Open problems in non-self-adjoint spectral theory

suggested by the participants of the conference series
Mathematical aspects of the physics with non-self-adjoint operators
and related meetings.

Some of the open problems have been published in the journal
Integral Equations and Operator Theory (Birkhauser).


2015 AIM meeting

2015-01-AIM:

updates
Completeness of eigenfunctions for Schroedinger operators with complex potentials by Y. Almog

Tumanov, S.: arXiv:1903.10594v2 [math.FA], 2019
2015-02-AIM: Magnetic Schroedinger operator by Y. Almog
2015-03-AIM: Schauder bases of periodic functions and multipliers by L. Boulton
2015-04-AIM: Numerical approximation of rigorous enclosures for the spectrum of J-self-adjoint operators by L. Boulton
2015-05-AIM: Non-self-adjoint inverse problems by A. Boumenir
2015-06-AIM: Isoresonant potentials by T. Christiansen
2015-07-AIM: Computations of the instability index for a non-self-adjoint operators by M. Chugunova
2015-08-AIM: Spectral radius and operator norm by M. Demuth
2015-09-AIM: Estimates for the resolvent near the spectrum by M. Demuth
2015-10-AIM:

updates
Davies' conjecture about approximate diagonalization by M. Embree

J. Banks, A. Kulkarni, S. Mukherjee, N. Srivastava, arXiv:1906.11819 [math.FA] (2019)
2015-11-AIM:

updates
Crouzeix' conjecture about the norm of matrix functions by M. Embree

2017 AIM workshop
2015-12-AIM:

updates
Laptev-Safronov conjecture by R. Frank

Bogli, S.: Comm. Math. Phys. 352 (2017), 629-639
2015-13-AIM: Tensor trick for perturbed operators by M. Hansmann
2015-14-AIM: Upper bounds on the norm of the resolvent by M. Hitrik
2015-15-AIM: Inverse spectral problems for non-self-adjoint operators, especially in the semiclassical limit by M. Hitrik
2015-16-AIM: Spectra for non-self-adjoint operators in the presence of symmetries by M. Hitrik
2015-17-AIM:

updates
Semiclassical pseudomodes of Schrodinger operators with discontinuous potentials by D. Krejcirik

Krejcirik, D. and Siegl, P.: J. Funct. Anal. 276 (2019), 2856-2900.
2015-18-AIM: Large-time behavior of the heat equation: subcriticality versus criticality by D. Krejcirik
2015-19-AIM:

updates
The cloaking effect in metamaterials: beyond ellipticity by D. Krejcirik

Cacciapuoti, C., Pankrashkin, K. and Posilicano, A.: J. Anal. Math. to appear
2015-20-AIM: Complex eigenvalues of indefinite pencil by M. Levitin
2015-21-AIM: Indefinite Sturm-Liouville by M. Levitin
2015-22-AIM: Zeros and poles of Nevanlinna functions by M. Marletta
2015-23-AIM: Schauder basis for Schroedinger operators with periodic boundary conditions by B. Mityagin
2015-24-AIM: Non-polynomial complex potentials by K. Shin
2015-25-AIM:

updates

Riesz basis for Schroedinger operator with complex potential by P. Siegl

Mityagin, B. and Siegl, P.: J. Anal. Math. to appear,
Krejcirik, D. and Siegl, P.: J. Funct. Anal. 276 (2019), 2856-2900.
2015-26-AIM: Spectral representation of two-point differential operators by D. A. Smith
2015-27-AIM: Open problem for complex-valued periodic potentials by R. Weikard

2013 ICMS meeting

2013-01-ICMS:
published version

updates
Spectral theory of Sturm-Liouville operators with indefinite weights by J. Behrndt
Behrndt, J.: Integr. Equ. Oper. Theory (2013) 77: 299.

Levitin, M and Seri, M: Oper. Matrices 10 (2016), 223-245.
2013-02-ICMS: Magnetic resonances and meromorphic families of non-self-adjoint operators by V. Bruneau
2013-03-ICMS: A Riemann-Hilbert approach to Toeplitz operators and the corona theorem by C. Camara
2013-04-ICMS: Stability of regularized shock solutions in coating flows by M. Chugunova
2013-05-ICMS: 1D Dirac operators with trigonometric polynomial potentials by P. Djakov
2013-06-ICMS:

The spectrum of second order multi-point problems by F. Genoud
presentation
2013-07-ICMS: The complex cubic oscillator: spectral instability and real spectrum by R. Henry
2013-08-ICMS:

