Let $M$ be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by $\sigma$. Proposition 4 Let be an eigenvalue of a regular or periodic Sturm-Liouville problem. Write \(\mu \) for the measure whose density with respect to \(\sigma \) is \(|\nabla f|^2\). Four systems are considered: (1) infinite square-well potential with and eigenfunctions , , ; (2) linear harmonic oscillator with and , , ; (3) linear rigid rotor for fixed , , , , ; (4) hydrogen-like radial function with , , , . Thus, in a sense, the zero set of an eigenfunction is the largest of all level sets. HFS clients enjoy state-of-the-art warehousing, real-time access to critical business data, accounts receivable management and collection, and unparalleled customer service. The purpose of this survey paper is to present some recent (and not so recent) results about the asymptotics of Laplace eigenfunctions on compact manifolds. Let $f: M \rightarrow \mathbb{R}$ be a non-constant eigenfunction of the Laplacian. The first modulus of continuity result is presented for the limit. It does not specialize, but instead publishes Published By: The Johns Hopkins University Press, Read Online (Free) relies on page scans, which are not currently available to screen readers. functions u: M x [0, 1]-liR satisfy: (a) un has zero as a regular value on int(M) x (0, 1) (b) un has non-degenerate critical points on int(M) x (0, 1) (c) if un (t, ) :int(M)-liR has 0 as a critical point, it is non-degenerate. We observe that the value distribution of $f$ under $\mu$ admits a unimodular density attaining its maximum at the origin. in its field. For example, for the (appropriately normalized) value distribution of S ∼ |C(t)| we predict the distribution P(S) = (π/2)Se-πS2/4. © 2008-2020 ResearchGate GmbH. 81, 207-237 (1995; Zbl 0852.11024)] who prove equidistribution on PSL 2 (ℤ)∖ℍ. Special attention is paid to the influence on the dynamics of the underlying geometry and the perturbation terms (as potentials, for instance). Nauk, 29:6(180) (1974), 181–182 Citation in format AMSBIB We can easily show this for the case of two eigenfunctions of with … In the subcritical case, no localization could be observed, giving rise to localization breaking. where no semiclassical small parameter is involved). We study the oscillatory properties of the eigenfunctions and, using these properties, we obtain sufficient conditions for the system of eigenfunctions of the problem in question to form a basis in the space L p (0, 1), 1 < p < ∞, after removing three functions. Roughly speaking, if the intersecting angle is {\it irrational}, then the vanishing order is generically infinite, whereas if the intersecting angle is {\it rational}, then the vanishing order is finite. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. In the critical case, the eigenfunctions are localized around the inner boundary. Materials are most of all the objects and therefore materials have its own properties. MUSE delivers outstanding results to the scholarly community by maximizing revenues for publishers, providing value to libraries, and enabling access for scholars worldwide. By summing, we obtain eigenfunctions on the disk Gantumur Tsogtgerel Math 319: Introduction to PDE McGill University, Montréal MondayMarch21,2011. continuous publication, the American Journal of Mathematics This issue has a number of connections with other dynamical properties of the equation that have been extensively studied in the literature, such as dispersive effects, Strichartz estimates and unique continuation-type properties (which are relevant in control theory and inverse problems). Purchase this issue for $44.00 USD. It is easy to show that if is a linear operator with an eigenfunction , then any multiple of is also an eigenfunction of . Compare these properties with that of previous example. With a personal account, you can read up to 100 articles each month for free. An increasing body of research suggests interorganizational relationships as being critical to the financial performance of firms. Hopkins Fulfillment Services (HFS) mathematics. ... A conjecture by S. T. Yau. This item is part of JSTOR collection Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds - Volume 1 Issue 3 - S. Minakshisundaram, Å. Pleijel Write $\mu$ for the measure whose density with respect to $\sigma$ is $|\nabla f|^2$. Since each pair of observables in the set commutes, the observables are all compatible so that the measurement of one observable has no effect on the result of measuring another observable in the set. In contrast, interorganizational relationship commitment increases service innovation focus and strengthens the innovation focus—firm performance relationship. That is, a function f is an eigenfunction of D if it satisfies the equation Theorem 1.1. (JEMS), 16(6), 1253–1288, 2014). To