Schedule and Abstracts in PDF Here

MATHEMATICAL ASPECTS OF THE PHYSICS WITH NON-SELF-ADJOINT OPERATORS: 10 YEARS AFTER MARSEILLE, MARCH 23 – 27, 2020

SCHEDULE AND ABSTRACTS

Schedule

Monday, 23rd March

08.15 - 08.30 Welcome & Opening 08.30 - 09.10 Beatrice Pelloni: The quantization of solution of linear evolution PDEs 09.15 - 09.40 Lucrezia Cossetti: Absence of eigenvalues of Schrödinger,Dirac and Pauli Hamiltonians via the method of multipliers 09.45 - 10.10 Frantiˇsek Stampach:ˇ On Lieb-Thirring inequlities for non-self-adjoint Jacobi matrices 10.10 - 10.40 Coffee Break 10.40 - 11.20 Filippo Gazzola: Long-time Dynamics of An Extensible Hinged-Free Plate 11.25 - 11.50 Sabine Bögli: Essential numerical ranges for linear operator pencils 11.55 - 12.20 Borbala Gerhat: Schur complement dominant operator matrices 12.30 - 14.30 Lunch 14.30 - 15.10 Ross Pinsky: Markov Search Problems for a Stationary Target 15.15 - 15.40 Uwe Günther: Asymptotic spectral scaling graphs for IR-truncated PT −symmetric model Hamiltonians with V = −(ix)2n+1 potentials and beyond 15.45 - 16.10 Sergiusz Kuzhel: Lax-Phillips Scattering Method in Studies of Non-self-adjoint SchrödingerOperators 16.10 - 16.40 Coffee Break 16.40 - 17:20 Nicolas Raymond: A first formula of pure magnetic tunnel effect 17.25 - 17.50 Cristian Cazacu: Weighted Hardy-Rellich type inequalities 17.55 - 18.20 Robin Lang: On the eigenvalues of the Robin Laplacian with a complex parameter 18.25 - 19.30 Poster Exhibition 19.30 Dinner

Date: March 4, 2020. Tuesday, 24th March

08.30 - 09.10 Alain Joye: Nonlinear Quantum Adiabatic Approximation 09.15 - 09.40 FrédéricHérau: Non-selfadjoint tools in kinetic theory 09.45 - 10.10 Martin Vogel: Almost sure Weyl asymptotics for nonselfadjoint Toeplitz operators 10.10 - 10.40 Coffee Break 10.40 - 11.20 Guy Bouchitté: A repulsive multi-marginal transport model in quantum chemistry 11.25 - 11.50 Iveta Semorádová: Spectral approximation of Schrödingeroperators with complex potentials: diverging eigenvalues 11.55 - 12.20 Juan Manuel PérezPardo: Quantum controllability of infinite dimensional quantum systems based on Quantum Graphs 12.30 - 13.45 Lunch 13.45 - 13.50 COST Action presentation COST Invited Talk 13.55 - 14.35 Amru Hussein: Non-self-adjoint graphs: spectra, similarity, semigroups COST Session I 14.40 - 15.05 MatˇejTuˇsek: Location of hot spots in thin curved strips 15.10 - 15.35 Michal Tichý: Large-time behaviour of the heat equation in sheared strips 15.40 - 16.05 Biagio Cassano: Location of eigenvalues of non-self-adjoint discrete Dirac operators 16.10 - 16:35 Tereza Kurimaiová: When damping cannot be detected from spectral data 16.35 - 17.05 Coffee Break 17.05 - 17:45 Michael Ruzhansky: Nonharmonic analysis of boundary value problems 17.50 - 18.50 Open Problem Session 19.30 Dinner

Wednesday, 25th March

08.30 - 09.10 Tanya Christiansen: Asymptotic location of resonances for Schrödingeroperators on infinite cylinders 09.15 - 09.40 Yaniv Almog: On the stability of laminar flows between plates 09.45 - 10.10 Sergey Tumanov: Completeness theorem for the system of eigenfunctions of the complex 2 2 2/3 Schrödingeroperator Lc = −d /dx + cx 10.10 - 10.40 Coffee Break 10.40 - 11.20 Albrecht Böttcher: Lattices from equiangular tight frames 11.25 - 11.50 Frank Rösler: On The Solvability Complexity Index for Unbounded Selfadjoint Operators and SchrödingerOperators 11.55 - 12.20 Duc Tho Nguyen Classical and semi-classical analysis of magnetic fields in two dimensions 12.30 - 14.30 Lunch 14.30 - 19:00 Free Afternoon 19.30 Dinner Thursday, 26th March

08.30 - 09.10 Ari Laptev: Symmetry Results in Two-Dimensional Inequalitiesfor AharonovBohm Mag- netic Fields 09.15 - 09.40 Yehuda Pinchover: How large can Hardy-weight be? 09.45 - 10.10 Idan Versano: On families of optimal Hardy-weights for linear second order elliptic operators 10.10 - 10.40 Coffee Break 10.40 - 11.20 Mariana Haragus: Linear stability of spectrally stable Lugiato-Lefever periodic waves 11.25 - 11.50 Catherine Drysdale: Complications regarding Amplitude Equations for Non-self-adjoint Operators 11.55 - 12.20 Tobias Bolle: Selectively non-self-adjoint linear fluid dynamics 12.30 - 14.15 Lunch COST Session II 14.15 - 14.40 Marjeta Kramar Fijavˇz: Linear Hyperbolic Systems on Networks 14.45 - 15.10 Luka Grubiˇsić: Contour integration methods fro sectorial operators 15.15 - 15.40 Ivica Nakić: Uncertainty relations for second order elliptic operators 15.45 - 16.10 Jean-Bernard Bru: LiebRobinson Bounds for MultiCommutators 16.10 - 16.40 Coffee Break 16.40 - 17:20 Denis Grebenkov: The Bloch-Torrey operator: eigenmodes localization, branching spectrum, asymptotic analysis, and applications 17.25 - 18.25 Open Problem Session 19.30 Conference Dinner

