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The pharmaceutical industry is currently implementing new manufacturing principles and modernizing the related processing solutions. A key element in this development is implementation of process analytical technologies (PAT) for measuring product quality in a real-time mode, ideally for a continuously operating processing line. Near-infrared (NIR) spectroscopy is widely used for this purpose, but has limited use for low concentration formulations, due to its inherent detection limit. Light-induced fluorescence (LIF) spectroscopy is a PAT tool that can be used to quantify low concentrations of active pharmaceutical ingredient, and recent development of instrumentation has made it available for in-line applications. In this study, the content of tryptophan in a dynamic powder flow could be measured as low as 0.10 w/w % with LIF spectroscopy with good accuracy of RMSEP = 0.008 w/w %. Both partial least squares regression and support vector machines (SVM) were investigated, but we found SVM to be the better option due to non-linearities between the calibration test and the in-line measurements. With the use of SVM, LIF spectroscopy is a promising candidate for low concentration applications where NIR is not suitable.

This is the third and the last paper in a series of papers on spectral and scattering theory for the Schrodinger operator on a manifold possessing an escape function, for example a manifold with asymptotically Euclidean and/or hyperbolic ends. Here we discuss the time-dependent scattering theory. A long-range perturbation is allowed, and scattering by obstacles, possibly non-smooth and/or unbounded in a certain way, is included in the theory. We also resolve a conjecture by Hempel–Post–Weder on cross-ends transmissions between two or more ends, formulated in a time-dependent manner.

We develop a new scheme of proofs for spectral theory of the [Formula: see text]-body Schrödinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich’s theorem, limiting absorption principle bounds, microlocal resolvent bounds, Hölder continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich’s theorem which is unified with exponential decay estimates studied previously only for [Formula: see text]-eigenfunctions. Each pair-potential is a sum of a long-range term with first-order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with ‘zeroth order’ operators. In particular, they do not rely on Mourre’s differential inequality technique.

We study exponential decay of eigenfunctions of self-adjoint higher order elliptic operators on Rd. We show that the possible (global) critical decay rates are determined algebraically. In addition we show absence of super-exponentially decaying eigenfunctions and a refined exponential upper bound.

For a class of long-range potentials, including ultra-strong perturbations of the attractive Coulomb potential in dimension $d\geq3$, we introduce a stationary scattering theory for Schr\"odinger operators which is regular at zero energy. In particular it is well defined at this energy, and we use it to establish a characterization there of the set of generalized eigenfunctions in an appropriately adapted Besov space generalizing parts of \cite{DS3}. Principal tools include global solutions to the eikonal equation and strong radiation condition bounds.

In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds of derivatives up to order one of the trace of this quantity. These conditions are shown to be optimal for existence and completeness of a wave operator. Our theory does not involve prescribed asymptotic behaviour of the metric at infinity (like asymptotic Euclidean or hyperbolic metrics studied previously in the literature). A consequence of the theory is spectral theory for the Laplace-Beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum.

We show an optimal version of the Rellich theorem for generalized many-body Schrodinger operators. It applies to singular potentials, in particular to a model for atoms and molecules with infinite mass and finite extent nuclei. Our proof relies on a Mourre estimate and a functional calculus localization technique., Comment: 11 pp

We study spatial analyticity properties of solutions of the Navier-Stokes equations and obtain new growth rate estimates for the analyticity radius. We also study stability properties of strong global solutions of the Navier-Stokes equations with data in H^r, r greater or equal 1/2, and prove a stability result for the analyticity radius., 34 pages

The ratio between mannitol and sucrose in a freeze-dried formulation has a major impact on the processing and the stability of a lyophilised product. The ratio of these common excipients influences a critical quality attribute of the system, namely the overall amount of water, due to the different nature of their water–solid interactions. For this experiment samples containing various ratios of mannitol and sucrose and several other additives were freeze-dried, stored under different conditions and measured by NIR. Different spectral pre-processing methods and wavelength selections were tested. Multivariate analysis was applied to correlate the Karl Fischer titration to the NIR spectra. It was found that standard normal variate (SNV) transformation of the wavenumber range 4200–7400 cm −1 yielded prediction errors close to the accepted measurement error of the Karl Fischer titration, when measuring samples of up to 5.5% (w/w) water. It was further found that there was a slight tendency for samples containing inorganic salts or histidine to be underestimated in the NIR measurements. However, no influence was found to be caused by the varying mannitol–sucrose ratio. By reducing the sample set to those samples containing up to around 2% of water, the error was found to be below the uncertainty originating from the reference method. Due to this it can no longer be determined whether the deviation originates from the NIR method or the reference method. It can therefore be concluded the NIR is a suitable tool for quantification of the water content in lyophilised samples with varying mannitol–sucrose ratios.

