Inverse Problems on Differential operators  Page description

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Details of project

  
Identifier
61311
Type K
Principal investigator Horváth, Miklós
Title in Hungarian Inverz feladatok differenciáloperátorokon
Title in English Inverse Problems on Differential operators
Keywords in Hungarian Sajátértékek, inverz spektrálelmélet, inverz szórás
Keywords in English Eigenvalues, inverse spectral theory, inverse scattering
Discipline
Mathematics (Council of Physical Sciences)100 %
Panel Mathematics and Computing Science
Department or equivalent Department of Mathematical Analysis (Budapest University of Technology and Economics)
Starting date 2006-02-01
Closing date 2011-01-31
Funding (in million HUF) 3.756
FTE (full time equivalent) 1.25
state closed project
Summary in Hungarian
Egy- és többváltozós Schrödinger operátorok inverz feladatai:

A/ A sajátértékek eloszlásának vizsgálata másod- és magasabbrendű közönséges differenciáloperátorokra.
Egyenlőtlenségek a sajátértékek között. Többváltozós Dirichlet-feladat sajátértékeinek eloszlása.

B/ Inverz sajátértékfeladat Schrödinger operátorra. Unicitást biztosító sajátértékhalmazok jellemzése
a számegyenesen. Izospektrális halmazok vizsgálata, tulajdonságainak leírása.

C/ Inverz szórási feladat a térben rögzített energia illetve impulzusmomentum mellett, a számegyenesen.
Stabilitás elvi és gyakorlati értelemben. Potenciál konstrukciója szórási adatokból.
Summary
Inverse problems for one- and multidimensional Schrödinger operators. More precisely:

A. Distribution of eigenvalues for ODE of second and higher order. Inequalities between eigenvalues.
Eigenvalue distribution for the Dirichlet Laplacian on a bounded domain.

B. Inverse Sturm-Liouville problem (i.e. inverse eigenvalue problem) for Schrödinger operators.
Description of the sets of uniqueness ( which can be produced by only one operator) on the real line.
Properties of the isospectral sets (e.g. connectedness).

C. Three-dimensional inverse scattering with fixed energy, inverse scattering on the line. Mathematical stability,
practical stability problems. Reconstruction of the potential from the scattering data.




 

Final report

  
Results in Hungarian
Inverz szórási feladatok sok műszaki, orvosdiagnosztikai, fizikai stb. alkalmazásban előfordulnak. A projektben a kvantummechanika inverz szórási feladatai közül vizsgáltunk néhányat. Bizonyítottunk unicitási tételeket, amelyek szerint a szórási adatokból bizonyos feltételek mellett a vizsgált objektum egyértelműen azonosítható. Ezzel lényegesen megjavítottunk korábban ismert eredményeket. Vizsgáltuk az inverz feladat stabilitását, tehát hogy a rekonstruált objektum mennyire érzékeny a szórási adatok kis változására. Kutattuk a szórási adatok belső, matematikai tulajdonságait, például a fixenergiás fázistolások sorozatának eloszlását, vagy a szórásamplitúdó komplex változós kiterjesztésének analitikus növekedési rendjét. Megjavítottunk ismert rekonstrukciós eljárásokat, melyek a Gelfand-Levitan módszer alapján állítják elő a potenciált a fázistolásokból és definiáltunk új rekonstrukciós eljárást, mely egy momentumfeladat minimális normájú megoldásán alapul. Az inverz sajátértékfeladat stabilitását is vizsgáltuk véges intervallumon Schrödinger operátorra, itt is megjavítva klasszikus eredményeket.
Results in English
Inverse scattering problems appear in many applications in engineering, medical diagnostics, Physics and so on. In this project we investigated some inverse scattering problems of quantum mechanics. We proved uniqueness theorems which state that the target object can be uniquely identified by scattering data. These are essential improvements of some former results. We investigated stability of inverse problems, that is the sensibility of reconstruction to small perturbation of scattering data. Research has been made to clarify the inner mathematical structure of scattering data, e.g. the distribution of fixed-energy phase shifts or the growth of analytic extensions of the scattering amplitude. We developed further some known reconstruction procedures, based on the Gelfand-Levitan method and defined a new one, based on the minimum norm solution of a moment problem. We also considered the stability of the inverse eigenvalue problem for the Schrödinger operator on finite intervals and obtained strengthening of some classical results.
Full text https://www.otka-palyazat.hu/download.php?type=zarobeszamolo&projektid=61311
Decision
Yes





 