Friday, 27th March

08.30 - 09.10 Charles Batty: Rates of decay of energy via operator semigroups and tauberian theorems 09.15 - 09.40 Tom Ter Elst: The Dirichlet problem without the maximum principle 09.45 - 10.10 Valentin Zagrebnov Trotter product formula for non-self-adjoint Gibbs semigroups 10.10 - 10.40 Coffee Break 10.40 - 11.20 Sonia Fliss: Scattering problem in general waveguides 11.25 - 11.50 Kirankumar Hiremath: Non-self-adjoint eigenvalue problem of optical bent waveguides 11.55 - 12.20 Archit Kulkarni: Gaussian Regularization of the Pseudospectrum and Davies’ Conjecture 12.30 - 14.30 Lunch 14.30 - 15.10 Nicolas Burq: Observability for SchrödingerGrushin operators 15.15 - 15.40 Irfan Glogić: Spectral properties of the wave operator in self-similar coordinates 15.45 - 16.10 Vladimir Lotoreichik: Optimization of lowest Robin eigenvalues on 2-manifolds and unbounded cones 16.10 - 16.40 Coffee Break 19.30 Dinner 1. Invited talks 1. Charles Batty Title: Rates of decay of energy via operator semigroups and tauberian theorems Abstract: An approach to studying decay of energy of damped waves and similar models involves considering a non-self-adjoint operator which generates an operator semigroup, and finding or estimating the norm of its resolvent on the imaginary axis. I will describe how the rate of decay can then be determined by applying recent results of tauberian type about operator semigroups.

2. Albrecht Böttcher Title: Lattices from equiangular tight frames Abstract: I take the liberty to leave the field of genuine operator theory and to move into linear algebra and (non-self-adjoint) matrix theory. The talk is about the question when certain matrices do generate a lattice, that is, a discrete subgroup of some finite-dimensional Euclidean space, and if this happens, which good properties these lattice have. The matrices considered come from equiangular tight frames. I promise a nice tour through some basics of equiangular lines, tight frames, and lattice theory. We will encounter lots of interesting vectors and matrices and enjoy some true treats in the intersection of discrete mathematics and finite-dimensional operator theory.

3. Guy Bouchitté Title: A repulsive multi-marginal transport model in quantum chemistry Abstract: An interesting issue in Density Functional Theory (DFT), an important branch of Quantum Chemistry, is to understand the asymptotic behavior as ε → 0 of the infimum problem d min εT (ρ) + C(ρ) − U(ρ): ρ ∈ PR where ε is a small parameter which depends on the Planck constant and - T (ρ) is the kinetic energy Z √ T (ρ) = |∇ ρ|2 dx; d R - C(ρ) describes the electron-electron interaction; - U(ρ) is the potential term Z U(ρ) = V (x)ρ dx; d R - P is the class of all probabilities over Rd. In this talk we propose a duality approach for the N-marginal repulsive cost C(ρ) which involves continuous functions vanishing at infinity and allows to consider minimizers ρ which are sub-probabilities (ionization). Then we give primal-dual necessary and sufficient optimality conditions for an optimal pair (ρ, ϕ) with • ϕ an Kantorovich potential for ρ (Optimal transport) √ • u = ρ a ground state for −ε∆ + (ϕ − V ). In a last part of the talk, we apply our results to the asymptotic as ε → 0 and, in some cases, we give evidence of a mass quantization effect for optimal sub-probabilities. This is a joined work with G. Buttazzo (Pisa), T.Champion (Toulon), L. De Pascale (Firenze).

4. Nicolas Burq Title: Observability for SchrödingerGrushin operators Abstract: This talk I will present some recent results on the two dimensional Grushin Schrödinger equation posed on a finite cylinder Ω = (−1, 1)x × Ty with Dirichlet boundary condition. We obtain the sharp observability by any horizontal strip, with the optimal time T∗ > 0 depending on the size of the strip. Consequently, we prove the exact controllability for Grushin Schrödinger equation. By exploiting the concentration of eigenfunctions of harmonic oscillator at x = 0, we also show that the observability fails for any T ≤ T∗. This is a joint work with Chenmin Sun (Cergy). 5. Tanya Christiansen Title: Asymptotic location of resonances for Schrödingeroperators on infinite cylinders Abstract: We consider Schrödingeroperators on the particular infinite cylinder X = R × S1, with bounded, compactly supported potential V . We show that in the high energy limit, resonances which lie “near” the physical sheet are close to points determined by the resonances of a Schrödingeroperator on the line, and discuss some refinements and applications.

6. Sonia Fliss Title: Scattering problem in general waveguides Abstract: We consider in this talk various time harmonic wave problems in infinite waveguides. The study of such problem present theoretical and numerical difficulties. From a theoretical point of view, one has to determine a radiation condition which characterizes the behaviour of the scattered waves at infinity, and which is essential to obtain well-posedness of the scattering problem. From a numerical point of view, in order to compute the solution, one has to construct transparent boundary conditions in order to restrict the computation in a bounded region. For the scalar problem in homogeneous waveguide involving the laplacian operator, the answer to these difficulties is well-known for a long time and it uses the self-adjointness of the transverse laplacian which leads to a modal decomposition of the solution. Actually, this case is extremely particular. Indeed, for general wave problems corresponding to elasticity, electromagnetism, plate models, in homogeneous or stratified waveguides, the same approach does not extend. Indeed, even if the underlying operator can be self-adjoint, the corresponding transverse one is not in general. As a consequence, the completeness of transverse modes is in general an open question and justifying a modal decomposition of the solution is not straightforward. A way to get it consists in using a more general approach, due to Kondratiev but also Nazarov and Plamenevskii, based on a Fourier transformation along the unbounded direction. The modal representation of the solution is then obtained by combining arguments of complex analysis with Fredholm theory in weighted Sobolev spaces. Finally, the previous theoretical approach can be used for numerical purposes to construct a new kind of transparent boundary conditions. This talk is based on joint work with Anne-Sophie Bonnet-Ben Dhia (POEMS, CNRS), Lau- rent Bourgeois (POEMS, Ensta) and Lucas Chesnel (Inria, CMAP).

