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We study and develop the stationary scattering theory for a class of one-body Stark Hamiltonians with short-range potentials, including the Coulomb potential, continuing our study in Adachi et al. (JDE 268: 5179–5206, 2020; Stationary scattering theory for 1-body Stark operators). The classical scattering orbits are parabolas parametrized by asymptotic orthogonal momenta, and the kernel of the (quantum) scattering matrix at a fixed energy is defined in these momenta. We show that the scattering matrix is a classical type pseudodifferential operator and compute the leading order singularities at the diagonal of its kernel. Our approach can be viewed as an adaption of the method of Isozaki-Kitada (Tokyo Univ. 35: 81–107, 1985) used for studying the scattering matrix for one-body Schrödinger operators without an external potential. It is more flexible and more informative than the more standard method used previously by Kvitsinsky-Kostrykin (Teoret. Mat. Fiz. 75(3): 416-430, 1988) for computing the leading order singularities of the kernel of the scattering matrix in the case of a constant external field (the Stark case). Our approach relies on Sommerfeld’s uniqueness result in Besov spaces, microlocal analysis as well as on classical phase space constructions.

Successful implementation of continuous manufacturing processes requires robust methods to assess and control product quality in a real-time mode. In this study, the residence time distribution of a continuous powder mixing process was investigated via pulse tracer experiments using near infrared spectroscopy for tracer detection in an in-line mode. The residence time distribution was modeled by applying the continuous stirred tank reactor in series model for achieving the tracer (paracetamol) concentration profiles. Partial least squares discriminant analysis and principal component analysis of the near infrared spectroscopy data were applied to investigate both supervised and unsupervised chemometric modeling approaches. Additionally, the mean residence time for three powder systems was measured with different process settings. It was found that a significant change in the mean residence time occurred when comparing powder systems with different flowability and mixing process settings. This study also confirmed that the partial least squares discriminant analysis applied as a supervised chemometric model enabled an efficient and fast estimate of the mean residence time based on pulse tracer experiments.

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 study spectral theory for the Schrodinger operator on manifolds possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends. Certain exterior domains for possibly unbounded obstacles are included. We prove Rellich's theorem, the limiting absorption principle, radiation condition bounds and the Sommerfeld uniqueness result, striving to extending and refining previously known spectral results on manifolds. The proofs are given by an extensive use of commutator arguments. These arguments have a classical spirit (essentially) not involving energy cutoffs or microlocal analysis and require, presumably, minimum regularity and decay properties of perturbations. This paper has interest of its own right, but it also serves as a basis for the stationary scattering theory developed fully in the sequel [19] .

We investigate spectral theory for a one-body Stark Hamiltonian under minimum regularity and decay conditions on the potential (actually allowing sub-linear growth at infinity). Our results include Rellich's theorem, the limiting absorption principle, radiation condition bounds and Sommerfeld's uniqueness, and most of these are stated and proved in sharp form employing Besov-type spaces. For the proofs we adopt a commutator scheme by Ito–Skibsted [13] . A feature of the paper is a special choice of an escape function related to parabolic coordinates, which conforms well with classical mechanics for the Stark Hamiltonian. The whole setting of the paper, such as the conjugate operator and the Besov-type spaces, is generated by this single escape function. We apply our results in the sequel paper [5] .

This is a short report on the authors’ recent work on spectral theory of N-body Schrodinger operators (Adachi et al., New Methods in Spectral Theory of N-Body Schrodinger Operators, arXiv:1804.07874 [math-ph]). We present Rellich’s theorem, the limiting absorption principle, microlocal resolvent bounds, and a microlocal Sommerfeld uniqueness result. Our assumptions on pair-potentials are minimal: Each of them consists 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. In addition, our setup also allows hard-core interactions. The proofs depend on a commutator scheme recently proposed by two of the authors (Ito and Skibsted, J. Funct. Anal. 278(9), 108449 (2020)).

Near infrared (NIR) spectroscopy is a well-established method for analysis of pharmaceutical products, and especially useful for process monitoring and control of continuous production due to high sample throughput. In this work, a previously established method called empirical target distribution optimization (ETDO) wherein reference sample values using information from model prediction of the calibration data was used as a tool to improve the performance of NIR partial least squares (PLS) models. Model performance was assessed using root mean square error (R2), bias and accuracy in prediction of test samples. A target value selection threshold was tested to assess the ETDO procedure for NIR analysis of powder samples. The amount of specific variation captured by the model was examined and compared for models calibrated with and without ETDO. The results reported in this work suggests that PLS models optimized with ETDO of reference values can provide more specific PLS models for NIR analysis for complex powder mixtures. In addition, the model optimization method could also be applied as a tool to verify the necessary amount of PLS components to produce robust models. The ETDO method presented in this work is an approach that could be applied in the development of continuous blending or tableting processes where robust in-line quantitative analysis of powder samples is needed.