List of publications

  
B. Apagyi and M. Horváth (editors): Proceedings of the International Conference on Inverse Quantum Scattering Theory, 27-31. August, Siofok, 2007, Modern Physics Letters B vol 22, No 23, 2008
M. Horvath and O. Safar: Inequalities between fixed energy phase shifts II, (manuscript), 2011
M. Horváth: Spectral shift functions in the fixed energy inverse scattering, (submitted), 2011
M. Horváth and M. Kiss: Stability of direct and inverse eigenvalue problems: the case of complex potentials, (submitted), 2011
M. Horváth: Inequalities between the fixed-energy phase shifts, Int. J. Comput. Sci. Math. 3(2010), 132-141., 2010
M. Horváth: Partial identification of the potential from phase shifts, J. Math. Anal. Appl., 2011
M. Horváth and M. Kiss: On the stability of inverse scattering with fixed energy, Inverse Problems 25(2009), 015011, 2009
M. Horváth and M. Kiss: Stability of direct and inverse eigenvalue problems for Schrödinger operators on finite intervals, International Mathematics Research Notices, Advance Access doi:10.1093/imrn/mp210, 11(2010), 2022-2063, 2010
B. Apagyi, M. Horváth and T. Pálmai: Simplified solutions of the Cox-Thompson inverse scattering method at fixed energy, J. Phys. A 41(23)(2008), 235305, 2008
B. Apagyi, M. Horváth and T. Pálmai: Semi-analytic equations to the Cox-Thompson inverse scattering method at fixed energy for special cases, Proceedings of the International Conference on Inverse Quantum Scattering Theory, 27-31. August, Siofok, 2007, Modern Physics Letters B vol 22, No 23, 2191-2199., 2008
B. Apagyi and M. Horváth: Solution of the inverse scattering problem at fixed energy for potentials being zero beyond a fixed radius, Proceedings of the International Conference on Inverse Quantum Scattering Theory, 27-31. August, Siofok, 2007, Modern Physics Letters B vol 22, No 23, 2137-2149., 2008
M. Horváth: Notes on the distribution of phase shifts, Proceedings of the International Conference on Inverse Quantum Scattering Theory, 27-31. August, Siofok, 2007, Modern Physics Letters B vol 22, No 23, 2163-2175, 2008, 2008
M. Horváth: Inverse problems for linear differential operators, DSc Dissertation, Budapest, 2007
Miklos Horvath: Properties of the scattering data of the 3D inverse scattering problem, INVERSE PROBLEMS, DESIGN AND OPTIMIZATION SYMPOSIUM, August 25-27, Joao Pessoa, Brazil, 2010
Miklos Horvath: Properties of the fixed energy phase shifts of the 3D inverse scattering problem, 11th International Workshop on Optimization and Inverse Problems in Electromagnetism OIPE 2010 14-18 September, Sofia, Bulgaria, 2010
Miklos Horvath: On the use of spectral theory in inverse scattering problems, Short courses and Workshop on Spectral Function Theory, March 14 to 19, Barcelona, Spain, 2011
Miklos Horvath: Stability of some inverse problems, ICNAAM 2010 International Conference of Numerical Analysis and Applied Mathematics, 19-25 September, Rhodes, Greece, 2010
M. Horváth: Some properties of the eigenvalues of Sturm-Liouville operators, International Congress of Mathematicians, August 22-30, 2006, Madrid, 2006
M. Horváth: On the distribution of phase shifts, International Conference on Inverse Quantum Scattering Theory, 27-31. August, Siofok (to appear), 2007
M. Horváth: Inverse problems for linear differential operators, Fourth International Conference of Applied Mathematics and Computing (Plovdiv, August 12-18, 2007), 2007
B. Apagyi, M. Horváth and T. Pálmai: Semi-analytic solution of the Cox-Thompson inverse scattering problem at fixed energy for special cases, International Conference on Inverse Quantum Scattering Theory, 27-31. August, Siofok, 2007
M. Horváth: On the stability of some inverse problems, 8th Colloquium on the Qualitative Theory of Differential Equations, 25-28. June 2007. Szeged, 2007
B. Apagyi and M. Horváth: Solution of the inverse scattering problem at fixed energy for potentials being zero beyond a fixed radius, International Conference on Inverse Quantum Scattering Theory, 27-31. August, Siofok, 2007
M. Horváth: Inverse eigenvalue problems and inverse scattering, Seminar of the Chair of Applied Mathematics and Numerics, LMU München, 2007
Miklos Horvath: Inverse scattering problem: new results concerning the phase shifts, Seminar, Institute for Theoretical Physics, Justus-Liebig-University Giessen, Germany, 28th October, 2010



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