Schedule for: 22w5115 - Mathematical aspects of the Physics with non-self-Adjoint Operators

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

The low energy behavior of the resolvent of a self-adjoint Schrödinger operator is particularly complicated in dimension two. In this talk we look at a new approach to studying this problem which has the benefit of working for a large class of perturbations of $-\Delta$ on $\mathbb{R}^2$, including some non-self-ajoint ones. This talk is based on work in progress with Kiril Datchev.

Originally arisen to understand the dispersive phenomenon, 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 talk we will see the developments of the method reviewing some recent results concerning self-adjoint and non self-adjoint Schrödinger operators in different settings, specifically we will focus on matrix-valued Schrödinger operators, relativistic operators of Pauli and Dirac types and Schrödinger operators on domains with boundary. We will show how this technique allows to detect physically natural repulsive and smallness conditions on the potentials to guarantee total absence of eigenvalues. The talk is based on joint works with L. Fanelli and D. Krejčířik.

Due to the matrix structure, spectral problems arising from systems of coupled linear partial differential equations can be intrinsically challenging. A successful approach motivated already on the level of scalar matrices is to relate the operator matrix (to be implemented in a product Hilbert space) to one of its Schur complements. In the unbounded operator setting, we introduce a robust abstract method to rigorously establish this connection and thereby extend previous approaches which use relative boundedness within the matrix entries. The cornerstones of our method are on one hand a Schur complement which is dominant in a suitable sense with respect to the entries. On the other hand, we employ a distributional approach and implement the matrix operations in larger spaces before restricting to the maximal domain in the underlying product Hilbert space. This allows us to essentially pass from a well-behaved representation of the Schur complement (for instance by its sesquilinear form) to a well-behaved representation of the operator matrix, and to establish an equivalence between the (point and essential) spectra of matrix and Schur complement. We illustrate this abstract framework and present a semigroup generation result for a wave equation with accretive (differential) damping in a weighted space. The dominant Schur complement therein is implemented as the form representation of a Schrödinger-type operator with accretive potential in a weighted space. With respect to previous results, this new method allows for a generalisation to naturally minimal assumptions regarding the employed form techniques.

In this talk, we will present a new, dynamical approach for establishing bounds for $L^p$-norms of low-lying eigenfunctions of non self-adjoint semiclassical pseudodifferential operators with double characteristics. Most notably, our main theorem improves the already known results in the case of pseudodifferential operators with analytic symbols, and it is an open question whether such bounds continue to hold under less restrictive regularity assumptions. The main ingredients in the proof are Fourier-Bros-Iagolnitzer (FBI) transform techniques and complex Hamilton-Jacobi theory.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Athena Swan in the UK and its impact in Mathematical Sciences.

The development of quantum information and computation technologies is deeply related with the ability to control quantum systems. The techniques of finite dimensional control have been successful in this development but have intrinsic limitations. In this talk I will present new perspectives that are enabled by control in infinite dimensional quantum systems. I will introduce results on the existence of solutions of the time-dependent Schrödinger equation and their stability and use them to prove controllability of some quantum systems.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator $(P,B)$ with degenerate mixed boundary conditions. By an optimal Hardy-weight for a subcritical operator we mean a nonzero nonnegative weight function $W$ such that $(P-W,B)$ is critical, and null-critical with respect to $W$. Our results rely on a recently developed criticality theory for positive solutions of the corresponding mixed boundary value problem.

I will review the results for these operators, when posed on bounded domains. The door to these results have been unlocked by the understanding of the behaviour of third-order boundary values problems. These problems have been studied over the last 20 years by means of the Unified Transform approach originally due to Fokas. In some non-self-adjoint cases, this approach yields a spectral diagonalisation of the operator. More generally, I will highlight the dependence of these problems on the specific boundary conditions and how this differs fundamentally from the even-order case. Novel and surprising examples arise for "Dirichlet-type" boundary conditions, as well as for quasi-periodic and time-periodic ones.