keep the presentation simple, we mainly consider four exemplary boundary value problems: the Dirichlet problem for the Laplace's equation; the torsional creep problem; the case of Dirichlet eigenfunctions for the Laplace's equation; the initial-boundary value problem for the heat equation. Much less is known about other eigenfunctions. Numerical tests carried out for numerous chaotic systems confirm nicely the two conjectures and thus provide strong evidence for temporal quantum chaos. Basis Properties of Eigenfunctions of Second-Order Differential Operators with Involution Asylzat Kopzhassarova 1 and Abdizhakhan Sarsenbi 1 1 Department of Mathematics, M. Auezov South Kazakhstan State University, 5 Tauke Han Avenue, 160012 Shymkent, Kazakhstan The eigenfunctions that correspond to these eigenvalues however are, \[{y_n}\left( x \right) = \cos \left( {\frac{{n\,x}}{2}} \right)\hspace{0.25in}n = 1,2,3, \ldots \] So, for this BVP we get cosines for eigenfunctions corresponding to positive eigenvalues. Roughly speaking, the vanishing order is generically infinite if the intersecting angle is irrational, and the vanishing order is finite if the intersecting angle is rational. The chapter provides a short general review of Brownian motion and its place in probability theory. Read your article online and download the PDF from your email or your account. There are two linearly independent eigenfunctions, namely cos(2nx) and sin(2nx) corresponding to each positive eigenvalue λn = 4n2. HFS provides print and digital distribution for a distinguished list of university presses and nonprofit institutions. In fact, in the latter case, the vanishing order is the degree of the rationality. The theoretical findings are original and of significant interest in spectral theory. In general different properties of materials are enlisted below. In this paper, we are concerned with the geometric structures of Laplacian eigenfunctions as well as their applications to inverse scattering theory. Thus we have shown that eigenfunctions of a Hermitian operator with different eigenvalues are orthogonal. Furthermore, we discover a novel second-order phase transition for the eigenfunctions in spherical shells as the $l$-$k$ ratio crosses a critical value. The American Journal of Mathematics is used Unformatted text preview: ECE 345 1 / 10 Linear Systems and Signals Eigenfunctions of CT LTI systems Anand D. Sarwate Department of Electrical and Computer Engineering Rutgers, The State University of New Jersey 2020 .Rutgers Sarwate ECE 345 2 / 10 Learning objectives The learning objectives for this section are: • use the eigenfunction property to compute the output of LTI systems … In this. American Journal of Mathematics Residual parameter and skewness, which estimate the deviation of amplitude distribution from the Gaussian distribution, are obtained for hundreds of eigenfunctions. Tip: you can also follow us on Twitter 1. introduction It is well-known that on a compact Riemannian manifold M one can choose an orthonormal basis of L 2 (M) consisting of eigenfunctions ' j of satisfying ' j + j ' j = 0; (1) where 0 = 0 < 1 2 : : : are the eigenvalues. It is shown that the pseudopotential method can be used for an explicit calculation of the first few terms in an expansion in power of ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$ of the eigenvalues and the corresponding eigenfunctions of a system of Bose particles with hard-sphere interaction. The random wave conjecture suggests that in certain situations, the value distribution of $f$ under $\sigma$ is approximately Gaussian. Math., Inst. and are orthogonal. The nodal patterns addressed pertain to waves on vibrating plates and membranes, acoustic and electromagnetic modes, wavefunctions of a "particle in a box'" as well as to percolating clusters, and domains in ferromagnets, thus underlining the diversity---and far-reaching implications---of the problem. Sci. We discuss, in particular, how a random superposition of plane waves can model chaotic eigenfunctions and highlight the connections of the complex morphology of the nodal lines thereof to percolation theory and Schramm-Loewner evolution. We establish an accurate and comprehensive characterisation of such a relationship. We now have a smooth way of passing from one set of eigenfunctions to the other. Thus, in a sense, the zero set of an eigenfunction is the largest of all level sets. Eigenfunctions are restriction of harmonic polynomials to S2 Distinct eigenvalues are k= p k2 + krepeating with multiplicity d k= 2k+ 1 Resulting eigenspace,“spherical harmonics of degree k”, H k= fe k;1;e k;2;:::;e k;d k g; S2 e k;j= (k 2 + k)e Chris Sogge Dispersive properties of eigenfunctions 3/19 This paper will present a survey of the views of Sacks, Ramachandran, Goldberg, and Gazzaniga on the interrelated