7. Filippo Gazzola Title: Long-time Dynamics of An Extensible Hinged-Free Plate Abstract: A partially hinged/partially free rectangular plate is considered, with an aim to ad- dress the stability of a suspension bridge immersed in a fluid flow. This leads to a plate evolution equation with an extensible nonlinearity, active in the span-wise coordinate, which is both semilinear and nonlocal. The flow (taken in the chord-wise direction) is represented through a piston-theoretic approximation, which provides both weak (frictional) dissipation, as well as non- conservative lower order terms. These terms yield a non self-adjoint operator whose ”spectral properties” allow to find physically relevant dynamics of the plate. The overall long-time behavior of the solutions is analyzed in some detail. This is joint work with D. Bonheure (Brussels), I. Lasiecka (Memphis), J. Webster (Baltimore).

8. Denis Grebenkov Title: The Bloch-Torrey operator: eigenmodes localization, branching spectrum, asymptotic analysis, and applications Abstract: In this talk, we present recent advances on the non-self-adjoint Bloch-Torrey operator Aq = −∆ + iqx, where ∆ is the Laplace operator in an Euclidean domain Ω with common boundary conditions, q is a real parameter, and x is any space coordinate. In 1D, it reduces to the complex Airy operator −d2/dx2 + iqx. We discuss the conjecture that the spectrum of the Bloch-Torrey operator for q 6= 0 is discrete in any domain, in particular unbounded domains, except for trivial cases (such as free space with no boundary). This conjecture was proven in 1D and, under certain constraints, in 2D. Additionally, an approximate eigenmode construction performed at high q indicates that the eigenmodes are localized near the boundary of the domain. The transition from delocalized to localized eigenmodes is associated with branching points in the spectrum, at which two real eigenvalues coalesce and become a conjugate pair. We also discuss the asymptotic behavior of eigenvalues in the high-q limit, as well as applications in magnetic resonance imaging. Collaboration with N. Moutal and B. Helﬀer.

References [1] S. D. Stoller, W. Happer, and F. J. Dyson, Phys. Rev. A, 44, 74597477 (1991). [2] N. Moutal, D. S. Grebenkov, K. Demberg, T. A. Kuder, J. Magn. Res., 305, 162174 (2019). [3] D. S. Grebenkov, B. Helffer, and R. Henry, SIAM J. Math. Anal., 49, 18441894 (2017). [4] Y. Almog, D. S. Grebenkov and B. Helffer, J. Math. Phys., 59, 041501 (2018). [5] D. S. Grebenkov and B. Helffer, SIAM J. Math. Anal., 50, 622676 (2018).

9. Mariana Haragus Title: Linear stability of spectrally stable Lugiato-Lefever periodic waves Abstract: The Lugiato-Lefever equation is a nonlinear Schr¨odinger-type equation with damping, detuning and driving, derived in nonlinear optics by Lugiato and Lefever (1987). While exten- sively studied in the physics literature, there are relatively few rigorous mathematical studies of this equation. In this talk I’ll discuss the linear asymptotic stability of spectrally stable periodic waves to perturbations which are localized, i.e., integrable on the real line. The analysis relies upon a Floquet/Bloch decomposition of the linear solution operator and careful estimates of the resulting semi-groups.

10. Amru Hussein (COST) Title: Non-self-adjoint graphs: spectra, similarity, semigroups Abstract: On ﬁnite metric graphs Laplace operators subject to general non-self-adjoint matching conditions imposed at graph vertices are considered. A regularity criterion is proposed and spectral properties of such regular operators are investigated, in particular similarity transforms to self-adjoint operators and generation of C0-semigroups. Concrete examples are discussed exhibiting that non-self-adjoint boundary conditions can yield to unexpected spectral features. The talk is based on joint works with David Krejˇciˇr´ık (Czech Technical University in Prague), Petr Siegl (Queen’s University Belfast) and Delio Mugnolo (FernUniversit¨atHagen).

11. Alain Joye Title: Nonlinear Quantum Adiabatic Approximation Abstract: We consider the adiabatic limit of a nonlinear Schr¨odinger equation in which the Hamiltonian depends on time and on a ﬁnite number of components of the wave function. We show the existence of instantaneous nonlinear eigenvectors and of solutions which remain close to those, up to a rapidly oscillating phase in the adiabatic regime. This involves revisiting the linear adiabatic theorem for non-self-adjoint generators related to the linearisation of the problem. Joint work with Clotilde Fermanian-Kammerer.

12. Ari Laptev Title: Symmetry Results in Two-Dimensional Inequalitiesfor AharonovBohm Magnetic Fields Abstract: This talk is devoted to the symmetry and symmetry breaking properties of a two- dimensional magnetic Schrödingeroperator involving an Aharonov-Bohm magnetic vector potential. We investigate the symmetry properties of the optimal potential for the corresponding magnetic Keller-Lieb-Thirring inequality and show that this potential is radially symmetric if the intensity of the magnetic field is below an explicit threshold, while symmetry is broken above a second threshold corresponding to a higher magnetic field.

13. Ross Pinsky Title: Markov Search Problems for a Stationary Target Abstract: Let a ∈ R denote an unknown stationary target with a known distribution µ ∈ P(R), the space of probability measures on R. We consider two different Markovian models in which a diffusive searcher with constant diffusion coefficient D, sets out from the origin to locate the target. Let Ta denote the random time at which the searcher finds the target. In one of the models, the searcher can choose a fixed drift b to use in the search; thus, in this case the search D d2 d is performed by a diffusion process generated by 2 dx2 + b dx . In the other model, the searcher has no drift, so it simply performs Brownian motion with diffusion constant D. However, it is equipped with an exponential clock with spatially dependent rate r(·), so that if it has failed to locate the target by the time the clock rings, then its position is reset to the origin and it (b) continues its search anew from there with the same rules. In the two models, we let E0 and (r) E0 respectively denote the expectation for the corresponding process. The expected search R (b) R (r) times for the two models are respectively E Ta µ(da) and E Ta µ(da). We would like R 0 R 0 to find the minimum of each of these respectively over the class of all drifts b and over the class of all clock rates r. The second model is non-self adjoint. In the case of constant rate r, it has been investigated in the mathematical physics literature.