Properties of the coefficients of an (elliptic) differential operator, together with domain geometry and boundary conditions, can cause some eigenvectors of the operator to be strongly spatially localized in relatively small regions of the domain. A better understanding of where such eigenvectors are likely to localize, and for which eigenvalues this localization occurs, is of practical interest in the design of certain structures having desired acoustic or electromagnetic properties. Over the past decade, advances have been made in the mathematical understanding of localization for certain classes of operators, and a few techniques have been put forward that reliably predict regions of localization for eigenvectors whose eigenvalues are low in the spectrum, and even provide reasonable estimates of the smallest eigenvalue having an eigenvector localized in a given region. However, there is significant room for development of computational techniques that may be needed in practice for specific design problems, and may also lead to more refined conjectures on the theoretical side. We describe an approach that we believe is better suited for a more thorough \textit{numerical} investigation of eigenvector localization, which allows for exploration deeper into the spectrum and provides clear quantitative control over how strongly localized an eigenvector must be within a region before it is accepted as such. We provide theoretical justification of the approach, as well as numerical results of a partial realization of the associated algorithm that serves as a proof-of-concept.

We analyze the Zakharov system, which describes Langmuir turbulence in plasma and the Benney model for interaction of short and long waves in resonant water waves. Our particular interest is in the periodic traveling waves, which we construct and study in detail. For the Zakharov system, we show that periodic dnoidal waves are spectrally stable for all natural values of the parameters. For the Benney system, we prove that the periodic dnoidal waves are spectrally stable with respect to perturbations of the same period. For a different parameter set, we construct snoidal waves of the Benney system, which exhibit instabilities in the same setup. Our results are the first instability results in this context. Our approach, which allows for a definite answer for the entire domain of parameters, relies on the instability index theory developed by [1,2,3]. Even though the linearized operators are explicit, our spectral analysis requires subtle investigation of matrix Schrödinger operators in the periodic context, revealing some interesting features. $$-------$$ REFERENCES $$\ $$ [1] T. M. Kapitula, P. G. Kevrekidis, B. Sandstede. "Counting eigenvalues via Krein signature in infinite-dimensional Hamitonial systems", Physica D, 3-4, (2004), p. 263--282. and Addendum: "Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems'' Phys. D 195 (2004), no. 3-4, 263--282. and Phys. D 201 (2005), no. 1-2, 199--201. $$\ $$ [2] Z. Lin, C. Zeng, "Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs", Mem. Amer. Math. Soc. 275 (2022), no. 1347. $$\ $$ [3] D. Pelinovsky, "Inertia law for spectral stability of solitary waves in coupled nonlinear Schrödinger equations." Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2055, p. 783--812.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Some important non-self-adjoint problems in the theory of composites include those with energy loss such as those described in the quasistatic limit at constant frequency by effective complex moduli including dielectric and viscoelastic ones, and others such as conduction in the presence of a magnetic field and convection enhanced diffusion. Powerful tools in analyzing these problems include utilizing the analytic properties of the effective moduli as a function of those of the component phases, and generalizations of a technique of Gibiansky and Cherkaev for converting these non-self-adjoint problems into self-adjoint ones. More recently we investigated wave propagation in certain space-time microstuctures exhibiting a type of PT symmetry and found stable wave propagation for a range of parameter values, but exponential blow-up in time outside this range. This talk reviews these results.

We present abstract results on the generation of $C_0$-semigroups by first order differential operators on $\mathrm{L}^p(\mathbb{R}_+\mathbb{C}^{\ell})\times \mathrm{L}^p([0,1],\mathbb{C}^m)$ with general boundary conditions. In many cases we are able to characterize the generation property in terms of the invertibility of a matrix associated to the boundary conditions. The abstract results are used to study well-posedness of transport equations on non-compact metric graphs.

It is now very well established that small random perturbations lead to probabilistic Weyl laws for the eigenvalue asymptotics of non-selfadjoint semiclassical pseudo-differential operators, Berezin-Toeplitz quantizations of compact Kähler manifolds and Toeplitz matrices. In this talk, I present recent a work in collaboration with Anirban Basak and Ofer Zeitouni on eigenvector localization and delocalization of large non-selfadjoint Toeplitz matrices with small random perturbations.

