Direct and inverse scattering problems for operator of order 4 on the line

Abstract. We study both the direct and the inverse scattering problems for a differential operator of order 4 on the line. In the direct scattering problem we start by forming a differential equation from our operator. By applying the fundamental solution of the differential operator we turn the differential equation into an integral equation, and proceed to solve it. Having found the solution to the integral equation we study the asymptotic behaviour of the solution. We conclude the study of the direct scattering problem by defining so called transmission and reflection coefficients, that are needed to solve the inverse scattering problem. In the inverse scattering problem we simplify the operator by choosing its coefficients suitably. It turns out that the operator includes one interesting potential function V. The inverse problem is formulated as follows: find and construct the jumps and singularities of the potential V. By using the reflection coefficient defined previously we define the so called inverse Born approximation V_B. We prove that the difference V-V_B is a continuous function. This means that the jumps and singularities of potential V can be found by calculating V_B.Suora ja käänteinen sirontaongelma neljännen kertaluvun operaattorille. Tiivistelmä. Työssä tutkitaan sekä suoraa että käänteistä sirontaongelmaa neljännen kertaluvun differentiaalioperaattorille. Suorassa sirontaongelmassa käytämme operaattoria muodostaaksemme differentiaaliyhtälön. Soveltaen operaattorin perusratkaisua, voimme muuntaa differentiaaliyhtälön integraaliyhtälöksi ja ratkaista sen. Kun integraaliyhtälön ratkaisu on löydetty, tutkimme sen asymptoottista käyttäytymistä. Päätämme suoran sirontaongelman tarkastelun määrittelemällä asymptoottien avulla ns. välitys- ja heijastuskertoimet, joita tarvitaan käänteisen sirontaongelman ratkaisussa. Käänteisessä sirontaongelmassa yksinkertaistamme operaattoria hieman. Käy ilmi, että operaattori sisältää tällöin yhden kiinnostavan potentiaalifunktion V. Käänteisen sirontaongelman asettelu on seuraava: etsi potentiaalin V mahdolliset hyppyepäjatkuvuudet ja singulariteetit, kun heijastuskerroin on tunnettu. Heijastuskertoimen avulla voimme määritellä ns. käänteisen Bornin approksimaation V_B. Osoitamme, että erotus V-V_B on jatkuva funktio. Tällöin potentiaalin V hypyt ja singulariteetit voidaan löytää laskemalla V_B

Direct and inverse scattering problems for perturbations of the biharmonic operator

Abstract This dissertation is a combination of four articles on the topic of scattering problems for a biharmonic operator. The operator of interest has two coefficients which may be complex-valued and singular. Each of the articles concerns a different aspect of the problem. Namely, the first article discusses the direct scattering problem in higher dimensions and culminates in a proof of Saito's formula, which yields a uniqueness result for the inverse scattering problem. The second paper is about a backscattering problem in two and three dimensions. We prove that the inverse Born approximation can be used to recover the singularities in the coefficients of the operator. The third article fills in an answer to the question about recovering the complex-valued coefficients in three dimensions that was left open in the second article. The final article studies the inverse scattering problem on the line for a quasi-linear operator.Tiivistelmä Väitöskirjatyö koostuu neljästä artikkelista, jotka käsittelevät sirontaongelmia biharmoniselle operaattorille. Työn kohteena olevalla operaattorilla on kaksi kerrointa, jotka voivat olla kompleksiarvoisia ja singulaarisia. Kukin artikkeli käsittelee sirontaongelmaa eri näkökulmasta. Ensimmäinen artikkeli koostuu pääasiassa suorasta sirontateoriasta korkeammissa ulottuvuuksissa huipentuen lopulta Saiton kaavan todistukseen, jonka seurauksena saadaan yksikäsitteisyystulos käänteiselle sirontaongelmalle. Toisen artikkelin aiheena on takaisinsirontaongelma kahdessa ja kolmessa ulottuvuudessa. Todistamme, että käänteistä Bornin approksimaatiota voidaan käyttää paikantamaan kertoimien mahdolliset singulariteetit. Kolmas artikkeli vastaa toisessa artikkelissa avoimeksi jääneeseen kysymykseen kompleksiarvoisien kertoimien rekonstruoimisesta kolmessa ulottuvuudessa. Viimeisessä artikkelissa tutkitaan käänteistä sirontaongelmaa kvasilineaariselle operaattorille yhdessä ulottuvuudessa

Recovery of singularities from a backscattering Born approximation for a biharmonic operator in 3D

Abstract We consider a backscattering Born approximation for a perturbed biharmonic operator in three space dimensions. Previous results on this approach for biharmonic operator use the fact that the coefficients are real-valued to obtain the reconstruction of singularities in the coefficients. In this text we drop the assumption about real-valued coefficients and also establish the recovery of singularities for complex coefficients. The proof uses mapping properties of the Radon transform

Inverse backscattering problem for perturbations of biharmonic operator

Abstract We consider the inverse backscattering problem for a biharmonic operator with two lower order perturbations in two and three dimensions. The inverse Born approximation is used to recover jumps and singularities of an unknown combination of potentials. Numerical examples are given to illustrate the practical usefulness of the method

Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line

Abstract We consider an inverse scattering problem of recovering the unknown coefficients of quasi-linearly perturbed biharmonic operator on the line. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove that the inverse Born approximation can be used to recover some essential information about the unknown coefficients from the knowledge of the reflection coefficient. This information is the jump discontinuities and the local singularities of the coefficients

Scattering problems for perturbations of the multidimensional biharmonic operator

Abstract Some scattering problems for the multidimensional biharmonic operator are studied. The operator is perturbed by first and zero order perturbations, which maybe complex-valued and singular. We show that the solutions to direct scattering problem satisfy a Lippmann-Schwinger equation, and that this integral equation has a unique solution in the weighted Sobolev space $H_{-δ}^2$. The main result of this paper is the proof of Saito’s formula, which can be used to prove a uniqueness theorem for the inverse scattering problem. The proof of Saito’s formula is based on norm estimates for the resolvent of the direct operator in $H_{-δ}^1$