14. Beatrice Pelloni Title: The quantization of solution of linear evolution PDEs Abstract: It has been known since the 1840s that the solution of periodic problems for the Schrdinger equation exhibits a peculiar phenomenon (called revival, or quantisation), at values of the time that are multiples of the period. In this talk, I will discuss and illustrate the extent of this phenomenon for a much larger class of operators and boundary conditions, including non self-adjoint cases.

15. Nicolas Raymond Title: A first formula of pure magnetic tunnel effect Abstract: The semiclassical magnetic Neumann Schrödingeroperator on a generic, smooth, bounded, and simply connected domain of the Euclidean plane is considered. When the domain has a symmetry axis, the semiclassical splitting of the first two eigenvalues is analyzed. The first explicit tunneling formula in a pure magnetic field is established. The analysis is based on a pseudo-differential reduction to the boundary and the proof of the first known optimal purely magnetic Agmon estimates. Joint work with V. Bonnaillie-Noëland FrédéricHérau.

16. Michael Ruzhansky Title: Nonharmonic analysis of boundary value problems Abstract: In this talk we describe the nonharmonic theory of pseudo-diﬀerential operators based on non-self-adjoint operators with discrete spectra.

2. Contributed talks 1. Yaniv Almog Title: On the stability of laminar flows between plates Abstract: We prove that a two-dimensional laminar flow between two plates (x1, x2) ∈ R+ × [−1, 1] given by v = (U, 0) is linearly stable in the large Reynolds number limit, when U ∈ C4[−1, 1] is strictly monotone and either |U 00| |U 00| (nearly Couette flow) or U 00 6= 0 in [−1, 1]. We assume no-slip conditions on the plates and an arbitrarily large (but fixed) period in the x1 direction. Similar results are obtained when the no-slip conditions on the plates are replaced by a fixed traction force condition. This research is a joint work with Bernard Helffer.

2. Sabine B¨ogli Title: Essential numerical ranges for linear operator pencils Abstract: We introduce the notion of essential numerical range for linear operator pencils, that is, for generalised eigenvalue problems Af = zBf with linear operators A and B and eigenvalue z (for B the identity operator we recover the usual eigenvalue equation). The essential numerical range is used to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. We apply the results to various diﬀerential operator pencils. This talk is based on joint work with Marco Marletta.

3. Tobias Bolle Title: Selectively non-self-adjoint linear fluid dynamics Abstract: For most canonic flow configurations, linearisation of the Navier-Stokes equations about some reference state yields an inhomogeneous Cauchy problem, associated with a non- self-adjoint linear operator. Traditionally, linearised dynamics served exclusively to determine asymptotic stability of the reference state by evaluation of the discrete spectrum. However, it turns out that non-self-adjoint operators enable much richer dynamics which differ greatly from those inferred from consideration of the discrete spectrum alone. Today analysis of linear fluid dynamics essentially relies on canonical decomposition of either the induced semi-group or the resolvent giving rise to transient energy amplification despite asymptotic decay of all eigenstates and pseudo-resonance outside the neighbourhood of the discrete spectrum, respectively. Due to this importance many problems in fluid dynamics are blanketly classified being strongly non- self-adjoint. On the example of canonical vortex flow, we show that the situation might be more subtle. We discuss global and local measures of non-self-adjointness to show that the linear operator is non-self-adjoint only in the neighbourhood of small parts of its spectrum while it is effectively self-adjoint for the rest. This finding enables us to restrict the range of pertinent dynamics considerably. Moreover, we show how local measures of non-self-adjointness and spectral projections can be related to the leading rank-1 operators in a canonical decomposition of the resolvent. Similar links can be established between low-rank representations of the resolvent and the semi-group. Despite being frequently observed for various flow configurations, our results seem to be original, at least with applications to fluid dynamics.

4. Jean-Bernard Bru (COST) Title: LiebRobinson Bounds for MultiCommutators Abstract: I will present a generalization the usual Lieb-Robinson bounds to multi-commutators. Lieb-Robinson bounds for multi-commutators are eﬀective mathematical tools to handle analytic aspects of the dynamics of quantum particles with interactions which are non-vanishing in the whole space and possibly time-dependent. To illustrate this, I will explain how the bounds for multi-commutators of order three yield existence of fundamental solutions for the corresponding non-autonomous initial value problems for observables of interacting fermions on lattices. They can also be used to study linear and non-linear responses of interacting fermions to external perturbations. All results also apply to quantum spin systems, with obvious modiﬁcations.

5. Biagio Cassano (COST) Title: Location of eigenvalues of non-self-adjoint discrete Dirac operators Abstract: We provide quantitative estimates on the location of eigenvalues of one-dimensional discrete Dirac operators with non-hermitian `1-potentials, and, as a corollary, we determine subsets of the essential spectrum that are free of embedded eigenvalues for small potentials. Moreover, we show that our results are sharp for eigenvalues in the complex plane outside a region. Further results and sharpness of the obtained spectral bounds are also discussed for `p-potentials, with 1 < p ≤ ∞. This is a joint work with O.O. Ibrogimov, D. Krejˇciˇr´ıkand F. Stampach.ˇ

6. Cristian Cazacu Title: Weighted Hardy-Rellich type inequalities Abstract: In this talk we discuss both Hardy and Hardy-Rellich inequalities which establish useful properties for diﬀerential operators with singular potentials, for instance for non-symmetric x·∇ β expressions of the form −∆ + λ |x|2 + |x|2 , λ, β ∈ R. In particular, we discuss some recent results by Gesztesy and Littlejohn (2018) and present some new extensions to weighted Hardy-Rellich type inequalities. We focus on the best constants and the existence/nonexistence of minimizers in the energy space. This talk is partially supported by a grant of Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P1-1.1-TE-2016-2233, within PNCDI III.