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.

Spectroscopic sensors play a key role in process analytical technology. Such sensors can be interfaced to almost any type of manufacturing process, and by rapid, precise, and nondestructive spectroscopic analysis, it is possible to predict analyte concentration, quantify physical parameters, or fingerprint process streams and raw materials. The speed of modern spectroscopic instruments allows real-time process monitoring and control, and the ongoing development of miniaturized instruments and wireless communication now allows instruments to be placed directly on process equipment, which previously was not possible, for example, on rotating drums for mixing processes. Spectroscopic analyzers provide a real-time window into manufacturing processes that enables process control and inspection of intermediary and final product quality. The use of spectroscopy will continue to gain momentum in industrial manufacturing, as it is irreplaceable for ultrarapid process monitoring and control, and for 100% internal quality control necessary for tomorrow's individualized food and medicine.

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 application of high throughput experimentation (HTE) in protein purification process development has created an analytical bottleneck. Using a new label-free and non-invasive methodology for analyzing multicomponent protein mixtures by means of spectral measurements, we show that the analytical throughput for selective protein quantification can be increased significantly. An analytical assay based on this new methodology was shown to generate very precise results. Further, the assay was successfully applied as analytics for a resin screening performed in HTE mode. The increase in analytical throughput was obtained without decreasing the level of information when compared to analytical chromatography. This proves its potential as a valuable analytical tool in conjugation with high throughput process development (HTPD). Further, fast selective protein quantification can enhance process control in a commercial production environment and, hence, minimize the need for off-line release analysis.

The possibility for determination of the water content in pharmaceutical samples by near-infrared (NIR) spectroscopy has been more widely investigated in the past few years. However, many studies claim that changes in sample composition will require the establishment of a new method. The aim of this study was several fold: firstly to investigate validation aspects of water content determination in samples with varying composition and furthermore to see if a model based solely on freeze-dried mannitol–sucrose mixtures can be established that will be able to predict water contents for samples containing proteins, excipients or having a lower density of freeze-dried solids. Samples were measured by NIR, standard normal variate (SNV) corrected and the obtained spectra were compared with the results from a conventional Karl–Fischer titration by means of multivariate analysis, namely principal component analysis (PCA) and partial least square regression (PLS). For the overall sample set, a highly linear correlation between the NIR and the Karl–Fischer method with a slope of 1.00, an R 2 value of 0.98 and a root mean square error of cross-validation (RMSECV) of 0.15% were found. In a second step samples solely consisting of mannitol and sucrose mixtures were used to build a calibration set, which resulted in a RMSECV of 0.16%. The prediction of the remaining samples, which included protein or excipient containing samples, as well as lower density samples, resulted in a root mean square error of prediction (RMSEP) of 0.19%. Thus the present study demonstrated, that a general model for the determination of the water content by NIR could be established, within the limits investigated.

We compute up to a compact term the zero-energy scattering matrix for a class of potentials asymptotically behaving as −γ|x|−μ with 0 0. It turns out to be the propagator for the wave equation on the sphere at time .

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.

Determination of homogeneous mixing of the active pharmaceutical ingredient (API) is an important in-process control within the manufacturing of solid dosage forms. In this paper two new near-infrared (NIR) based methods were presented; a qualitative and a quantitative method. Both methods are based on the calculation of net analyte signal (NAS) models which were very easy to develop, specific with respect to the API and required no additional reference analysis. Using a well-mixed batch as a 'golden standard' batch, control charts were developed and used for monitoring the homogeneity of other batches with NIR. The methods were fast, easy to use, non-destructive and provided statistical tests of homogeneity. A mixing study was characterized with the two methods and the methods were validated by comparison with traditional HPLC analysis.

Let V 1 : S n−1 → ℝ be a Morse function and define V 0(x) = V 1(x/|x|). We consider the scattering theory of the Hamiltonian in L 2(ℝ n ), n ≥ 2, where V is a short-range perturbation of V 0. We introduce two types of wave operators for channels corresponding to local minima of V 1 and prove completeness of these wave operators in the appropriate energy ranges.

We develop an extension of the abstract Mourre theory which consecutively is used to prove spectral properties of various systems coupled to a massless bosonic field. Our models include the spin-boson model and the standard model of quantum electrodynamics for a non-relativistic atom considered recently in [8] and [3] respectively.