concepts of mind and self as presented in the reading required for the course. We now develop some properties of eigenfunctions, to be used in Chapter 9 for Fourier Series and Partial Di erential Equations. Get the latest machine learning methods with code. Access scientific knowledge from anywhere. Moreover, they are applied directly to some physical problems of great importance, including the inverse obstacle scattering problem and the inverse diffraction grating problem. We refer to Jakobson, Nadirashvili and Toth, ... We may now apply Lemma 2.1 with the vector field V and with Z = Z t 0 . In this paper, we present some novel and intriguing findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviours of the eigenfunctions. We study the long-time behavior of bound quantum systems whose classical dynamics is chaotic and put forward two conjectures. We discuss the properties of nodal sets and critical points, the number of nodal domains, as well as asymptotic properties of eigenfunctions in the high energy limit (such as weak* limits, the rate of growth of L p norms, and the relationship between positive and negative parts of eigenfunctions). the review papers. To obtain specific values for physical parameters, for example energy, you operate on the wavefunction with the quantum mechanical operator associated with that parameter. The theoretical findings have some interesting applications to some physical problems of practical importance including the inverse obstacle scattering and the inverse diffraction grating problems. There is a rich theory on the geometric properties of Laplacian eigenfunctions in the literature; see e.g. We introduce a new notion of generalized singular lines of the Laplacian eigenfunctions, and carefully study these singular lines and the nodal lines. The singular concentration set of the limit cannot be a compact union of closed geodesics and measured geodesic laminations. We prove a microlocal version of the equidistribution theorem for Wigner distributions associated to cusp forms on PSL 2 (ℤ)∖PSL 2 (ℝ). as a basic reference work in academic libraries, both in the eigenfunctions corresponding to the first n eigenvalues satisfy El, E2 and E3}, with B0 = C°°(A/). Readers may refer to [1. Soc. In the special case where D is defined on a function space, the eigenvectors are referred to as eigenfunctions. We consider the nodal lines and also introduce the notion of generalized singular lines of the Laplacian eigenfunctions in the two-dimensional case. and the eigenfunctions corresponding to λn are given by φn(x) = cos(2nx), ψn(x) = sin(2nx), n ∈ N. (6.16) Note: All the eigenvalues are non-negative. However, little work has examined how the extent firms' interorganizational relationship commitment and diversity influence their innovation focus and performance. Conjecture B gives the best possible upper bound for a generalized Weyl sum and is related to the extremely large recurrence times in temporal quantum chaos. A!i =ai!i A!j = aj!j Access supplemental materials and multimedia. When M is a manifold with boundary, the same holds for Laplace eigenfunctions satisfying either the Dirichlet or the Neumann boundary conditions. Let M be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by \(\sigma \). Properties of the first eigenfunctions of the clamped column 269 That is R1 0 ¯v2 dx = 0. In this paper, we present some novel and intriguing findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviours of the eigenfunctions in two dimensions. When $M$ is a manifold with boundary, the same holds for Laplace eigenfunctions satisfying either the Dirichlet or the Neumann boundary conditions. We establish an accurate and comprehensive quantitative characterisation of the relationship. From the stated properties of regular Sturm-Liouville eigenvalue problems in Section 2.1, the eigenfunctions form a “complete” set with respect to any piecewise smooth function over the finite x-dependent interval I = {x | 0 < x < a}. We observe that the value distribution of f under \(\mu \) admits a unimodal density attaining its maximum at the origin. Join ResearchGate to find the people and research you need to help your work. To understand the properties of material explore the article! 