EXPRO grant project

New challenges for spectral theory:
geometry, advanced materials and complex fields

2020-2024

The ultimate goal of the project is to develop unconventional tools in spectral theory in order to tackle various newly born, or more classical but recently revived, open problems in mathematics and physics. Among the variety of problems, I intend to particularly consider hot open questions in: spectral geometry of optimal shapes and eigenfunction properties; mathematical models of modern nanostructures, graphene and metamaterials; new concepts in quantum mechanics with non-self-adjoint operators, Schrödinger and Dirac operators with complex potentials, damped wave systems, non-standard stochastic processes and asymptotic distribution of eigenvalues of structured matrices. These apparently unrelated problems are in fact interlinked and a cross-fertilisation of ideas and techniques will constitute an important part of the project. As examples of the synergy, I propose to develop the method of multipliers to become a standard tool in spectral theory of differential operators with complex coefficients and explain the cloaking effect in metamaterials by operator-theoretic methods.

Vibrating circular membrane with attractive boundary conditions

Research team

researchers	doctoral students	undergraduate students
David Krejčiřík (PI) František Štampach Borbala Mercedes Gerhat (since Jan 2022) Miguel Monsalve López (Jan - Aug 2021) Duc Tho Nguyen (since Jan 2020) Nico Michel Schiavone (Nov 2021 - Mar 2022) Michele Zaccaron (Mar - Aug 2023)	Aitor Balmaseda (Oct 2020 - Mar 2021) Lukáš Heriban (since Oct 2022) David Kramár (since Oct 2023) Tereza Kurimaiová (Jan - Jun 2020) Romana Kvasničková (since Oct 2021) Iveta Semorádová (Aug - Dec 2020) Patrik Šnauko (since Oct 2022) Michal Tichý (Jan - Jun 2020) Tuyen Vu (since May 2021)	Tomáš Faikl (since Nov 2021) Jan Havel (since May 2022) Lukáš Heriban (Jan 2020 - Sep 2022) Michaela Jaklinová (Jan 2020 - Jun 2021) David Kramár (Jan 2020 - Sep 2023) Romana Kvasničková (Jan 2020 - Oct 2021) Alexandra Ridziková (Sep 2020 - May 2022)

Publications

Geometry:

T. Vu:
Spectral inequality for Dirac right triangles;
J. Math. Phys. 64 (2023) 041502.

W. Borrelli, Ph. Briet, D. Krejcirik and T. Ourmieres-Bonafos:
Spectral properties of relativistic quantum waveguides;
Ann. Henri Poincare 23 (2022) 4069-4114.

Ph. Briet and D. Krejcirik:
Spectral optimisation of Dirac rectangles;
J. Math. Phys. 63 (2022) 013502.

S. Kondej, D. Krejcirik and J. Kriz:
Soft quantum waveguides with an explicit cut-locus;
J. Phys. A: Math. Theor. 54 (2021) 30LT01.

M. Tichy:
The asymptotic behaviour of the heat equation in a sheared unbounded strip;
J. Differential Equations 297 (2021) 575-600.

D. Krejcirik, V. Lotoreichik, K. Pankrashkin and M. Tusek:
Spectral analysis of the multi-dimensional diffusion operator with random jumps from the boundary;
J. Evol. Equ. 21 (2021) 1651-1675.

Advanced materials:

T. G. Pedersen, H. Cornean, D. Krejcirik, N. Raymond and E. Stockmeyer:
Stark-localization as a probe of nanostructure geometry;
New. J. Phys 24 (2022) 093005.

H. Cornean, D. Krejcirik, T. G. Pedersen, N. Raymond and E. Stockmeyer:
On the two-dimensional quantum confined Stark effect in strong electric fields;
SIAM J. Math. Anal. 54 (2022) 2114-2127.

D. Krejcirik and P. Antunes:
Bound states in semi-Dirac semi-metals;
Phys. Lett. A 386 (2021) 126991.

T. Kalvoda and F. Stampach:
New family of symmetric orthogonal polynomials and a solvable model of a kinetic spin chain;
J. Math. Phys. 61 (2020) 103305.