7. Lucrezia Cossetti Title: Absence of eigenvalues of Schrödinger,Dirac and Pauli Hamiltonians via the method of multipliers Abstract: Originally arisen to understand characterizing properties connected with dispersive phenomena, in the last decades the method of multipliers has been recognized as a useful tool in Spectral Theory, in particular in connection with proof of absence of point spectrum for both self- adjoint and non self-adjoint operators. In this seminar we will see the developments of the method reviewing some recent results concerning self-adjoint and non self-adjoint Schrödingeroperators in different settings, specifically both when the configuration space is the whole Euclidean space Rd and when we restrict to domains with boundaries. We will show how this technique allows to detect physically natural repulsive and smallness conditions on the potentials which guarantee total absence of eigenvalues. Some very recent results concerning Pauli and Dirac operators will be also presented. The talk is based on joint works with L. Fanelli and D. Krejcirik. 8. Catherine Drysdale Title: Complications regarding Amplitude Equations for Non-self-adjoint Operators Abstract: Owing to the interaction between modes, difficulties arise in creating amplitude equations where non-normality and nonlinearity is present in the original system. For example, if amplitude equations are made via weakly nonlinear analysis, then approximating via the criti- cal mode only (least stable eigenvalue) does not work at higher orders where the mixing of the modes needs to be taken into consideration. However, using a different homogenisation technique, namely stochastic singular perturbation theory of authors like Papanicalaou [1], Blmker et al [2], where noise is applied to the stable modes only, then the linear operator in question is no longer non-self-adjoint. Although, the difficulty of the problem shifts to showing that we can use a Rigged Hilbert Space construction. If the original problem in a Hilbert space H, we force the main operator of our problem to be Hilbert-Schmidt by choosing our noise in a dense subspace S of H. We demonstrate this on the Complex-Ginsburg-Landau equation with cubic nonlinearity.

References [1] G.C. Papanicolaou, Some Probabilistic Problems and Methods in Singular Perturbations. Rocky Mountain J. Math 6 (1976), 653-674. [2] D. Blomker¨ and W. Mohammed, Amplitude Equations for SPDEs with Cubic Nonlinearities. Stochastics, An International Journal of Probability and Stochastic Processes, 2011.

9. Tom ter Elst Title: The Dirichlet problem without the maximum principle Abstract: The maximum principle plays an important role for the solution of the Dirichlet problem. Now consider the Dirichlet problem with respect to an elliptic operator d d d X X X A = − ∂k akl ∂l − ∂k bk + ck ∂k + c0 k,l=1 k=1 k=1 d on a suﬃciently regular open set Ω ⊂ R , where akl, ck ∈ L∞(Ω, R) and bk, c0 ∈ L∞(Ω, C). Suppose that the associated operator on L2(Ω) with Dirichlet boundary conditions is invertible. Note that in general this operator does not satisfy the maximum principle. Nevertheless, we 1 show that for all ϕ ∈ C(∂Ω) there exists a unique u ∈ C(Ω) ∩ Hloc(Ω) such that u|∂Ω = ϕ and Au = 0. In the case when Ω has a Lipschitz boundary and ϕ ∈ C(Ω) ∩ H1/2(Ω), then we show that u coincides with the variational solution in H1(Ω). This is joint work with Wolfgang Arendt [1].

References [1] Arendt, W. and Elst, A. F. M. ter, The Dirichlet problem without the maximum principle. Annales de l’Institut Fourier 69 (2019), 763–782.

10. Borbala Gerhat Title: Schur complement dominant operator matrices Abstract: In mathematical physics, matrix (differential) operators arise naturally in applications as coupled systems of partial differential equations. Up to now, the spectral analysis of such problems has typically been tackled by means of perturbation theory. We propose to view operator matrices in a more general setting, which allows our results to fully abstain from any perturbative argument. Rather than requiring the matrix to act in a Hilbert space H, we extend its action to a suitable distributional triple D ⊂ H ⊂ D0 and restrict it to its maximal domain in H. The crucial point in our approach is the choice of the spaces D and D0 which are essentially determined by the Schur complement of the matrix. We show spectral equivalence between the resulting operator matrix in H and its Schur complement, eventually implying closedness and non-empty resolvent set of the matrix. Finally, we apply our abstract results to the damped wave equation with possibly unbounded and singular damping, as well as to more general second order matrix differential operators with singular coefficients. By means of our methods, the previously used regularity assumptions can be lowered substantially in both cases. 11. Irfan Glogić Title: Spectral properties of the wave operator in self-similar coordinates Abstract: Numerical simulations indicate that generic blowup in supercritical wave equations is self-similar. For stability analysis, it is customary to pass to coordinates which are adapted to the self-similar nature of the underlying solution. This in turn leads to difficult non-selfadjoint spectral problems. We illustrate this procedure on supercritical wave maps and we discuss both numerical and analytical approaches to the resolution of the underlying spectral problem.

12. Luka Grubiˇsi´c(COST) Title: Contour integration methods fro sectorial operators Abstract: In this talk we present an algorithm for approximating a separated cluster of semisimple eigenvalues of a sectorial operator. The technique is based on functional calculus based on numerical contour integration and resolvent estimates. We also discuss issues related to spectral pollution. We will present numerical examples where both ﬁnite element as well as spectral approximations of the resolvent are utilized. This is a joint work with J. Gopalakrishnan, J. Ovall and B. Parker.

13. Uwe Günther Title: Asymptotic spectral scaling graphs for IR-truncated PT −symmetric model Hamiltonians with V = −(ix)2n+1 potentials and beyond Abstract: The PT −symmetric V = ix3 model over the real line, x ∈ R, is IR truncated and considered as Sturm-Liouville problem over a finite interval x ∈ [−L, L] ⊂ R. Structures hid- den in the Airy function setup of the V = −ix model are combined with WKB techniques developed by Bender and Jones in 2012 for the derivation of the real part of the spectrum of the (ix3, x ∈ [−1, 1]) model. Via WKB and Stokes graph analysis, the location of the complex spectral branches of the ix3 model as well as those of more general V = −(ix)2n+1 models over x ∈ [−L, L] ⊂ R are obtained. Splitting the related action functions into purely real scale fac- tors and scale invariant integrals allows to extract underlying asymptotic spectral scaling graphs R ⊂ C. These (structurally very simple) scaling graphs remain invariant under rescalings and cutoff-independent so that the IR limit L → ∞ can be formally taken. Moreover an increasing L can be associated with an R−constrained spectral UV→IR renormalization group flow on R. It is shown that the eigenvalues of the IR-complete (V = ix3, x ∈ R) model can be bijectively mapped onto a finite segment of R asymptotically approaching a (scale invariant) PT phase transition region. In this way, a simple heuristic picture and complementary explanation for the unboundedness of projector norms and C−operator for the ix3 model are provided and the lack of quasi-Hermiticity of the ix3 Hamiltonian over R appears physically plausible. Possible directions of further research are briefly sketched. The talk is partially based on arXiv:1901.08526 (a joint work with F. Stefani).