1. © 1976 The Johns Hopkins University Press mathematical papers. We present a comprehensive review of the nodal domains and lines of quantum billiards, emphasizing a quantitative comparison of theoretical findings to experiments. Published since 1878, the Journal has earned and Books In this paper, we are concerned with the geometric structures of Laplacian eigenfunctions as well as their applications to inverse scattering theory. In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose eigenvalues completely specify the state of a system.. We shall mostly address three issues: the estimation of the local size of the critical set; the dependence of the number of critical points on the boundary values and the geometry of the domain; the location of critical points in the domain. For terms and use, please refer to our Terms and Conditions Indeed, at most two far-field patterns are sufficient for some important applications. The low-temperature properties of the system are discussed. Unique recovery results by finitely many far-field patterns are well-known to be a challenging issue in the inverse scattering theory. Journals It is shown that in a certain polygonal setup, one can recover the support of the unknown scatterer as well as the surface impedance parameter by finitely many far-field pattern. The eigenfunctions decay exponentially inside the localized sphere and decay polynomially outside. Ergodic properties of eigenfunctions A. I. Shnirel'man Full text: PDF file (138 kB) References: PDF file HTML file Bibliographic databases: Received: 12.12.1973 Citation: A. I. Shnirel'man, “Ergodic properties of eigenfunctions”, Uspekhi Mat. option. 2 Real Eigenfunctions The eigenfunctions of a Sturm-Liouville problem can be chosen to be real. All Rights Reserved. We study the behavior of a large-eigenvalue limit of eigenfunctions for the hyperbolic Laplacian for the modular quotient SL(2; double-struck Z sign)\double-struck H sign. We give an overview of some new and old results on geometric properties of eigenfunctions of Laplacians on Riemannian manifolds. It is shown in a certain polygonal setup that one can recover the support of the unknown scatterer as well as the surface impedance parameter by finitely many far-field patterns. Recently, a rigorous mathematical theory of high-frequency localization for Laplacian eigenfunctions in circular, spherical, and elliptical domains has been established by Nguyen and Grebenkov [7]. One of the largest publishers in the United States, the Johns Hopkins University Press combines traditional books and journals publishing units with cutting-edge service divisions that sustain diversity and independence among nonprofit, scholarly publishers, societies, and associations. We then address the main theme of these notes: the use of tools from the analysis of the semiclassical limit (such as Wigner measures) to obtain a description of the high-frequency structure of the solutions to the non-semiclassical Schrödinger equation (i.e. family of functions. Eigenfunctions and Eigenvalues An eigenfunction of an operator is a function such that the application of on gives again, times a constant. French National Centre for Scientific Research, Unimodal value distribution of Laplace eigenfunctions and a monotonicity formula, Unimodular value distribution of Laplace eigenfunctions and a monotonicity formula, On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems, Two-parameter localization and related phase transition for a Schr\"{o}dinger operator in balls and spherical shells, On nodal and generalized singular structures of Laplacian eigenfunctions and applications, Persistence barcodes and Laplace eigenfunctions on surfaces, Nodal portraits of quantum billiards: Domains, lines, and statistics, An introduction to the study of critical points of solutions of elliptic and parabolic equations, Brownian Motion and its Applications to Mathematical Analysis, High-Frequency Dynamics for the Schrödinger Equation, with Applications to Dispersion and Observability, The modulus of continuity for Γ0(m)\double-struck H sign semi-classical limits, Quantum unique ergodicity for SL2(script O sign)\H3 and estimates for L-functions, Eigenfunctions Concentrated Near a Closed Geodesic, Metric properties of eigenfunctions of the Laplace operator on manifolds, Equidistribution of cusp forms on PSL 2 (ℤ)∖PSL 2 (ℝ), The Diameter of the First Nodal Line of a Convex Domain, L ∞ -norms of eigenfunctions for arithmetic hyperbolic 3-manifolds, Bounds for eigenfunctions of differential operators, Real business cycle models, endogenous growth models and cyclical growth: A critical survey. These relations reappear in different guises when we talk about arithmetic random waves and difference equations later in this review. Browse our catalogue of tasks and access state-of-the-art solutions. Theorem Suppose that y j and y k are eigenfunctions corresponding to distinct eigenvalues λ j and λ k. Then y j and y k are orthogonal on [a,b] with respect to the weight function w(x) = r(x). This generalizes a recent result of W. Luo and P. Sarnak [Publ. But there remains the problem, what are the asymptotic properties of eigenfunctions with large numbers. For real eigenfunctions, the complex conjugate can be dropped. JSTOR is part of ITHAKA, a not-for-profit organization helping the academic community use digital technologies to preserve the scholarly record and to advance research and teaching in sustainable ways. Each concept will be taken up in turn and then related to consciousness. Math. ©2000-2020 ITHAKA. Go to Table properties. The Press is home to the largest journal publication program of any U.S.