Complex fields:

T. Nguyen Duc:
Pseudomodes for biharmonic operators with complex potentials;
SIAM J. Math. Anal. 55 (2023) 6580-6624

M. C. Camara and D. Krejcirik:
Complex-self-adjointness;
Anal. Math. Phys. 13 (2023) art. no. 6.

I. Semorádová and P. Siegl:
Diverging Eigenvalues in Domain Truncations of Schrödinger Operators with Complex Potentials;
SIAM J. Math. Anal. 54 (2022) 5064-5101.

N.-A. Lai and N.M. Schiavone:
Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture;
Math. Z. 301 (2022) 3369-3393.

H. Mizutani and N. M. Schiavone:
Spectral enclosures for Dirac operators perturbed by rigid potentials;
Rev. Math. Phys. 34 (2022) 2250023.

L. Heriban and M. Tusek:
Non-self-adjoint relativistic point interaction in one dimension;
J. Math. Anal. Appl. 516 (2022) 126536.

D. Krejcirik, A. Laptev and F. Stampach:
Spectral enclosures and stability for non-self-adjoint discrete Schrodinger operators on the half-line;
Bull. London. Math. Soc. 54 (2022) 2379-2403.

M. Hansmann and D. Krejcirik:
The abstract Birman-Schwinger principle and spectral stability;
J. Anal. Math. 148 (2022) 361-398.

D. Krejcirik and T. Nguyen Duc:
Pseudomodes for non-self-adjoint Dirac operators;
J. Funct. Anal. 282 (2022) 109440.

D. Krejcirik and F. Stampach:
A sharp form of the discrete Hardy inequality and the Keller-Pinchover-Pogorzelski inequality;
Amer. Math. Monthly 129 (2022) 281-283.

F. Stampach and P. Stovicek:
On diagonalizable quantum weighted Hankel matrices;
chapter to the book series Toeplitz Operators and Random Matrices: In Memory of Harold Widom, Operator Theory: Advances and Applications, 289, Birkhäuser, 2022.

F. Stampach:
Asymptotic spectral properties of the Hilbert L-matrix;
SIAM J. Matrix Anal. Appl. 43 (2022) 1658-1679.

F. Stampach:
The Hilbert L-matrix;
J. Funct. Anal. 282 (2022) 1-46.

P. D'Ancona, L. Fanelli, D. Krejcirik, N. M. Schiavone:
Localization of eigenvalues for non-self-adjoint Dirac and Klein-Gordon operators;
Nonlinear Anal. 214 (2022) 112565.

D. Kramar:
The collapse of quasi-self-adjointness at the exceptional points of a PT-symmetric model with complex Robin boundary conditions;
J. Phys. A: Math. Theor. 54 (2021) 415202.

S. Bogli and F. Stampach:
On Lieb-Thirring inequalities for one-dimensional non-self-adjoint Jacobi and Schrödinger operators;
J. Spectr. Theory 11 (2021), 1391-1413.2

O. O. Ibrogimov, D. Krejcirik and A. Laptev:
Sharp bounds for eigenvalues of biharmonic operators with complex potentials in low dimensions;
Math. Nachr. 294 (2021) 1333-1349.

P. Blaschke and F. Stampach:
The asymptotic zero distribution of Lommel polynomials as polynomials of their order with a variable complex argument;
J. Math. Anal. Appl. 490 (2020) 124238.

D. Krejcirik and T. Kurimaiova:
From Lieb-Thirring inequalities to spectral enclosures for the damped wave equation;
Integral Equations Operator Theory 92 (2020) 47.

Organised conferences

3-7 Jun 2024	CIRM conference: Mathematical aspects of the physics with non-self-adjoint operators; Marseille, France.
10-15 Jul 2022	BIRS workshop: Mathematical aspects of the physics with non-self-adjoint operators; Banff, Canada.
1-5 Feb 2021 ~~7-11 Dec 2020~~ ~~23-27 Mar 2020~~	CIRM conference: Mathematical aspects of the physics with non-self-adjoint operators: 10 years after; Marseille, France.

Last modified: 9 November 2023