In recent years an efficient framework was established to prove scattering for nonlinear dispersive equations, based on the combination of concentration-compactness principles and induction on energy arguments. Originally developed by Kenig and Merle, the framework has been adapted to several equations with constant coefficients. The presence of potential perturbations or variable coefficients introduces new difficulties due to unisotropy. In this talk I shall report on some new results, obtained in collaboration with Angelo Zanni (Roma), concerning scattering for a defocusing, subcritical NLS in one space dimension, with fully variable coefficients.

We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficients are thus included.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Let $\Omega$ be a bounded domain of $R^d$ with Lipschitz boundary $\Gamma$. We define the Dirichlet-to-Neumann operator $\mathcal{N}$ on $L_2(\Gamma)$ associated with a second-order elliptic operator $$A = -\sum_{k,j=1}^d \partial_k (c_{kl} \, \partial_l) + \sum_{k=1}^d (b_k \, \partial_k - \partial_k( c_k \cdot)) + a_0.$$ We prove a criterion for invariance of a closed convex set under the action of the semigroup of $\mathcal{N}$. Roughly speaking, it says that if the semigroup generated by $-A$, endowed with Neumann boundary conditions, leaves invariant a closed convex set of $L_2(\Omega)$, then the `trace' of this convex set is invariant for the semigroup of $\mathcal{N}$. We use this invariance to prove a criterion for the domination of semigroups of two Dirichlet-to-Neumann operators. We apply this criterion to prove the diamagnetic inequality for such operators on $L_2(\Gamma)$. $$\ $$ This is joint work with El Maati Ouhabaz.

We consider the Real non-self-adjoint Ginzburg Landau (RnsaGL) equation where the linear operator gives rise to a quasi-basis structure. This quasi-basis structure allows us to look at the transfer of energy between eigenmodes in a system that is both non-normal and nonlinear. The importance of non-normality versus nonlinearity has been a topic of debate concerning where difficulty modelling Fluid Mechanics systems lies. Although only a toy model, the quasi-basis structure allows us to illustrate the ramifications of non-normality and nonlinearity contemporaneously in contrast to tools such as the plotting of pseudospectra that only illuminates the linear phenomena.

The Solvability Complexity Index (SCI) Hierarchy is a classification of the complexity of problems that take infinitely long to compute. The simplest example is computing the spectrum of an infinite matrix: one can ask whether there exists an algorithm which reads ever increasing finite sections and approximates its spectrum correctly (being spectrally exact and with no pollution). It turns out that this is impossible: an algorithm that can compute the spectrum of any such infinite matrix requires exactly three limits, hence we say that this problem has $\mathrm{SCI}=3$ [3,4]. $$\ $$ In this talk I will present recent results that apply this theory to the computation of scattering resonances: both quantum [2] and classical [1]. In both cases it is shown that a single limiting procedure suffices, i.e. $\mathrm{SCI}=1$, by constructing explicit implementable algorithms. In the case of quantum scattering by a compactly supported potential this is done by obtaining quantitative estimates of the bordered resolvent. For scattering by a compact obstacle in $R^d$ this is done by expressing the Dirichlet-to-Neumann operator as a compact perturbation of the identity and approximating this compact perturbation. $$-------$$ REFERENCES $$\ $$ [1] Jonathan Ben-Artzi, Marco Marletta and Frank Rösler. "Computing the Sound of the Sea in a Seashell", Found. Comput. Math. (FoCM) 22 (2022) 697--731. $$\ $$ [2] Jonathan Ben-Artzi, Marco Marletta and Frank Rösler. "Computing Scattering Resonances", J. Eur. Math. Soc. (JEMS) to appear. $$\ $$ [3] Jonathan Ben-Artzi, Matthew Colbrook, Anders Hansen, Olavi Nevalinna and Markus Seidel. "Computing Spectra -- On the Solvability Complexity Index Hierarchy and Towers of Algorithms" arXiv:1508.03280. $$\ $$ [4] Anders Hansen. "On the solvability complexity index, the $n$-pseudospectrum and approximations of spectra of operators" J. Amer. Math. Soc. (JAMS) 24 (2011) 81--124.