Perturbation determinant and the Evans function by I. Karambal
presentation
2013-09-ICMS: Differential operators with exotic boundary conditions arising in probability theory by M. Kolb
2013-10-ICMS: The cloaking effect in metamaterials: beyond ellipticity by D. Krejcirik
2013-11-ICMS: Computing spectral determinants for multi-dimensional non-selfadjoint operators by S. Malham
2013-12-ICMS: Alternative proofs of maximal and non maximal subelliptic estimates for geometric Kramers-Fokker-Planck equations by F. Nier
2013-13-ICMS:

Two open problems from linear stability analysis in hydrodynamics by C. Tretter
presentation
2013-14-ICMS:

The rate of exponential resolvent growth for quadratic operators by J. Viola
presentation
2013-15-ICMS:

Eigenvalue statistics for a model non-self-adjoint semiclassical differential operator with a small random perturbation by M. Vogel
presentation
2013-16-ICMS:

Real resonances of Schroedinger operators with complex-valued potentials by X. P. Wang
presentation

2012 OTAMP meeting

2012-01-OTAMP: Basis properties of the eigenvectors of a PT-symmetric Sturm-Liouville operator by L. Boulton
2012-02-OTAMP: Curved Dirichlet waveguides in strong magnetic field by P. Exner
2012-03-OTAMP: Large-time behaviour of the heat equation: subcriticality versus criticality by D. Krejcirik
2012-04-OTAMP: "Strange" functions in the Laplace operator domains and their relation to the eigenfunctions by J. Kriz
2012-05-OTAMP: Asymptotics of solution of a certain ODE by A. Luger
2012-06-OTAMP: On the invertibility of convolution operators on cones by H. Mascarenhas
2012-07-OTAMP: Self-adjoint Jacobi operators possessing generalized eigenvectors with the strongly increasing phase sequence: do they exist? by M. Moszynski

2010 ESF meeting

2010-01-ESF:
published version

updates
Open problems regarding applying non-self-adjoint operator techniques to the p-Laplace non-linear operator in one dimension by L. Boulton
Boulton, L.: Integr. Equ. Oper. Theory (2012) 74: 1.

Edmunds, D. E. , Gurka, P. and Lang, J.: Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), 20140221.
2010-02-ESF: The Weyl law for resonances of operators with variable coefficients on graphs by E. B. Davies
2010-03-ESF:
published version
Decay law regularity by P. Exner
Exner, P.: Integr. Equ. Oper. Theory (2012) 72: 1.
2010-04-ESF:
published version
Complex symmetric operators by S. R. Garcia
Garcia, S.R.: Integr. Equ. Oper. Theory (2012) 72: 3.
2010-05-ESF: The de Sitter group representations: open questions by J.-P. Gazeau
2010-06-ESF:
published version
The spectrum of the Hilbert matrix as an operator on $\ell_p$ by M. Gonzalez
Gonzalez, M.: Integr. Equ. Oper. Theory (2012) 74: 3.
2010-07-ESF:
published version
PT-symmetric waveguides and the lack of variational techniques by D. Krejcirik
Krejcirik, D.: Integr. Equ. Oper. Theory (2012) 73: 1.
2010-08-ESF: The large-time behaviour of the heat kernel: subcriticality versus criticality by D. Krejcirik
2010-09-ESF:
published version
The growth order of a solution of a non-self-adjoint differential equation by M. Marletta
Marletta, M.: Integr. Equ. Oper. Theory (2012) 72: 5.
2010-10-ESF:
published version
PT-symmetric Laplace-Beltrami operator in the strip on a sphere by P. Siegl
Siegl, P.: Integr. Equ. Oper. Theory (2012) 73: 5.
2010-11-ESF: Some open problems related to the Weyl asymptotics for non-self-adjoint operators by J. Sjostrand
2010-12-ESF:
published version
Uniform time-decay of semigroups of contractions by X. P. Wang
Wang, X.P.: Integr. Equ. Oper. Theory (2012) 73: 3.
2010-13-ESF:
published version
Matrix Hamiltonians with a chance of being complex symmetric by M. Znojil
Znojil, M.: Integr. Equ. Oper. Theory (2012) 74: 5.
2010-14-ESF: Open problem in non-selfadjoint linear second-order ODE's in complex domain: the proof of the reality of spectra for quantum toboggans by M. Znojil

Other

2012-01-OTHER:
published version
Lieb-Thirring Type Inequalities for Schroedinger Operators with a Complex-Valued Potential by M. Demuth and M. Hansmann and G. Katriel
Demuth, M., Hansmann, M. and Katriel, G.: Integr. Equ. Oper. Theory (2013) 75: 1.
2012-02-OTHER:
published version
Combinatorial Identities Related to Eigen-Function Decompositions of Hill operators: Open Questions by P. Djakov and B. Mityagin
Djakov, P. and Mityagin, B.: Integr. Equ. Oper. Theory (2013) 75: 7.


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