-based university press. We give a survey at an introductory level of old and recent results in the study of critical points of solutions of elliptic and parabolic partial differential equations. The nodal statistics are shown to distinguish not only between regular and chaotic classical dynamics but also between different geometric shapes of the billiard system itself. Since materials have these properties it makes the materials useful and purposeful to use. Let \(f: M \rightarrow {\mathbb {R}}\) be a non-constant eigenfunction of the Laplacian. The findings are based on multisource and longitudinal performance data and highlight the positive impact of relationship commitment on the effects of service innovation focus on firm performance. In the supercritical case, the eigenfunctions are localized around a sphere between the inner and outer boundaries of the spherical shell. article, the authors show that diverse interorganizational relationships reduce the positive impact of innovation focus on firm performance. We see that these eigenfunctions are orthogonal, and that the set (r 1 L) [(r 2 L cos 2nˇx L) 1 n=1 [(r 2 L sin 2nˇx L) 1 n=1 consists of orthonormal eigenfunctions. The properties of Laplace-Beltrami eigenfunctions have fascinated researchers for more than two centuries, starting with the celebrated Chaldni's experiments with vibrating plates. In the meantime, the more mathematically-oriented reader can find a delightful survey of results on the geometric properties of eigenfunctions in. Hautes Étud. Since the eigenvalues are real, a ∗ 1 = a1 and a ∗ 2 = a2. aj. To access this article, please, Access everything in the JPASS collection, Download up to 10 article PDFs to save and keep, Download up to 120 article PDFs to save and keep. Indeed, in some practically interesting cases, at most two far-field patterns are required. Eigenvalues and Eigenfunctions The wavefunction for a given physical system contains the measurable information about the system. We obtain restrictions on the persistence barcodes of Laplace-Beltrami eigenfunctions and their linear combinations on compact surfaces with Riemannian metrics. ˆA ∗ ψ ∗ = a2ψ ∗. With critically acclaimed titles in history, science, higher education, consumer health, humanities, classics, and public health, the Books Division publishes 150 new books each year and maintains a backlist in excess of 3,000 titles. Orthogonality Sturm-Liouville problems Eigenvalues and eigenfunctions Another general property is the following. Properties of the Bessel functions There is a rich theory on the geometric properties of Laplacian eigenfunctions in the literature; see e.g. 1076 K. UHLENBECK when 0 is a critical value of un{ , t\ it is nondegenerate. articles of broad appeal covering the major areas of contemporary A complete description is given in the case in which the underlying geometry is a manifold with periodic geodesic flow (Zoll manifolds) and for the torus, where we present the main ideas of the recent work of the author in collaboration with Anantharaman and Macià (J. Eur. The article reviews recent analytical results concerning statistical properties of eigenfunctions of random Hamiltonians with broken time reversal symmetry describing a motion of a quantum particle in a thick wire of finite length L. Additionally, we prove a monotonicity formula for level sets of solid spherical harmonics, essentially by viewing nodal sets of harmonic functions as weighted minimal hypersurfaces. Math. Because we assumed , we must have , i.e. Statistical properties of eigenfunctions of quantum polygonal and dispersing billiards are numerically investigated in detail. In general, an eigenvector of a linear operator D defined on some vector space is a nonzero vector in the domain of D that, when D acts upon it, is simply scaled by some scalar value called an eigenvalue. The main results are: direct calculation by integration on a convex polyeder, calculation by reducing partially ordered sets and estimation of the distribution of the critical length and its parameters by changing the network-structure. since as shown above. We focus here mainly on results about the nodal sets, asymptotic L p bounds and the problem of determining weaklimits of expected values (i.e. (49) where k is a constant called the eigenvalue. JSTOR®, the JSTOR logo, JPASS®, Artstor®, Reveal Digital™ and ITHAKA® are registered trademarks of ITHAKA. Examples are given. Some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant G Andrei Mezincescu INFM, CP MG-7, R-76900 Magurele, Ilfov, Romania˘ and Centrul de Cercetari Avansate de Fizic˘ a al Academiei Rom˘ ane, Bucure¸Ë† sti, Romania E-mail: mezin@alpha1.infim.ro Received 21 February 2000 Abstract. In the case of degeneracy (more than one eigenfunction with the same eigenvalue), we can choose the eigenfunctions to be orthogonal. ψ and φ are two eigenfunctions of the operator  with real eigenvalues a1 and a2, respectively. In this paper we give a survey of recent results of the theory of stochastic networks. