# Inverse Problems With Nondecreasing Potentials

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### Subject Index to Volumes 91 100 (1998)

Fourth order Sturm Liouville problems 96 91 Fourth-order di⁄erence equation 92 103 Fre chet derivative 94 13 Free boundary problems 94 55 Free boundary problems 98 191 Freud weights 99 219 Freud weight 99 463 Fundamental biharmonic problem 91 231 G-condition 92 59 GMERR method 98 49 Galerkin Þnite element method 97 81 Gamma function 99 167

### Nonlinear Diffusion with Fractional Laplacian Operators

of a state law p = f(u),where f is a nondecreasing scalar function, which is linear when the ﬂow is isothermal, and a power, i.e., f(u)=cum−1 with c>0 and m>1, if it is adiabatic. The linear relationship happens also in the simpliﬁed description of water inﬁl-tration in an almost horizontal soil layer according to Boussinesq. In both

### New Additions to the Toolkit for Forward/Inverse Problems in

of the potentials on the heart. The inverse algorithms implemented in SCIRun that we will discuss in this section use different source models and characterizations of the potentials during the QRS seg-ment. The ﬁrst one, implements an FEM-based inverse that imposes sparsity on the spatial gradient of the poten-tials.

### AMS Mathematics Subject Classiﬁcation (2010): arXiv:2009

to the following inverse spectral problem. Inverse Problem 1.3. Given the spectral data S, construct the potentials {qj}m j=1 and the coeﬃcient h. The uniqueness of Inverse Problem 1.3 solution follows, in particular, from the results of [14,15,38,40]. In the papers [15,21,40], a constructive solution of this inverse problem

### Ernest K. Ryu Seoul National University Mathematical and

Operator Inverse The inverse operator of ﬂ: ﬂ 1 = f(y;x) j(x;y) 2ﬂg ﬂ 1 is always well de ned (ﬂ 1 need not be single-valued). Graﬂ ﬂ 1 (ﬂ 1) = ﬂ and domﬂ 1 = rangeﬂ ﬂ 1is not an inverse in the usual sense since ﬂ ﬂ 6= › possible.

### Resonance absolute quantum reﬂection at selected energies

created by using the inverse problem formalism (exact solutions generalizing Eqs. (1, 2). So we get a new class of potentials corresponding to the exactly solvable models with resonance reﬂection. It is worth to mention that tails of BSEC-potentials on a ≤ x ≥ ∞ for any a are responsible for the wave conﬁnement.

### Correlation inequalities for quantum spin systems with

is a monotone nondecreasing function of A independent of the values of all the other s.Asa consequence, we have that for A 0 P A Av J A A U A=0. 15 However, for A=0, the two random variables J A and A U are independent, Av J A A A=0 =Av J A Av A A=0 =0, 16 where the last equality comes from having chosen distributions with Av J A =0. It also

### ON INVERSE SCATTERING FOR THE KLEIN-GORDON EQUATION

and inverse scattering problem for the equation fju = 0 perturbed by an obstacle, and to [8] for a more physically oriented approach to scattering. Both [6] and [8] have an excellent list of references, to which we refer for more information about scattering and inverse scattering problems. Finally, we want to mention, and thank

### www.aimsciences.org

MATHEMATICAL CONTROL ANDdoi:10.3934/mcrf.2020014 RELATED FIELDS Volume 10, Number 3, September 2020 pp. 643{667 LIPSCHITZ STABILITY FOR SOME COUPLED DEGENERATE

### Partial Differential Equations - GBV

a. Nonlinear Eigenvalue Problems, b. The Method of Lyapunov-Schmidt. c. Bifurcation from a Simple Eigenvalue. 13.2 The Method of Sub- and Supersolutions 366 a. Barriers for a Semilinear Equation. b. Monotone Iteration. c. Application with Uniformly Bounded f(x,u). d. Application with f(x,u), Nondecreasing in u. 13.3 The Variational Method 372 a.

