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).

2024 CIRM meeting

2024-01-CIRM:	Quasimodes for the Stationary Damped Wave Operator: Sharp Decay from Schrödinger Observability by P. Kleinhenz
2024-02-CIRM: arxiv version	Violation of locality for Schrödinger operators with complex potentials by J.-C. Cuenin and R. L. Frank J.-C. Cuenin and R. L. Frank: arXiv:2409.11285v1 [math.SP]

2015 AIM meeting

2015-01-AIM: updates	Completeness of eigenfunctions for Schroedinger operators with complex potentials by Y. Almog S. N. Tumanov: J. Funct. Anal. 280 (2021) 108820 S. N. Tumanov: Mat. Zametki, 109 (2021) 797-800
2015-02-AIM:	Magnetic Schroedinger operator by Y. Almog
2015-03-AIM: updates	Schauder bases of periodic functions and multipliers by L. Boulton B. Mityagin: Funct. Anal. Its Appl. 51 (2017) 236-239
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 and N. Srivastava: Comm. Pure Appl. Math. 74 (2021) 2114-2131
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. L. Frank S. Bogli: Comm. Math. Phys. 352 (2017) 629-639 J.-C. Cuenin: Comm. Math. Phys. 376 (2020) 2147-2160 S. Bogli and J.-C. Cuenin: arXiv:2109.06135 [math.SP]
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 D. Krejcirik and P. Siegl: 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 J. Behrndt and D. Krejcirik: J. Anal. Math. 134 (2018) 501-522 C. Cacciapuoti, K. Pankrashkin and A. Posilicano: J. Anal. Math. 139 (2019) 155-177
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 B. Mityagin and P. Siegl: J. Anal. Math. 139 (2019) 83-119 D. Krejcirik and P. Siegl: 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 J. Behrndt: Integr. Equ. Oper. Theory 77 (2013) 299 M. Levitin and M. Seri: Oper. Matrices 10 (2016) 223-245 J. Behrndt, P. Schmitz, C. Trunk: Proc. Am. Math. Soc. 146 (2018) 3935-3942 J.-C. Cuenin, O. O. Ibrogimov: J. Funct. Anal. 280 (2021) 108804
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: updates	Two open problems from linear stability analysis in hydrodynamics by C. Tretter presentation B. Gerhat, O. O. Ibrogimov and P.Siegl: Commun. Math. Phys. 392, (2022) 377–397
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 L. Boulton: Integr. Equ. Oper. Theory 74 (2012) 1 D. E. Edmunds, P. Gurka and J. Lang: 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 P. Exner: Integr. Equ. Oper. Theory 72 (2012) 1
2010-04-ESF: published version	Complex symmetric operators by S. R. Garcia S. R. Garcia: Integr. Equ. Oper. Theory 72 (2012) 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 M. Gonzalez: Integr. Equ. Oper. Theory 74 (2012) 3
2010-07-ESF: published version	PT-symmetric waveguides and the lack of variational techniques by D. Krejcirik D. Krejcirik: Integr. Equ. Oper. Theory 73 (2012) 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 M. Marletta: Integr. Equ. Oper. Theory 72 (2012) 5
2010-10-ESF: published version	PT-symmetric Laplace-Beltrami operator in the strip on a sphere by P. Siegl P. Siegl: Integr. Equ. Oper. Theory 73 (2012) 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 X. P. Wang: Integr. Equ. Oper. Theory 73 (2012) 3
2010-13-ESF: published version	Matrix Hamiltonians with a chance of being complex symmetric by M. Znojil M. Znojil: Integr. Equ. Oper. Theory 74 (2012) 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 updates	Lieb-Thirring Type Inequalities for Schroedinger Operators with a Complex-Valued Potential by M. Demuth and M. Hansmann and G. Katriel M. Demuth, M. Hansmann, and G. Katriel: Integr. Equ. Oper. Theory 75 (2013) 1 S. Bogli and F. Stampach: J. Spec. Theory to appear, arXiv:2004.09794 [math.SP]
2012-02-OTHER: published version	Combinatorial Identities Related to Eigen-Function Decompositions of Hill operators: Open Questions by P. Djakov and B. Mityagin P. Djakov and B. Mityagin: Integr. Equ. Oper. Theory 75 (2013) 7

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Last updated: 17 February 2022