Romanian version here.

Grant PN-II-RU-TE- 2014-4-0007, Contract 109 din 01/10/2015 financed by CNCS-UEFISCDI, ANCS

Host institution IMAR.

Title: Analysis of Schrodinger equations

Abstract. In this project we consider deterministic and stochastic Schrödinger equations on metric graphs. By using fine tools from harmonic analysis and stochastic analysis we study the dispersion property in terms of the topology of the considered structure. We analyze the well-posedness of nonlinear models and study some of their qualitative properties: long time behavior, solitons, waves propagation, blow-up, scattering. We investigate the influence of the graph topology on the behavior of the solutions, the existence of maximizers for Strichartz –like estimates and the way they depend on the topology. Discrete models of the above equations will be considered both from the numerical as the theoretical point of view.

There are few research directions

A. Schrödinger equations on graphs, Stochastic Schrödinger equation on graphs.

B. Discrete Schrödinger equations. Discrete stochastic Schrödinger equations.

TEAM

1. Liviu Ignat

2. Cristi Cazacu (postdoc)

3. Diana Stan (postdoc)

4. Iulian Cimpean (PhD student)

5. Cristian Bereanu (postdoc)

6. Oana Lupascu (postdoc)

7. Andreea Grecu (master student)

8. Maria Farcaseanu (PhD student)

9. Mihai Mihailescu (postdoc)

1. Emilian Paraicu (master student) - former member

2. Dana Bereanu (postdoc) - former member

Report 2015, PDF

Budget, PDF

Report 2016, PDF

Report 2017 Final, PDF

Papers

1. M. Farcaseanu, M. Mihailescu: The continuity of the first eigenvalue for a family of degenerate eigenvalue problems, Archiv der Mathematik, 107 (2016), 659-667, PDF

2. M. Farcaseanu, M. Mihailescu, and D. Stancu-Dumitru: Perturbed fractional eigenvalue problems, Discrete and Continuous Dynamical Systems - Series A 37 (2017), 6243-6255, PDF

3. Ignat, T. Ignat, Long time behavior for a nonlocal convection diffusion equation, J. Math. Anal. Appl.455, (1), 816{831, (2017), PDF

4. C. Bereanu, M. Zamora, Periodic solutions for indefinite singular perturbations of the relativistic acceleration, Proc. Royal Soc. Edinburgh - A, 12 p, accepted, PDF

5. O. Lupascu, V. Stanciulescu, Numerical Solution for the Non-linear Dirichlet Problem of a Branching Process, Complex Analysis and Operator Theory, PDF

6. M. Farcaseanu, M. Mihailescu, and D. Stancu-Dumitru: On the convergence of the sequence of solutions for a family of eigenvalue problems, Mathematical Methods in the Applied Sciences, Volume 40, Issue 18, December 2017, Pages 6919–6926 , PDF

7. I. Cimpean, L. Beznea, Quasimartingales associated to Markov processes, accepted TAMS, PDF

8. L. Ignat, A. Pozo, A splitting method for the augmented Burgers equation, BIT Numerical Mathematics, 2018, Volume 58, Issue 1, pp 73–102, PDF

9. Cazacu, L. Ignat and A. Pazoto, Null control lability of the Kuramoto-Sivashinsky equation on star-shaped trees, first revision SIAM Control and Optimization, PDF

10. L. Ignat, D. Stan, Asymptotic behaviour for fractional difusion - convection equations, accepted London Mathematical Society, PDF