### Inverse Problem for Periodic 'Weighted' Operators

There are various methods of solving inverse problems. We briefly describe a direct approach from [KK1], based on a theorem from non-linear functional analysis. We recall definitions. Suppose that H, H 1 are real separable Hilbert spaces with norms &}&, &}& 1. The derivative of a map f: H H 1 at a point y # H is a bounded linear map

### Analysis of the Criteria of Activation-Based Inverse

The inverse problem of electrocardiography (ECG) is to estimate source parameters on or in the heart given a geometric and conductivity model of the torso volume and observed electric potentials on the body surface. Activation-based in-verse ECG models the functional sources of the heart at any location on the heart surface (synthesized by

### Deep learning for inverse problems in quantum mechanics

Inverse problems are common in science, for instance to find the potential that scatters particles in a certain way [1], which can be used to solve for protein structures using X-ray scattering [2]. In medical physics there are many inverse problems in computerized tomography and magnetic resonance imaging [3].

### www.nsm.buffalo.edu

Physics LettersA382 (2018) 2632 2637 Contents lists available at ScienceDirect Physics Letters A. www.elsevier.com/locate/pla. Imaginary eigenvalues of Zakharov

### INVERSE N-BODY SCATTERING WITH THE TIME-DEPENDENT HARTREE

Inverse N-body scattering problems ask to determine the interaction potential and the external potential from the scattering states of particles. Such inverse problems have been extensively studied for N-body Schr odinger equations with no external potentials (Enss and Weder [5]; Novikov [16]; Wang [26,27]; Vasy [24]; Uhlmann and Vasy [20{22]).

### A small collection of open problems

Keywords: Unique continuation, inverse problems, elliptic equations, parabolic equa-tions. MS Classi cation 2010: 35A02, 35J15, 35K10, 35R30. 1. Introduction Here I collect a number of open problems that I have struggled with, and which I believe maintain some interest. I hope this collection may stimulate younger minds.

### ELLIPTIC INVERSE PROBLEM - AMS

Uniqueness in the two-dimensional inverse scattering problem at fixed energy, for exponentially decaying potentials assumed sufficiently small, was obtained by Novikov ([No]). More recently, in [Is Su] a global uniqueness result was proved assuming the scattering amplitude given at a finite (sufficiently large) number of energies.

### www.researchgate.net

arXiv:1810.07818v6 [math.SP] 2 Dec 2019 A RIEMANN HILBERT PROBLEM APPROACH TO PERIODIC INFINITE GAP HILL OPERATORS AND THE KORTEWEG DE VRIES EQUATION KENNETH T-R MCLAUGHLIN AN

### Electronic Journal of Di erential Equations, Vol. 2014 (2014

controllability and inverse problems of degenerate parabolic equations are studied in [10, 12, 13, 14, 27, 32], and for the coupled degenerate parabolic systems in [1, 2, 7, 11, 26]. In these papers, the degeneracy considered is at the boundary of the spatial domain. After the pioneering works [19, 20], there has been substantial progress in un-

### An Extension of Hedberg's Convolution Inequality and - CORE

potentials 55I) f L npr nyap Rn.F Cf55 pn,1-p-nra,1.4 a where C is a constant independent of f, immediately follows from inequal-ity 1.3 , thanks to the boundedness of the operator M on Lp Rn., the latter being a consequence of the Marcinkiewicz interpolation theorem. Our basic result is an optimal Orlicz-space version of inequality 1.3

### INVERSE PROBLEM FOR DIFFERENTIAL PENCILS WITH INCOMPLETELY

INVERSE PROBLEM FOR DIFFERENTIAL PENCILS WITH INCOMPLETELY SPECTRAL INFORMATION Yongxia Guo and Guangsheng Wei* Abstract. In this paper we are concerned with the inverse spectral problems for energy-dependent Sturm-Liouville problems (that is, differential pencils) defined on interval[0,1]with two potentials known on a subinterval[a1,a2] ⊂ [0

### University of Alabama at Birmingham

1276 M Marletta and R Weikard strengthened using interpolation space estimates if one has further aprioriinformation about the potential to be recovered; however, without such inf

### Fixed energy inverse scattering

Abstract. For potentials that decay faster than any exponential the uniqueness of the solution to the inverse scattering problem with ﬁxed-energy data is proved. 1. Introduction. We study the inverse scattering problem in potential scattering at ﬁxed energy. Let q(x), x∈ Rn be a (short range) potential and denote by A(ω,θ,k) the related

### Stability for an inverse resonance problem

2007 Inverse Problems 23 1677 this paper we give an estimate for the difference of two potentials (one of them monotone nondecreasing.