14. Kiran Hiremath Title: Non-self-adjoint eigenvalue problem of optical bent waveguides Abstract: Nowadays, Optics-Photonics based technology is vigorously explored as the next step to microelectronics based technology. Many research groups around the world are working on various aspects of all-optical signal processing and all-optical circuits. Optical waveguides are one of the basic circuit components of such optical circuits. In the case of waveguides, one needs to know so-called guided modes and their propagation constants. Various form of straight waveguides and optical ﬁbres are studied quite well in term of their mathematical models, numerical simulation schemes and actual fabrications. Mathematically, optical straight waveguides are modelled as self-adjoint eigenvalue problem on the unbounded domain, e.g. for one dimensional waveguides, it is studied on the interval (−∞, ∞). Here the guided modes appear as eigenfunctions corresponding to real eigenvalues. Another essential form of optical waveguides is bent waveguides. These bent waveguides are mathematically modelled as eigenvalue problem with complex-valued eigenvalues. In our earlier semi-analytic studies of optical bent waveguides, we have shown that these complex-valued eigenvalues are dependent on bending radius of the bent waveguide. Moreover, as the bending radius tends to 1, these complex-valued eigenvalues tend to real-valued eigenvalues, and the bent waveguide behaves like the corresponding straight waveguide. In this work, we mathematically show that the optical bent waveguide eigenvalue problem is a non-self adjoint eigenvalue problem on the unbounded domain with complex eigenvalues. We show that the imaginary part of these eigenvalues is negative, conﬁrming its manifestation in terms of lossy/leaky nature of the bent waveguide modes. We also show that as the bending radius tends to1, these eigenvalues become real and the corresponding non-self adjoint eigenvalue problem of bent waveguides becomes the self-adjoint eigenvalue problem of straight waveguides. By posing the eigenvalue problem in the function theoretic setting, we will also discuss mathematical properties of the bent waveguide eigenvalue problem.

15. FrédéricHérau Title: Non-selfadjoint tools in kinetic theory Abstract: Non-selfadjoint tools are now currently used in the study of kinetic equations, in order to get existence, regularization as well as time behavior results for various models. In this talk we will try to give an overview of some recent results and particular points in this direction.

16. Marjeta Kramar Fijavˇz(COST) Title: Linear Hyperbolic Systems on Networks Abstract: We consider hyperbolic systems of linear partial diﬀerential equations on 1-dimensional structures that we interpret as networks. On each edge of the network we take an evolution equation of the form ∂u ∂u e (t, x) = M (x) e (t, x) + N (x)u (t, x) (1) ∂t e ∂x e e

where ue is a vector-valued function of size ke ∈ {1, 2, 3,... }, Me and Ne are matrix-valued functions of size ke × ke and le is the length of the edge e. We couple equations (1) for diﬀerent edges e via boundary conditions. Our goal is to characterise boundary conditions that yield a well-posed problem on the network. We also study some qualitative properties of the solutions. Our framework covers a large variety of models, like 1D-Maxwell equations in cable networks or linearised Saint-Venant model for water dynamics in sediment-ﬁlled canals, for example. This is a joint work with Delio Mugnolo (Hagen) and Serge Nicaise (Valenciennes).

17. Archit Kulkarni Title: Gaussian Regularization of the Pseudospectrum and Davies’ Conjecture Abstract: A diagonalizable matrix has linearly independent eigenvectors. Since the set of non- diagonalizable matrices has measure zero, every matrix is a limit of diagonalizable matrices. We prove a quantitative version of this fact: every n x n complex matrix is within distance delta in the operator norm of a matrix whose eigenvectors have condition number poly(n)/delta, con- ﬁrming a conjecture of E. B. Davies. The proof is based on regularizing the pseudospectrum of an arbitrary matrix with a complex Gaussian perturbation. Joint work with J. Banks, S. Mukherjee, N. Srivastava.

18. Tereza Kurimaiov´a(COST) Title: When damping cannot be detected from spectral data Abstract: Based on an abstract result by Tosio Kato from 1966, we derive new results about the similarity of the operator associated with the damped wave equation to the Laplace operator in some sense.

19. Sergiusz Kuzhel Title: Lax-Phillips Scattering Method in Studies of Non-self-adjoint SchrödingerOperators Abstract: Problems of spectral theory of non-self-adjoint operators attract steady interests of experts in different field of mathematics and physics. In the past twenty years, this interest grew considerably due to the recent progress in theoretical physics of PT-symmetric (pseudo- Hermitian) Hamiltonians [1]. The present talk contains a contribution in this field inspired by the Lax-Phillips scattering theory [2]. We show how to investigate the spectral properties of some non-self-adjoint Schrödingeroperators with the use of an operator-theoretical interpretation of the Lax-Phillips scattering theory developed in [3]. Examples of application of this method can be found in [4,5,6]. References [1] C. M. Bender et al, PT-symmetry in Quantum and Classical Physics, World Scientific, 2019. [2] P. Lax and R. Phillips, Scattering Theory, Second edition, Academic Press, New York, 1989. [3] S. Kuzhel, On the inverse problem in the Lax-Phillips scattering theory method for a class of operator-differential equations, St. Petersburg Math. J, 13 (2002), 41-56. [4] S. Albeverio and S. Kuzhel, On elements of the Lax-Phillips scattering scheme for PT- symmetric operators, J. Phys. A 45 (2012) [5] P. A. Cojuhari and S. Kuzhel, Lax-Phillips scattering theory for PT-symmetric p-perturbed operators J. Math. Phys. 53 (2012) 073514. [6] P. A. Cojuhari, A. Grod and S. Kuzhel, On the S-matrix of Schrödingeroperators with non-symmetric zero-range potentials, J. Phys. A 47 (2014) 315201.