The increasing use of NIR and process data acquisition systems in industry facilitates the realisation of real time release (RTR). RTR is a system of pharmaceutical product release that provides assurance of the intended product quality based on the information collected during the manufacturing process and on the compliance with specific GMP requirements.1 RTR is product and process specific. The development of a RTR system begins with risk assessment and involves the use of designed experiments to determine how important process parameters afect critical quality attributes of a final product. Through process understanding and control of process variability, consistent quality may be achieved, and the need for final quality tests reduced or eliminated. In this column the authors summarise their approach to using NIR for successful RTR.

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.

The inclusion of hyperspectral imaging systems in the manufacturing and development of pharmaceutical products is allowing a successful improvement in the quality control of solid dosage forms. The correct distribution not only of active pharmaceutical ingredient (API) but also of the rest of excipients is essential to assure the correct behavior of the tablet when ingested. This is especially relevant in tablets with low content of potent APIs, in which the prescribed intake dosage frequently corresponds to half-a-tablet. Therefore, the aim of this work is to study the surface distribution of the compounds in tablets with low API content. The proposed procedure includes the scanning of the tablet surface using near infrared hyperspectral spectroscopy in association with multivariate curve resolution (MCR) techniques to obtain selective pictures for each individual compound and to allow the fast assessment of their distribution in the measured surface. As an example, a set of commercial Lorazepam tablets (approximately 1% mass fraction of API, and four excipients) were analyzed. The results obtained show the capacity of the proposed methodology as an expedite approach to evaluate the uniformity of the contents between and within tablets. A method to estimate the homogeneity distribution of API in the two halves of the tablet is also proposed.

Noninvasive near-infrared (NIR) and Raman spectroscopies were applied to provide a fast and efficient insight into the formation of different mannitol solid forms occurring in freeze-dried formulations. Multivariate data analysis clearly showed the formation of δ-mannitol in the presence of protein, whereas β-mannitol was observed in the absence of protein.The multivariate analysis of the NIR spectra also gave an indication for the formation of mannitol hemihydrate in the absence of protein. Spectroscopic techniques in combination with multivariate analysis can be applied for fast screening and identification of excipient solid-state properties in pharmaceutical formulations. © 2011 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 100:2871–2875, 2011

We prove smoothness of the 2-cluster-2-cluster and 2-cluster-N-cluster scattering amplitudes under a general short range condition on the potential and under a discreteness assumption on the channel energies.

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.

Near-infrared chemical imaging (NIR-CI) is the fusion of near-infrared spectroscopy and image analysis. It can be used to visualize the spatial distribution of the chemical compounds in a sample (providing a chemical image). Each sample measurement generates a hyperspectral data cube containing thousands of spectra. An important part of a NIR-CI analysis is the data processing of the hyperspectral data cube. The aim of this study was to compare the ability of different commonly used calibration methods to generate accurate chemical images. Three common calibration approaches were compared: (1) using single wavenumber, (2) using classical least squares regression (CLS) and (3) using partial least squares regression (PLS1). Each method was evaluated using two different preprocessing methods. A calibration data set of tablets with five constituents was used for analysis. Chemical images of the active pharmaceutical ingredient (API) and the two major excipients cellulose and lactose in the formulation were made. The accuracy of the generated chemical images was evaluated by the concentration prediction ability. The most accurate predictions for all three compounds were generated by PLS1. The drawback of PLS1 is that it requires a calibration data set and CLS, which does not require a calibration data set, therefore proved to be an excellent alternative. CLS also generated accurate predictions and only requires the pure compound spectrum of each constituent in the sample. All three calibration approaches were found applicable for hyperspectral image analysis but their relevance of use depends on the purpose of analysis and type of data set. As expected, the single wavenumber method was primarily found useful for compounds with a distinct spectral band that was not overlapped by bands of other constituents. This paper also provides guidance for hyperspectral image (or NIR-CI) analysis describing each of the typical steps involved.

In this note we give an account on some recent results on the smoothness of quantum mechanical N-body scattering amplitudes [S1]. We have results for the 2-cluster—2-cluster and 2-cluster—N-cluster amplitudes under a short range condition on the potential and in addition under a discreteness assumption on the 2-cluster channel energies. This gives a rather complete picture for N = 3 while a number of interesting cases remain to be treated for N > 3. To explain our results and what could be expected for other scattering amplitudes we state the following conjecture. (All notations are given in precise terms below.)

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.