11. C. Bereanu, P. D. Torres, A variational approach for the Neumann problem in some FLRW spacetimes, submitted, PDF

Talks

• (1) Cristian Bereanu, Periodic solutions for singular perturbations of the relativistic ac- celeration Workshop on Geometry and PDEs 2016, 10-11 /06/ 2016, Univ. de Vest din Timisoara, Romania.
• (2) Cristian Bereanu, Periodic solutions for singular perturbations of the relativistic ac- celeration, Departamento de Matematica Aplicada, University of Granada, Spain, 8/11/2016
• (3) Cristian Cazacu, Schr ̈odinger operators with singular potentials: some properties and applications, “6th International Conference on Mathematics and Informatics”, Targu Mures, Romania, 07-09/09/2017
• (4) Cristian Cazacu, Controllability results for Kuramoto-Sivashinsky models on star- shaped trees, Workshop “Stability and Control in Infinite Dimensional Systems”, IMAR, Romania, 19/10/2016.
• (5) Iulian Cimpean, From excessive functions to semimartingales on Dirichlet spaces, XIII-eme colloque Franco-Roumaine de mathematique appliques, Iasi, Romania, au- gust 25-29, 2016.
• (6) Iulian Cimpean, Semimartingale functionals of Markov processes: a semigroup approach, 4Th Summer school on Levy processes, Universite de Lille 1, Franta, iulie 18-22, 2016.
• (7) Iulian Cimpean, Semimartingales associated to Markov processes: a semigroup approach, The 14th Romanian-Finnish Seminar, Bucuresti, Romania, iunie 20-24, 2016
• (8) Iulian Cimpean, Semimartingale functionals of Markov processes: a semigroup approach, The 6th Workshop Series on Mathematics, University of Pitesti, iunie 3-4, 2016.
• (9) Iulian Cimpean, Talk: Semimartingale functionals associated to Dirichlet forms, Wor- kshop for Young Researchers in Mathematics 6th edition, Constanta, mai 19-22, 2016.
• (10) Iulian Cimpean, Talk: On the Schrodinger equation with white noise dispersion on graphs , Workshop for young researchers in mathematics 7th edition, 17-20 May, 2017, Bucharest.
• (11) Maria F ̆arc ̆a ̧seanu, Workshop Applied Mathematics: Methods and Modeling, May 7-8, 2017, Craiova, Romania, title of the talk: Perturbed fractional eigenvalue problems;
• (12) Maria F ̆arc ̆a ̧seanu, Workshop for Young Researchers in Mathematics, May 17-20, 2017, Institute of Mathematics ”Simion Stoilow” of the Romanian Academy, Bucharest, Romania, title of the talk: On a family of torsional creep problems involving rapidly growing operators in divergence form
• (13) Maria F ̆arc ̆a ̧seanu, 6th International Conference on Mathematics and Informatics, September 7-9, 2017, Tˆargu Mure ̧s/Marosv ́as ́arhely, Romania, On a family of torsional creep problems involving rapidly growing operators in divergence form
• (14) Maria F ̆arc ̆a ̧seanu, International conference TMSF 2017, August 27-30, 2017, Institute of Mathematics and Informatics - Bulgarian Academy of Sciences (IMI - BAS), Sofia, Bulgaria, title of the talk: Perturbed fractional eigenvalue problems
• (15) Andreea Grecu, Dispersive and Strichartz Estimates for the Solution of the Schr ̈odin- ger Equation on a Graph with Cycle at XIII-eme Colloque Franco-Roumain de Ma- thematiques Appliquees, August 25th-August 29th, 2016, Iasi, Romania
• (16) Andreea Grecu, Dispersive Properties for the Linear Schr ̈odinger Equation on Star- Graphs (Poster), Insubria Summer School in Mathematical Physics: Spectral and scattering theory: from selfadjoint operators to boundary value problems, 18-22 Sep- tember, 2017, Como, Italy
• (17) Andreea Grecu, Dispersive Properties for the Linear Schr ̈odinger Equation on Star- Graphs. Application to NLS, VII Partial Differential Equations, Optimal Design and numerics, August 20-September 1, 2017, Benasque, Spain
• (18) Andreea Grecu, Dispersive Properties for the Linear Schr ̈odinger Equation on Star- Graphs. Application to NLS, Nonlinear Partial Differential Equations on Graphs, June 18-24, 2017, Mathematisches Forschungsinstitut Oberwolfach, Germany
• (19) Andreea Grecu, Dispersive Properties for the Linear Schr ̈odinger Equation on Star- Graphs, Workshop in Geometry and PDEs, June 13-14, Timisoara, Romania
• (20) Andreea Grecu, Dispersive Properties for the Linear Schr ̈odinger Equation on Star- Graphs, Workshop for Young Researchers in Mathematics, 7th Edition, May 17-20, 2017, Bucharest, Romania
• (21) Laplacians on metric graphs, Happy PDEs Days, 8-9 December, 2016, Bucharest, Romania
• (22) Liviu Ignat, Kuramoto-Sivashinsky equation on a strar-shaped tree. A controllability result, Nonlinear Partial Differential Equations on Graphs, Oberwolfach, 18-24 June 2017, Germany
• (23) Liviu Ignat, Asymptotic behaviour for fractional diffusion- convection equations, Wor- kshop on Nonlinear Analysis on the Occasion of the 65th Birthday of Patrizia Pucci, May 25-27, 2017, Babe ̧s-Bolyai University, Cluj-Napoca
• (24) Liviu Ignat, Asymptotic behaviour for fractional diffusion- convection equations, In- ternational Conference on Elliptic and Parabolic Problems, Gaeta, 22-26 May 2017, Gaeta, Italy
• (25) Liviu Ignat, ”Flash” Dispersion on Trees, French-American Conference on Nonlinear Dispersive PDEs June 12 - 16, 2017, CIRM, France
• (26) Oana Lupascu, Numerical solution for the non-linear Dirichlet problem of a branching process, TOSCA Seminar, La Petite Pierre, France, October 26–28, 2016, invited talk.
• (27) Oana Lupascu, Branching properties for measure-valued Markov process and appli- cations, XIII-eme Colloque Franco-Roumain de Mathematiques Appliquees, in the Section ”Maths et planete Terre”, Ia ̧si, August 25–28, 2016, invited talk.
• (28) Oana Lupascu, Measure-valued branching processes and applications, Summer School on L ́evy Processes, Lille, France, July 18-22, 2016.
• (29) Oana Lupascu, Non-linear Dirichlet problem of a branching process: a probabilistic numerical approach, The 14th Romanian-Finnish Seminar, Bucharest, June 20-24, 2016.
• (30) Oana Lupascu, Branching processes: their nonlinear PDEs and applications, Workshop for Young Researchers in Mathematics - 6th Edition, Constan ̧ta, May 19-22, 2016, invited talk.
• (31) Mihai Mih ̆ailescu, Seminar of the Department of Mathematics and Statistics, March 14, 2017, The College of Arts and Sciences, American University of Sharjah, United Arab Emirates, title of the talk: Inhomogeneous torsional creep problems;
• (32) Mihai Mih ̆ailescu, Workshop on Nonlinear Analysis on the Occasion of the 65th Birthday of Patrizia Pucci, May 25-27, 2017, Babe ̧s-Bolyai University, Cluj- Napoca, Romania, title of the talk: Typical and nontypical eigenvalue problems for some classes of differential operators;
• (33) Mihai Mih ̆ailescu 6th International Conference on Mathematics and Infor- matics, September 7-9, 2017, Tˆargu Mure ̧s/Marosv ́as ́arhely, Romania, title of the talk: Classification of isolated singularities for inhomogeneous operators in divergence form; http://mitis.ro/mathinfo/2017/
• (34) Diana Stan, Nonlocal porous medium type equations, Seminar of the group MATHE- MATICAL ANALYSIS AND APPLICATIONS of the University of the Basque Co- untry EHU/UPV, Bilbao, Spain, 4 February 2016.
• (35) Diana Stan, Recent results on nonlocal diffusion equations, BASQUE CENTER OF APPLIED MATHEMATICS scientific seminar, Bilbao, Spain, 16 February 2016
• (36) Diana Stan, Asymptotic behaviour for fractional diffusion-convection equations, IMAR, Say/Happy PDEs Seminar, 20/09/2017
• (37) Diana Stan, Asymptotic behavior of fractional diffusion-convection equations., Seminar of the Mathematics Department, Norwegian University of Technology NTNU, Trondheim, Norway, April 26, 2017.
• (38) Diana Stan, Asymptotic behavior of fractional diffusion-convection equations., Congreso de jovenes investigadores, 4-8 sep, Valencia, 2017