20. Robin Lang Title: On the eigenvalues of the Robin Laplacian with a complex parameter α Abstract: We study the spectrum of the Robin Laplacian −∆Ω with a complex parameter α on a bounded Lipschitz domain Ω ⊂ Rd. We establish a number of properties, in particular regarding the analytic dependence of eigenvalues and eigenspaces on α ∈ C as well as basis properties of the eigenfunctions. Using estimates on the numerical range of the associated operator we give bounds and asymptotics for the eigenvalues as functions of α, which lead to new eigenvalue bounds even in the self-adjoint case α ∈ R. For the asymptotics of the eigenvalues as α → ∞, in place of the variational min-max characterisation of the eigenvalues and Dirichlet-Neumann bracketing techniques commonly used in the self-adjoint case, we exploit the duality between the eigenvalues of the Robin Laplacian and the eigenvalues of the Dirichlet-to-Neumann map to show that every Robin eigenvalue either diverges to ∞ in C or converges to a point in the Dirichlet spectrum.

21. Vladimir Lotoreichik Title: Optimization of lowest Robin eigenvalues on 2-manifolds and unbounded cones Abstract: In this talk, we will focus on optimization for the lowest eigenvalue of the Robin Laplacian with a negative boundary parameter on a compact, smooth, simply-connected, two- dimensional manifold with C2-boundary of a fixed length. The main novelty compared to the better-understood Euclidean case is that the eigenvalue is optimized in the sub-class of manifolds, for which the Gauss curvature satisfies the pointwise inequality K ≤ K◦ for a fixed constant K◦ ≥ 0. This constraint on the curvature naturally enters into the problem. Our main result can be concisely formulated as follows: the geodesic disk on the manifold of the constant Gauss curvature K◦ is a maximizer. Moreover, we will discuss a result on the optimization of the lowest Robin eigenvalue on an unbounded three-dimensional Euclidean cone Λ with a C2-smooth, simply-connected cross- section Λ ∩ S2 of a fixed perimeter. We prove that the cone with a circular cross-section is a maximizer. This talk is based on a joint work with Magda Khalile.

22. Ivica Naki´c(COST) Title: Uncertainty relations for second order elliptic operators Abstract: In the talk we consider second order elliptic operators on Rd. We derive uncertainty relations (aka spectral inequalities) for the spectral subspaces obtained by Riesz projections on compact, separated parts of the spectrum, when the sampling set is equidistributed. The obtained uncertainty relations are very precise with respect to the coarseness scale and the radius deﬁning the equidistributed set. We indicate the applicability of these relations in various areas of analysis of PDE. This is a joint work with Martin Tautenhahn.

23. Duc Tho Nguyen Title: Classical and semi-classical analysis of magnetic fields in two dimensions Abstract: The talk is combined by two parts: Classical mechanics and semi-classical analysis, especially in the presence of magnetic fields. In classical mechanics, we use Hamiltonian dynamics to describe the movement of a charged particle in a domain affected by the magnetic fields. We focus on two classical physical problems: the confinement and the scattering problem. In semi- classical analysis, we study the spectral problem of the magnetic Laplacian at the semi-classical level in two-dimensional domains: on a compact Riemannian manifold with boundary and on R2. Under the assumption that the magnetic field has a unique positive and non-degenerate minimum, we can describe the eigenfunctions by WKB method. The exponential decay of the eigenfunctions is also mentioned in the presentation. Furthermore, the eigenfunctions are very well approximated in an exponentially weighted space.

24. Juan Manuel PérezPardo Title: Quantum controllability of infinite dimensional quantum systems based on Quantum Graphs Abstract: The development of quantum information processing and quantum computation goes hand in hand with the ability of addressing and manipulating quantum systems. Quantum Control Theory has provided a successful framework, both theoretical and experimental, to design and develop the control of such systems. In particular, for finite dimensional quantum systems or finite dimensional approximations to them. The theory for infinite dimensional systems is much less developed. In this talk I propose a scheme of infinite dimensional quantum control on quantum graphs based on interacting with the system by changing the self-adjoint boundary conditions. I will show the existence of solutions of the time-dependent Schrödingerequation, the stability of the solutions and the (approximate) controllability of the state of a quantum system by modifying the boundary conditions on generic quantum graphs.

25. Yehuda Pinchover Title: How large can Hardy-weight be? Abstract: In the ﬁrst part of the talk we will discuss the existence of optimal Hardy-type inequalities with ’as large as possible’ Hardy-weight for a general second-order elliptic operator deﬁned on a noncompact Riemannian manifold, while the second part of the talk will be devoted to a sharp answer to the question: “How large can Hardy-weight be?”

26. Frank Rösler Title: On The Solvability Complexity Index for Unbounded Selfadjoint Operators and Schrödinger Operators Abstract: We study the solvability complexity index (SCI) for unbounded selfadjoint operators on separable Hilbert spaces and perturbations thereof. In particular, we show that if the extended essential spectrum of a selfadjoint operator is convex, then the SCI for computing its spectrum is equal to 1. This result is then extended to relatively compact perturbations of such operators and applied to Schrödingeroperators with compactly supported (complex valued) potentials to obtain SCI=1 in this case, as well.

27. Iveta Semorádová Title: Spectral approximation of Schrödingeroperators with complex potentials: diverging eigenvalues Abstract: Domain truncations for Schrödingeroperators with complex potentials are known to be spectrally exact. However, several examples suggest that additional eigenvalues escaping to infinity seem to be a generic feature. We explain the latter by a deeper analysis of truncated operators and proving a generalized norm resolvent convergence of transformed truncated operators.