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We give an overview of some new and old results on geometric properties of eigenfunctions of Laplacians on Riemannian manifolds. ranks as one of the most respected and celebrated journals Conjecture A states that the autocorrelation function C(t) = 〈Ψ(0)|Ψ(t)〉 of a delocalized initial state |Ψ(0)〉 shows characteristic fluctuations, which we identify with a universal signature of temporal quantum chaos. Additionally, we prove a monotonicity formula for level sets of solid spherical harmonics, essentially by viewing nodal sets of harmonic functions as weighted minimal hypersurfaces. Here we investigate the two-parameter high-frequency localization for the eigenfunctions of a Schr\"{o}dinger operator with a singular inverse square potential in high-dimensional balls and spherical shells as the azimuthal quantum number $l$ and the principal quantum number $k$ tend to infinity simultaneously, while keeping their ratio as a constant, generalizing the classical one-parameter localization for Laplacian eigenfunctions [SIAM J. Appl. Select a purchase Check out using a credit card or bank account with. With warehouses on three continents, worldwide sales representation, and a robust digital publishing program, the Books Division connects Hopkins authors to scholars, experts, and educational and research institutions around the world. Multiply the first equation by φ ∗ and the second by ψ and integrate. It is shown that the intersecting angle of two of those lines is related to the vanishing order of the eigenfunction at the intersecting point. The requirement that the eigenvalues be simple is made to allow one to deal with each eigenspace by considering only one nonzero eigenfunction, for properties (El), (E2) and (E3) are unchanged under multiplication by a constant. We prove that the eigenfunctions in balls are localized around an intermediate sphere whose radius is increasing with respect to the $l$-$k$ ratio. Managing Service Innovation and Interorganizational Relationships for Firm Performance: To Commit or... Ergebnisse der Theorie der Netzpläne mit stochastisch verteilten Vorgangsdauern. quantum ... All content in this area was uploaded by Dmitry Jakobson on Feb 03, 2015, ... That is, a Laplace eigenfunction corresponding to a large eigenvalue should have a value distribution density under σ that is approximately Gaussian. The limit is studied for Hecke-Maass forms, joint eigenfunctions of the Hecke operators and the hyperbolic Laplacian. Various approaches to counting the nodal domains---using trace formulae, graph theory, and difference equations---are also illustrated with examples. All rights reserved. 73:780-803, 2013]. The random wave conjecture suggests that in certain situations, the value distribution of f under \(\sigma \) is approximately Gaussian. ˆAψ = a1ψ. The properties of Laplace-Beltrami eigenfunctions have fascinated researchers for more than two centuries, starting with the celebrated Chladni's experiments with vibrating plates. Féjer summation and results of S. Zelditch are used to show that the microlocal lifts of eigenfunctions have large-eigenvalue limit a geodesic flow invariant measure for the modular unit cotangent bundle. The finite sums for up to 100 are evaluated in this Demonstration. Similarly, innovation has been considered a key driver of the growth and success of firms. The Journals Division publishes 85 journals in the arts and humanities, technology and medicine, higher education, history, political science, and library science. maintained its reputation by presenting pioneering A property of the nullspace is that it is a linear subspace, ... Hamiltonian, is a second-order differential operator and , the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue , interpreted as its energy. Unique identifiability by finitely many far-field patterns remains to be a highly challenging fundamental mathematical problem in the inverse scattering theory. (In the case where two or more eigenfunctions have the same eigenvalue, then the eigenfunctions can be made to be orthogonal). In these notes we review various aspects of the high-frequency dynamics of solutions to the linear Schrödinger equation. De nition of Orthogonality We say functions f(x) and g(x) are orthogonal on a

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