Research stages.

• (1) Cristian Bereanu, Univ. din Granada, Spain, 04. 11. 2016 - 18. 11. 2016.
• (2) Cristian Bereanu, Univ. Granada. Spain, 01 - 31/03/ 2017
• (3) Cristian Cazacu, 01/08/2016-10/08/2016 - Group of Mathematical Physics, Depart-ment of Theoretical Physics, Nuclear Physics Institute, Prague, Czech Republic (invited by Prof. David Krejˇciˇr ́ık).
• (4) Oana Lupascu, University Paris 13, France, scientific collaboration with Prof. Ioan R. Ionescu, two weeks, July 2016
• (5) Oana Lupascu, University Paris 1, France, scientific collaboration with Prof. Ciprian A. Tudor, October-November 2016, the flight ticket was supported from this project.
• (6) Mihai Mih ̆ailescu, American University of Sharjah, United Arab Emirates, 10/03/2017-18/03/2017.
• (7) Mihai Mih ̆ailescu, Universitatea Babes-Bolyai, Cluj-Napoca, Romania, 24/05/2017-27/05/2017.
• (8) Mihai Mih ̆ailescu, Loyola University Chicago, US, 19/06/2017-02/07/2017.
• (9) Liviu Ignat, Basque Center for Applied Mathematics, 23/10-04/11/2016
• (10) Liviu Ignat, Universidad de Alicante, 1-11/07/2017

Participants to Summer schools/Workhops.

• (1) Cristian Cazacu, 29/06/2016-07/07/2016- “Regional Romanian-French Summer School in Applied Mathematics”, Sinaia, Romania.
• (2) Cristian Cazacu, 05 -09/06/2017- “Mathematical aspects of the physics with nonself- adjoint operators”, Marseille, France.
  • (3) Iulian Cimpean, Minischool on Recent Trends in PDEs, March 21-30, 2017, Madrid, Spain
  • (4) Maria F ̆arc ̆a ̧seanu: Regional Romanian-French Summer School in Applied Mathema- tics, July 2 -11, 2017, Sinaia, Romania.
  • (5) Andreea Grecu, The 6th Workshop Series on Mathematics, June 3rd-June 4th, 2016, Pitesti, Romania
  • (6) Andreea Grecu, The 14th Romanian-Finnish Seminar (International Conference on Complex Analysis and Related Topics), June 20th-June 24th, 2016, Bucharest, Ro- mania
  • (7) Andreea Grecu, Harmonic Analysis, Spectral Theory and PDEs, 12-15 September, 2017, Rome, Italy
  • (8) Andreea Grecu, Regional Romanian-French Summer School in Applied Mathematics, July 2 -11, 2017, Sinaia, Romania.
  • (9) Andreea Grecu, Minischool on Recent Trends in PDEs, March 21-30, 2017, Madrid, Spain
  • (10) Emilian Paraicu, Workshop for Young Researchers in Mathematics - 6th Edition, Constan ̧ta, May 19-22, 2016