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


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

Davies' conjecture about approximate diagonalization by M. Embree

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

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

2017 AIM workshop

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

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

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


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

published version

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

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

Perturbation determinant and the Evans function by I. Karambal
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

Two open problems from linear stability analysis in hydrodynamics by C. Tretter

The rate of exponential resolvent growth for quadratic operators by J. Viola

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

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

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

published version

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
published version
Decay law regularity by P. Exner
Exner, P.: Integr. Equ. Oper. Theory (2012) 72: 1.
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
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.
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
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.
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
published version
Uniform time-decay of semigroups of contractions by X. P. Wang
Wang, X.P.: Integr. Equ. Oper. Theory (2012) 73: 3.
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


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.
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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