested with a homemade laser-based stereolithography (SLA) setup. The high light absorbance of BaTiO3 in the spectral range of 350–410 nm makes this material hardly usable with most commercial SLA 3D printers. The typical polymerization depth of BaTiO3 ceramic pastes in this spectral range hardly reaches 30–50 µm for 40 vol % powder loading. A spectral change to 465 nm was realized in this work via a robot-based experimental SLA setup to improve the 3D printing efficiency. The ceramic paste was prepared from a preconditioned commercial BaTiO3 powder and used for 3D printing. The paste’s polymerization was investigated with variation of powder fraction (10–55 vol %), speed of a laser beam (1–10 mm/s, at constant laser power), and a hatching spacing (100–1000 µm). The polymerization depths of over 100 µm were routinely reached with the 465 nm SLA for pastes having 55 vol % powder loading. The spectral shift from 350–410 nm spectral region to 465 nm reduced the light absorption by BaTiO3 and remedied the photopolymerization process, emphasizing the importance of comprehensive optical analysis of prospective powders in SLA technology. Two multi-layered objects were 3D-printed to demonstrate the positive effect of the spectral shift.

Ultrafast spectroscopy of C-H vibrations in pathogenic bacteria in 3-μm spectral range

Dynamic optical density spectra were obtained under multipulse excitation of bacterial cultures of S. aureus and P. aeruginosa by 3 μm mid-infrared ultrashort laser pulses, corresponding to the vibrational excitation of the C–H bonds of the bacterial cell. These spectra demonstrated pronounced laser intensity-dependent blue spectral shift, presumably associated with the breaking of hydrogen bonds, which are responsible for the formation of secondary and tertiary protein structures.

Thermo-optical properties of silicon nitride Mach-Zehnder interferometer for the on-chip quantum random number generator

Here we study the thermo-optical properties of an on-chip silicon nitride Mach-Zehnder interferometer (MZI). The spectral shift of the MZI is associated with a change in the chip temperature. This can be explained by a change in the splitting ratio of the directional couplers, as well as a significant change in phase difference between waveguide arms. We experimentally found a phase shift of 2π when heated by 1.67 °C and changes in resonant wavelength at different temperatures (dλ/dT) equal 12.0 pm/°C, theoretically obtained a formula for an arbitrary splitting ratio of the directional couplers in an MZI, and determined the temperature stability required to the device operation inside a quantum cryptography system.

Trace singularities in obstacle scattering and the Poisson relation for the relative trace

We consider the case of scattering by several obstacles in $${\mathbb {R}}^d$$ R d , $$d \ge 2$$ d ≥ 2 for the Laplace operator $$\Delta $$ Δ with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators $$\Delta _1$$ Δ 1 and $$\Delta _2$$ Δ 2 obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative operator $$g(\Delta ) - g(\Delta _1) - g(\Delta _2) + g(\Delta _0)$$ g ( Δ ) - g ( Δ 1 ) - g ( Δ 2 ) + g ( Δ 0 ) was introduced in Hanisch, Waters and one of the authors in (A relative trace formula for obstacle scattering. arXiv:2002.07291, 2020) and shown to be trace-class for a large class of functions g, including certain functions of polynomial growth. When g is sufficiently regular at zero and fast decaying at infinity then, by the Birman–Krein formula, this trace can be computed from the relative spectral shift function $$\xi _\mathrm {rel}(\lambda ) = -\frac{1}{\pi } {\text {Im}}(\Xi (\lambda ))$$ ξ rel ( λ ) = - 1 π Im ( Ξ ( λ ) ) , where $$\Xi (\lambda )$$ Ξ ( λ ) is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of $$\xi _\mathrm {rel}$$ ξ rel . In particular we prove that $${\hat{\xi }}_\mathrm {rel}$$ ξ ^ rel is real-analytic near zero and we relate the decay of $$\Xi (\lambda )$$ Ξ ( λ ) along the imaginary axis to the first wave-trace invariant of the shortest bouncing ball orbit between the obstacles. The function $$\Xi (\lambda )$$ Ξ ( λ ) is important in the physics of quantum fields as it determines the Casimir interactions between the objects.

Paper-Based Multiplexed Colorimetric Device for the Simultaneous Detection of Salivary Biomarkers

In this study, we describe a monolithic and fully integrated paper-based device for the simultaneous detection of three prognostic biomarkers in saliva. The pattern of the proposed multiplexed device is designed with a central sample deposition zone and three identical arms, each containing a pre-treatment and test zone. Its one-step fabrication is realized by CO2 laser cutting, providing remarkable parallelization and rapidity (ca. 5 s/device). The colorimetric detection is based on the sensitive and selective target-induced reshaping of plasmonic multibranched gold nanoparticles, which exhibit a clear spectral shift (and blue-to-pink color change) in case of non-physiological concentrations of the three salivary biomarkers. A rapid and multiplexed naked-eye or smartphone-based readout of the colorimetric response is achieved within 10 min. A prototype kit for POCT testing is also reported, providing robustness and easy handling of the device.

Theoretical study of perovskite nanowires optical response to hydrogen halides vapor exposure

Highly sensitive detection of harmful to the human health and environment hydrogen halide vapors is one of the key problems for the chemical industry. The available electrochemical and optical sensors most often show no selectivity to different hydrogen halides and can be produced via costly high-tech fabrication. In contrast to them, CsPbBr3 perovskite nanowires (NWs) exhibiting laser generation are capable of selective and precise detecting for HCl and HI. Exposure of a single NW to these analytes results in an anion exchange that modifies the chemical composition of the NW and therefore invokes a small spectral shift of the laser peak. Herein we propose a theoretical model describing such an optical response. Taking into account that the anion exchange occurs at the surface of the NW and initiates the formation of a core-shell structure, we perform numerical estimation of the eigenmode spectral position for different thicknesses of the chlorine-and iodine-rich shell. Calculations reveal that even a 10 nm shell causes a noticeable spectral shift of 0.81 and 0.63 nm for eigenmode in CsPbBr3-CsPbCl3 and CsPbBr3-CsPbl3 core-shell NWs, respectively.