### Non-Equilibrium Dynamics of One-dimensional Infinite Particle

1 Institute for Problems of Information Transmission , SU-111024 Moscow E-24 USSR 2 Mathematical Institute , Budapest Hungary Abstract. An infinite system of Newton's equation of motion is considered for one-dimensional particles interacting by a finite-range hard-core potential of singularity like an inverse power of distance between the hard

### arXiv:1307.1924v1 [math.SP] 7 Jul 2013

form of the associated transformation between two inverse problems, in particular, the function spaces for the impedance functions and the potentials. Thus there has

### CARLEMAN ESTIMATES AND NULL CONTROLLABILITY OF DEGENERATE

mechanics or linearized combustion problems. Furthermore, while in [5, 27] only the existence of a solution for the uniformly (a>0) parabolic problem with singular potential is considered, in [13, 26] the authors analyze in detail the question of whether it is possible to control heat equations involving singular inverse-square potentials.

### Direct and Inverse Problems for Differential Systems

Direct and Inverse Problems for Differential Systems Connected with Dirac Systems and Related Factorization Problems Damir Z. Arov & Harry Dym ABSTRACT. Uniqueness theorems for inverse problems for canon ical differential systems of the form y'(t, A) = iAy(t,A)H(t)J when H ( f ) = X ( t ) NX ( t ) * for appropriately restricted X ( t ) and

### Additional Exercises for Convex Optimization

his nondecreasing in the ith argument, and g iis convex his nonincreasing in the ith argument, and g iis concave g iis a ne. Show that f is convex. (This composition rule subsumes all the ones given in the book, and is the one used in software systems such as CVX.) You can assume that domh= Rk; the result

### Spaces and - ASC

tentials, thus completely solving the related direct and inverse scattering problems. Using the inverse scattering transform approach [1], we then describe all solutions of the Korteweg{de Vries (KdV) equa-tion whose initial pro le is an integrable re ectionless potential. Such solutions stay integrable and re

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May 16, 2019 Inverse Problems 7KHVFDWWHULQJWUDQVIRUPIRUWKH%HQMDPLQ 2QR HTXDWLRQ To cite this article: R R Coifman and M V Wickerhauser 1990 Inverse Problems 6 825 View the article online for u

### University at Buffalo

JID:PLA AID:25197 /SCO Doctopic: Nonlinear science [m5G; v1.240; Prn:4/07/2018; 9:38] P.1(1-6) Physics LettersA ( ) Contents lists available

### METHODS OF ALGEBRAIC GEOMETRY IN THE THEORY OF NON-LINEAR

in the solution of all the following inverse problems. We consider the function yL x - ^- ) Ψ(Χ, У, t, Ρ) = 0. It satisfies all the requirements defining the Akhiezer function except one. The expansion of the regular factor for an exponent in P o begins with Oik'1). From the uniqueness of ψ(χ, у, t, Ρ) it follows that this function

### SEMILINEAR ELLIPTIC PDE S WITH A SINGULAR POTENTIAL

semilinear elliptic pde s with a singular potential 975 Observe that given any > 0, if a satisﬁes (0.1), then the space H associated with a:= (1−)a coincides with H1 0(Ω).So that in the generic case, our

### NULL CONTROLLABILITY OF DEGENERATE/SINGULAR PARABOLIC EQUATIONS

though many problems that are relevant for applications are described by parabolic equations degenerating at the boundary of the space domain. For instance, in [2,6,7,16], the reader will nd a motivating example of a Crocco-type equation coming from the study of the velocity eld of a laminar ow on a at plate.

### u′′ x u a xm a xm arXiv:math/0502522v1 [math.SP] 24 Feb 2005

Then we apply these to the inverse spectral problem, reconstructing some coeﬃcients of polynomial potentials from asymptotic expansions of the eigenvalues. Preprint. 1. Introduction In this paper, we study non-self-adjoint Schro¨dinger operators in L2([0,+∞)), with monic