The solution to a large class of linear dispersive PDEs under periodic boundary conditions, including for example the free linear Schrödinger equation and the Airy PDE, exhibits a dichotomy at rational and irrational times. At rational times, the solution is decomposed in a finite number of translated copies of the initial condition. Equivalently, when the initial function has a jump discontinuity, then the solution also exhibits finitely many jump discontinuities. On the other hand, at irrational times the solution is known to become a continuous, but nowhere differentiable function. These two effects form the revival and fractalisation phenomenon at rational and irrational times respectively. In this talk, we will give an overview emphasising on the phenomenon of revivals and discuss recent extensions outside the classical theory.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, I summarize recent developments of an encounter-based approach to diffusion-controlled reactions, which is based on the concept of the boundary local time [1,2,3]. While conventional theories of restricted diffusion towards a reactive target in a confining Euclidean domain rely on the spectral properties of the Laplace operator with mixed boundary conditions, the new approach involves a version of the Dirichlet-to-Neumann operator. Some probabilistic interpretations and applications of this approach will be provided. Even though such pseudo-differential operators have been studied in the past, relations between the spectrum of the Dirichlet-to-Neumann operator and the geometric structure of the domain remain poorly understood, especially in non-self-adjoint settings. Several open problems will be announced. $$--------$$ REFERENCES $$\ $$ [1] D. S. Grebenkov. "Paradigm Shift in Diffusion-Mediated Surface Phenomena", Phys. Rev. Lett. 125 (2020) 078102. $$\ $$ [2] D. S. Grebenkov "Surface Hopping Propagator: An Alternative Approach to Diffusion-Influenced Reactions" Phys. Rev. E 102 (2020) 032125. $$\ $$ [3] D. S. Grebenkov "An encounter-based approach for restricted diffusion with a gradient drift," J. Phys. A: Math. Theor. 55(2022) 045203.

The Lugiato-Lefever equation is a nonlinear Schrödinger-type equation with damping, detuning and driving, derived in nonlinear optics by Lugiato and Lefever (1987). In a previous contribution we have studied the linear asymptotic stability of spectrally stable periodic waves to perturbations which are localized, i.e., integrable on the real line. In this work, we show how the results found at the linear level can be used to analyse the asymp- totic nonlinear stability of these periodic waves. We obtain decay rates for localized perturbations which are precisely the same as the ones found for the linear problem.

The purpose is to revisit the proof of the Gearhart-Prüss-Huang-Greiner theorem for a semigroup $S(t)$, following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on $\Vert S(t) \Vert$ in terms of bounds on the resolvent of the generator. Our aim is to present new improvements, partially motivated by a paper of D. Wei and discuss the optimality of our results. This is a common work with J. Sjöstrand and more recently also with J. Viola.

We study the generator $G$ of the one-dimensional damped wave equation with unbounded damping. We show that the norm of the corresponding resolvent operator, $\Psi(\lambda) := \| (G - \lambda)^{-1} \|$, is approximately constant as $|\lambda| \to +\infty$ on vertical strips of bounded width contained in the closure of the left-hand side complex semi-plane, $\overline{\mathbb{C}}_{-} := \{\lambda \in \mathbb{C}: \operatorname{Re}\lambda \le 0\}$. We shall outline the main ideas behind the proof, which rests on a precise asymptotic analysis of the norm of the inverse of $T(\lambda)$, the quadratic operator associated with $G$, and illustrate our analysis with some examples.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will present some recent developments in the study of the phenomena of dispersive quantization (the Talbot effect) and revival in linear and nonlinear dispersive systems, including the effects of boundary conditions, coupling dispersive systems to Lamb oscillators, new manifestations in integro-differential equations modeling interface dynamics, and a study of the Fermi-Pasta-Ulam-Tsingou problem for coupled nonlinear oscillators.

In this talk, we present the proof of the following theorem: let M be a complete non-compact Riemannian manifold whose curvature goes to zero at infinity, then its essential spectrum of the Laplacian on differential forms is a connected set. In particular, we study the case of form spectrum when the manifold is collapsing at infinity. This is joint with Nelia Charalambous.

We analyze the microlocal structure of the semi-classical scattering ampli- tude for Schrodinger operators with a strong magnetic and a strong electric fields at nontrapping energies. For this purpose we develop a framework and establish some of the properties of semi-classical-Fourier-integral-operator- valued pseudodifferential operators and prove that the scattering amplitude is given by such an operator.