Real time release (RTR) of products is a new paradigm in the pharmaceutical industry. An RTR system assures that when the last manufacturing step is passed all the final release criteria are met. Various types of models can be used within the RTR framework. For each RTR system, the monitoring capability, control capability and RTR capability need to be tested. This paper presents some practical examples within the RTR framework using near-infrared and process data obtained from a tablet manufacturing process.

We prove Besov space bounds of the resolvent at low energies in any dimension for a class of potentials that are negative and obey a virial condition with these conditions imposed at infinity only. We do not require spherical symmetry. The class of potentials includes in dimension $\geq3$ the attractive Coulomb potential. There are two boundary values of the resolvent at zero energy which we characterize by radiation conditions. These radiation conditions are zero energy versions of the well-known Sommerfeld radiation condition., Comment: 22 pages

We study structural stability of smoothness of the maximal solution to the geometric eikonal equation on (Rd, G), d \geq 2. This is within the framework of order zero metrics G. For a subclass we show existence, stability as well as precise asymptotics for derivatives of the solution. These results are applicable for examples from Schr\"odinger operator theory., Comment: 35 pages

We prove asymptotic completeness for short- and long-rangeN-body Stark Hamiltonians with local singularities of at most Coulomb type. Our results include the usual models for atoms and molecules.

We prove that the spectrum for a large class ofN-body Stark Hamiltonians is purely absolutely continuous. We need slow decay at infinity and local singularities of at most Coulomb type. In particular our results include the usual models for atoms and molecules.

We show absence of positive eigenvalues for generalized 2-body hard- core Schroedinger operators under the condition of bounded strictly convex obstacles. A scheme for showing absence of positive eigenvalues for generalized N -body hard-core Schroedinger operators, N \geq 2, is presented. This scheme involves high energy resolvent estimates, and for N = 2 it is implemented by a Mourre commutator type method. A particular example is the Helium atom with the assumption of infinite mass and finite extent nucleus.

We study in dimension d ⩾ 2 low-energy spectral and scattering asymptotics for two-body d-dimensional Schrodinger operators with a radially symmetric potential falling off like − γ r − 2 , γ > 0 . We consider angular momentum sectors, labelled by l = 0 , 1 , … , for which γ > ( l + d / 2 − 1 ) 2 . In each such sector the reduced Schrodinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic formula for the phase shift.

We study regularity of bound states pertaining to embedded eigenvalues of a self-adjoint operator $H$, with respect to an auxiliary operator $A$ that is conjugate to $H$ in the sense of Mourre. We work within the framework of singular Mourre theory which enables us to deal with confined massless Pauli-Fierz models, our primary example, and many-body AC-Stark Hamiltonians. In the simpler context of regular Mourre theory our results boils down to an improvement of results obtained recently in \cite{CGH}., 70 pages

We study second order perturbation theory for embedded eigenvalues of an abstract class of self-adjoint operators. Using an extension of the Mourre theory, under assumptions on the regularity of bound states with respect to a conjugate operator, we prove upper semicontinuity of the point spectrum and establish the Fermi Golden Rule criterion. Our results apply to massless Pauli-Fierz Hamiltonians for arbitrary coupling., Comment: 30 pages, 2 figures

In this paper we study absence of embedded eigenvalues for Schr\"odinger operators on non-compact connected Riemannian manifolds. A principal example is given by a manifold with an end (possibly more than one) in which geodesic coordinates are naturally defined. In this case one of our geometric conditions is a positive lower bound of the second fundamental form of angular submanifolds at infinity inside the end. Another condition may be viewed (at least in a special case) as being a bound of the trace of this quantity, while similarly, a third one as being a bound of the derivative of this trace. In addition to geometric bounds we need conditions on the potential, a regularity property of the domain of the Schr\"odinger operator and the unique continuation property. Examples include ends endowed with asymptotic Euclidean or hyperbolic metrics studied previously in the literature.

Excipient selection is critically affecting the processing and the stability of a lyophilised product. Near infra-red (NIR) spectroscopy was applied to investigate freeze-dried samples containing varying ratios of the commonly used excipients mannitol and sucrose. Further variation in the formulation was achieved by adding NaCl, CaCl(2) and histidine and by exposing the samples to different conditions. Untreated NIR spectra are strongly affected by the physical nature of samples and can thus be useful for detecting production outliers. Applying standard normal variate (SNV) transformation highlights chemical information. The obtained NIR spectra of the freeze-dried samples were clustered by principal component analysis (PCA) after applying SNV correction in the range from 4200 to 7400cm(-1) (1350-2380nm). Relative humidity under storage and the mannitol/sucrose ratio were clearly represented in the first two principal components, while influence of other excipients was observed in the 3rd and 4th principal component. It was investigated whether this could be due to an influence of the excipients on the mannitol crystallisation behavior. Performing PCA with two principal components of SNV-corrected spectra in the range 4200-4500cm(-1) (2220-1380nm) led to the following observation: while the 1st principal component closely resembled the spectra of beta-mannitol, the 2nd principal component contained additional features that were not attributable to beta-mannitol but correlated well to the main absorbance band of delta-mannitol and mannitol hemihydrate. Therefore, it seems feasible that NIR can analyse versatile freeze-dried samples and classify these according to composition, water content and solid-state properties.