28. Frantiˇsek Stampachˇ Title: On Lieb-Thirring inequlities for non-self-adjoint Jacobi matrices Abstract: The Lieb-Thirring inequlities for self-adjoint Jacobi matrices were proved by Hundert- mark and Simon in 2002. A near generalization of these inequalities applicable to non-self-adjoint Jacobi matrices was discovered by Hansmann and Katriel in 2011. In the self-adjoint case, the generalized inequality implies a slightly weaker result then the classical one of Hundertmark and Simon, however. This circumstance led Hansmann and Katriel to formulate a conjecture and another open problem concerning validity boundaries of the Lieb-Thirring inequlities when extended to non-self-adjoint Jacobi matrices. The aim of this talk is to give answers to these open problems. The talk is based on a joint work with Sabine Bögli. 29. Michal Tichý(COST) Title: Large-time behaviour of the heat equation in sheared strips Abstract: We consider the Dirichlet Laplacian in an unbounded planar domain built by trans- lating a segment oriented in a constant direction along an infinite curve. We show that the associated heat semigroup admits an extra polynomial decay rate with respect to the straight strip. We use the method of self-similar variables and weighted Sobolev spaces.

30. Sergey Tumanov Title: Completeness theorem for the system of eigenfunctions of the complex Schrödingeroperator 2 2 2/3 Lc = −d /dx + cx Abstract: We consider the operator 2 d α Lc,α = − + cx dx2 on the semi-axis in L2(R+), with Dirichlet boundary conditions and a constant c ∈ C. ∗ −1 It is well known ([1], Lemma V.6.1) that the resolvent R(λ) = (Lc,α − λ) decays outside ∗ the closed sector Λ = {arg λ ∈ [0, − arg c]}—the numerical range of Lc,α. It is also known ∗ [2] that the operator Lc,α is of the order of 2α/(2 + α). If f ∈ L2(R+) is orthogonal to all ∗ −1 eigenfunctions of Lc,α, then the vector function R(λ)f = (Lc,α − λ) f, with values in L2(R+), is an entire function with the order of growth 2α/(2 + α) (see §4 [3]). It follows from the Phragmen–Lindelöfprinciple that if the central angle of the sector Λ is less than 2πα/(2 + α) (i.e., | arg c| < 2πα/(2 + α)), then R(λ)f ≡ 0, and thus f ≡ 0. This proves the completeness of the system of eigenfunctions of Lc,α. These ideas originate from the work of Keldysh [4]. Of course, this approach does not provide information about the behavior of R(λ)f inside Λ, where, a priori, R(λ)f increases exponentially. However, if instead of the vector function R(λ)f, we consider the scalar entire function (R(λ)f)(x), fixing an arbitrary point x ≥ 0, then this function may be decaying in a wider sector than C \ Λ. This observation turns out to be decisive for the proof of the completeness theorem under weaker conditions on the argument of c. The proposed approach is likely to make it possible to solve the following problem: prove that, for each α ∈ (0, 2), there exists ε > 0, such that the eigenfunctions of Lc,α form a complete system, for all c: | arg c| < 2απ/(2 + α) + ε. We consider the case of α = 2/3. Thus, our main theorem is dedicated to the operator 2 d 2/3 Lc = − + cx , (2) dx2 and is stated as follows:

Theorem 2.1. There exists θ0 ∈ (π/10, π/9) such that, for all c ∈ C: | arg c| < π/2 + θ0, the system of eigenfunctions of the operator Lc is complete in L2(R+). Our study is motivated by the work of Savchuk and Shkalikov [2], who propose the following hypothesis. For operators on the semi-axis of the form 2 d α Lc,α = − + cx , (3) dx2 2 2 α there exists α0 < 2/3 such that the system of eigenfunctions of Li,α = −d /dx +ix is complete in L2(R+), for all α ∈ (α0, 2/3]. The completeness problem for Li,α has not yet been solved, even for α = 2/3. In 2015, Almog [5] identiﬁed this as an open problem. Of course, our theorem solves the problem for α = 2/3.

References [1] Gohberg I. C., Krein M. G., Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space. (Translations of Mathematical Monographs), AMS, 1969. Nauka, Moscow, 1965, translation from the Russian. [2] Savchuk A. M., Shkalikov A. A., Spectral Properties of the Complex Airy Operator on the Half-Line, Funct. Anal. Appl., 51:1 (2017), 66–79. [3] Shkalikov A. A., Perturbations of self-adjoint and normal operators with discrete spectrum, Russian Math. Surveys, 71:5 (2016), 907–964. [4] Keldysh M. V. On eigenvalues and eigenfunctions of some classes of non-selfadjoint equations, Reports of the Academy of Sciences of the USSR, 77:1 (1951), 11–14. [5] Materials of the workshop ”Mathematical aspects of physics with non-self-adjoint operators”. List of open problems. https://aimath.org/pastworkshops/nonselfadjoint.html

31. MatˇejTuˇsek(COST) Title: Location of hot spots in thin curved strips Abstract: The maxima and minima of Neumann eigenfunctions of thin tubular neighbourhoods of curves on surfaces are located in terms of the maxima and minima of Neumann eigenfunctions of the underlying curves. In particular, the hot spots conjecture for a new large class of domains (possibly non-convex and non-Euclidean) is proved.The talk is based on a joint work with D. Krejˇciˇr´ık.

32. Idan Versano Title: On families of optimal Hardy-weights for linear second order elliptic operators Abstract: We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we ﬁrst characterize all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators , and then apply this result to obtain families of optimal Hardy inequalities for general linear second-order elliptic operators in higher dimensions.

33. Martin Vogel Title: Almost sure Weyl asymptotics for nonselfadjoint Toeplitz operators Abstract: The spectra of nonselfadjoint linear operators can be very unstable and sensitive to small perturbations. This phenomenon is usually referred to as ”pseudospectral eﬀect”. To explore this spectral instability we study the spectra of small random perturbations of nonselfadjoint operators in the case of Toeplitz matrices and in the case of the Toeplitz quantization of complex-valued functions on the torus. We show that almost surely the eigenvalues follow a Weyl law. (Partly based on joint work with J. Sj¨ostrand)

34. Valentin Zagrebnov Title: Trotter product formula for non-self-adjoint Gibbs semigroups Abstract: In this lecture I present an analytic extension method for holomorphic families of the Trotter product formula approximants. It allows to prove the trace-norm convergence of this formula for Gibbs semigroups. Besides that it covers the case when the involved non-self-adjoint generators are not subordinated.