In this talk I will give an overview of classical results for the damped wave equation and then focus on the torus with damping that does not satisfy the geometric control condition. This setup is interesting because two dampings with the same support can have different decay rates. The behavior of the damping near the boundary of its support determines these decay rates, but it is not clear what properties are of utmost importance. Leading candidates are Hölder regularity and a "derivative bound condition" in which the size of the damping controls the size of the gradient of the damping. I will discuss results in two model cases, polynomial damping and oscillatory damping, that give a framework in which a resolvent estimate or quasimode construction for the associated non-self-adjoint stationary operator will determine which of these two properties determine the sharp decay rate.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider the stability of symmetric flows in a two-dimensional channel (including the Poiseuille flow). In 2015 Grenier, Guo, and Nguyen have established instability of these flows in a particular region of the parameter space, affirming formal asymptotics results from the 1960’s. We prove that these flows are stable outside this region in parameter space. More precisely we show that the Orr-Sommerfeld operator $${\mathcal B} =\Big(-\frac{d^2}{dx^2}+i\beta(U+i\lambda)\Big)\Big(\frac{d^2}{dx^2}-\alpha^2\Big) -i\beta U^{\prime\prime}$$ Defined on $$D({\mathcal B})=\{u\in H^4(0,1)\,,\, u^\prime(0)=u^{(3)}(0)=0 \mbox{ and }\, u(1)=u^\prime(1)=0\}.$$ Is bounded on the half-plane $\Re \lambda \geq 0$ for $\alpha \gg \beta^{-1/10}$ or $\alpha \ll \beta^{-1/6}$

I will report on recent progress concerning eigenvalues of Schrödinger operators with complex potentials. In the introduction I will give some details on the recent counterexample to the Laptev-Safronov conjecture. In the main part of the talk I will show how the decay assumptions can be weakened under randomization or sparsification of the potential. This is based partly on joint works with Sabine Bögli and Konstantin Merz.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

Optical waveguides play an important role in carrying a high-speed signal from one place to another. Straight waveguides and bent waveguides are the frequently used waveguide structures [1]. These waveguides are mathematically studied as an eigenvalue problem to understand the propagation of optical waves in them [2]. In our earlier work, the eigenvalue problem corresponding to the bent waveguide was found to be a non-self-adjoint problem with a discontinuous differential coefficient containing a bend radius parameter defined on the semi-infinite domain [3]. It has complex eigenvalues. When the bend radius tends to infinity, the bend waveguide eigenvalue problem behaves like the straight waveguide eigenvalue problem. In terms of the underlying mathematical model, when this parameter tends to infinity, the non-self-adjoint eigenvalue problem changes into a self-adjoint problem on the infinite domain with real eigenvalues. $$\ $$ There are not many general predictions about the properties of non-self-adjoint operators, such as the nature of eigenvalues, eigenfunctions, etc [4]. We investigated the bent waveguide eigenvalue problem focusing more on the behavior of its operator, eigenvalues, and eigenfunctions. We found that the corresponding operator is compact, it has a finite number of eigenvalues, and the eigenfunctions corresponding to the distinct eigenvalues are orthogonal. This work can answer a question such as Does a bounded operator defined on infinite-dimensional Banach space have a finite number of eigenvalues or not? We also studied the asymptotic behavior of the eigenfunction for this problem. $$--------$$ REFERENCES $$\ $$ [1] A. Ghatak. Optics. Tata McGraw-Hill Education, 2005. $$\ $$ [2] K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna and Čtyroký. "Analytic approach to dielectric optical bent slab waveguides". Optical and Quantum Electronics, 37 (2005) 37--61. $$\ $$ [3] R. Kumar and K. R. Hiremath. "Non-self-adjointness of bent optical waveguide eigenvalue problem". Journal of Mathematical Analysis and Applications 12 (2022) 126024. $$\ $$ [4] E. B. Davies. "Non-self-adjoint differential operators". Bulletin of the London Mathematical Society 34 (2002) 513--532.

In this talk, we discuss about the spectrum and the resolvent of the non-self-adjoint operator $$L=\frac{d^{4}}{dx^4}+i\ sign(x)$$ Where $sign(x)$ is the sign function. As a by-product, the pseudospectrum at infinity can be revealed by estimating the resolvent which can be computed explicitly.