For a class of negative slowly decaying potentials, including V(x):=−γ|x|−μ with 0<μwhole continuous spectrum of the Hamiltonian, including the energy 0. We show that the modified scattering matrices S(λ) are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces. These results are used to derive (oscillatory) asymptotics of the standard short-range and Dollard type S-matrices for the subclasses of potentials where both kinds of S-matrices are defined. For potentials whose leading part is −γ|x|−μ we show that the location of singularities of the kernel of S(λ) experiences an abrupt change from passing from positive energies λ to the limiting energy λ=0. This change corresponds to the behaviour of the classical orbits. Under stronger conditions one can extract the leading term of the asymptotics of the kernel of S(λ) at its singularities.

We prove propagation estimates (of strong type) for long-rangeN-body Hamiltonians. Emphasis is on phase-space analysis in the free channel region.

We consider Hamiltonian systems of a certain class with unstable orbits moving to infinity. We prove a theorem showing that analogous quantum states do not exist. This theorem is applied to Schrodinger operators with potentials of degree zero which are Morse when restricted to the unit sphere.

We consider the long time behavior of a quantum particle in a 2D magnetic field which is homogeneous of degree -1. If the field never vanishes, above a certain energy the associated classical dynamical system has a globally attracting periodic orbit in a reduced phase space. For that energy regime, we construct a simple approximate evolution based on this attractor, and prove that it completely describes the quantum dynamics of our system. Udgivelsesdato: JUN 1 We consider the long time behavior of a quantum particle in a 2D magnetic field which is homogeneous of degree -1. If the field never vanishes, above a certain energy the associated classical dynamical system has a globally attracting periodic orbit in a reduced phase space. For that energy regime, we construct a simple approximate evolution based on this attractor, and prove that it completely describes the quantum dynamics of our system.

We present a limiting absorption principle at zero energy for two-body Schrodinger operators with a long-range potential having a positive virial at infinity. Furthermore, we prove existence of limits (in weighted spaces), as the spectral parameter tends to zero, of all powers of the resolvent. The principal tools of proof are absence of eigenvalue at zero, singular Mourre theory and microlocal estimates. Some elements of the proof will be explained. 1. Statement of main results We give an account of some recent results on asymptotic expansion at zero of the resolvent R(ζ) = (H − ζ) of a two-body Schrodinger operator H = −∆+ V on L(R); see [5] for details. It is well-known, see [18], [12] and the more recent work [13] in which further references can be found, that if V (x) = O(|x|−(2+ ) with > 0 then such an asymptotic expansion exists. For the ‘long-range’ case, V (x) = O(|x|−μ) with μ R, (1.1) and a similar positive virial condition. For such potentials we prove complete asymptotic expansions (in weighted spaces)

Net analyte signal statistical quality control (NAS-SQC) is a new methodology to perform multivariate product quality monitoring based on the net analyte signal approach. The main advantage of NAS-SQC is that the systematic variation in the product due to the analyte (or property) of interest is separated from the remaining systematic variation due to all other compounds in the matrix. This enhances the ability to flag products out of statistical control. Using control charts, the analyte content, variation of other compounds, and residual variation can be monitored. As an example, NAS-SQC is used to appreciate the control content uniformity of a commercially available pharmaceutical tablet product measured with near-infrared spectroscopy. Using the NAS chart, the active pharmaceutical ingredient (API) content is easily monitored for new tablets. However, since quality is a multivariate property, other quality parameters of the tablets are also monitored simultaneously. It will be demonstrated that, besides the API content, the water content of the tablets as well as the homogeneity of the other compounds is monitored. © 2005 American Chemical Society.

Preprocessing of near-infrared spectra to remove unwanted, i.e., non-related spectral variation and selection of informative wavelengths is considered to be a crucial step prior to the construction of a quantitative calibration model. The standard methodology when comparing various preprocessing techniques and selecting different wavelengths is to compare prediction statistics computed with an independent set of data not used to make the actual calibration model. When the errors of reference value are large, no such values are available at all, or only a limited number of samples are available, other methods exist to evaluate the preprocessing method and wavelength selection. In this work we present a new indicator (SE) that only requires blank sample spectra, i.e., spectra of samples that are mixtures of the interfering constituents (everything except the analyte), a pure analyte spectrum, or alternatively, a sample spectrum where the analyte is present. The indicator is based on computing the net analyte signal of the analyte and the total error, i.e., instrumental noise and bias. By comparing the indicator values when different preprocessing techniques and wavelength selections are applied to the spectra, the optimal preprocessing technique and the optimal wavelength selection can be determined without knowledge of reference values, i.e., it minimizes the non-related spectral variation. The SE indicator is compared to two other indicators that also use net analyte signal computations. To demonstrate the feasibility of the SE indicator, two near-infrared spectral data sets from the pharmaceutical industry were used, i.e., diffuse reflectance spectra of powder samples and transmission spectra of tablets. Especially in pharmaceutical spectroscopic applications, it is expected beforehand that the non-related spectral variation is rather large and it is important to remove it. The indicator gave excellent results with respect to wavelength selection and optimal preprocessing. The SE indicator performs better than the two other indicators, and it is also applicable to other situations where the Beer–Lambert law is valid.

We prove a limiting absorption principle at zero energy for two-body Schr\"odinger operators with long-range potentials having a positive virial at infinity. More precisely, we establish a complete asymptotic expansion of the resolvent in weighted spaces when the spectral parameter varies in cones; one of the two branches of boundary for the cones being given by the positive real axis. The principal tools are absence of eigenvalue at zero, singular Mourre theory and microlocal estimates., Comment: 45 pages

We study the notion of asymptotic velocity for a class of perturbed timedependent quadratic Hamiltonians. In particular we give a sufficient condition for absolute continuity.

We consider a dilation-analytic Schrodinger operator represented (by the Mellin transform) in the space H M≔{f: R →h|f is measurable and ∫∞−∞‖f(λ)‖2hdλ R 3 . In this representation a notion of resonance eigenfucntions is defined by using a certain Gelfand triple. We find an isomorphic connection between the space of resonance eigenfunctions and the space N(HM(θ) − z0), Im θ > − 1 2 Arg z 0 , where N(HM(θ) − z0) is the space of eigenfunctions associated with a resonance z0 and the θ-dilated operator HM(θ) in the space H M .

We discuss the perturbation of continuum eigenvalues without analyticity assumptions. Among our results, we show that generally a small perturbation removes these eigenvalues in accordance with Fermi's Golden Rule. Thus, generically (in a Baire category sense), the Schrodinger operator has no embedded non-threshold eigenvalues. I. Introduction It is well known (R-Sl) that a one-body Schrodinger operator — A + V(x), where V is sufficiently well behaved at infinity, cannot have eigenvalues λ embedded in the continuous spectrum (except possibly at threshold, λ = 0). The situation is quite different in the JV-body problem where continuum eigenvalues not only can exist, but do indeed exist in important physical situations: The operator Ho = -Δ1-Δ2-2/\x1\-2/\x2\ in L2{U6) (describing the Helium atom without electronic repulsion) has eigenvalues embedded in the continuous spectrum. While this example has an obvious symmetry, such symmetry is not necessary for the existence of embedded eigenvalues. An example in (F-H-HO-HO) can be modified to produce an embedded eigenvalue where no symmetry is apparent. In (Howl, 2) and (SI), analyticity assumptions are made which allow the treatment of embedded eigenvalues using the perturbation theory developed for use with isolated eigenvalues. The major idea in this theory is that when a small perturbation βW is added to the Schrodinger operator H, the continuum eigenvalue Eo turns into a "resonance," E0(β), which, while not necessarily an eigenvalue of H + βW, is a pole in the analytic continuation of certain matrix elements (φ,(H + βW — z)~ιφ) of the resolvent. The function E0(β) is analytic in β for \β\ small. E0(β) has an imaginary part which appears first to second order in β: τ d2E0(β)

We prove that the number of zeros of the s-wave Jost function (defined for potentials of the form γrβ exp(− ζrβ), γ ≠ 0, ζ > 0, β > − 2 and α > 1) is infinite. The asymptotic density and distribution of the zeros are given. Furthermore we find that the number of (two-body) dilation-analytic resonances, associated with V(r) = γr exp (− ζr 2 ), ¦γ¦> ( 2 π 1 2 ) ζ 3 2 , is infinite.

For a quantum mechanical two-bodys-wave resonance we prove that the evolution of square integrable approximations of the Gamow function is outgoing and exponentially damped. An error estimate is given in terms of resonance energy and explicity. We obtain the Breit-Wigner form. The results are used in an α-decay model to prove general validity of the exponential decay law for periods of several lifetimes.

We consider two-body Schrödinger operators with multiplicative, exponentially decreasing and form-compact potentials. It is proved that the evolution of sharp cutoff approximations of a resonance function is outgoing and exponentially damped. Except for the choices of cutoff radii, shown to be determined by outgoing asymptotics, an explicit error estimate is given in terms of time variable, resonance energy, and width.