Popular Lecture - Facultad de MatemÃ¡ticas. Pontificia Universidad ...

Spectral Days, Santiago **de** Chile, September 20 – 24, 2010 1**Popular** **Lecture**Seismic imaging: **de**tection of subsurface structuresMarcus Carlsson, Purdue University, USAE-mail: mcarlsso@math.purdue.eduThis talk is a brief overview of a very large area, directed to non-experts. The mainpurpose of seismic imaging is to find the local travel speed of acoustic waves as a function of2 or 3 spacial dimensions (the velocity mo**de**l). Typically one is interested in a region suchas a few kilometers below a given area on the earth’s surface, but it could also be the entireglobe. Obviously, the smaller the volume to be investigated, the better the resolution. Themain tool is to set of an acoustic wave (vibration) at a point location, (generated e.g. byexplosions, air-guns or earthquakes), and then measure the waves that gets reflected to thesurface. This is an “inverse problem” with many applications, e.g. exploration of naturalresources, since the wave speed is highly **de**pen**de**nt on the material; gas, liquid, rock. Theterm inverse problem arise as follows: Suppose we knew the velocity mo**de**l c(x), then thewave-equation( )1c 2 (x) ∂2 t − △ u(x, t) = δ 0 (t)δ 0 (x − x s )(+boundary conditions & initial conditions),(where δ 0 is the dirac distribution at 0, x s is the location of the ”explosion” at time 0 and△ is the Laplace operator acting on the spacial variables), would give us the vibration u ofthe earth as a function of space x and time t. In particular, we could calculate the measured”data” d, which is the restriction of u to the surface. We now consi**de**r the inverse problem,namely given d, how to reconstruct c.We **de**scribe standard ways of simplifying the problem, starting with the single scatteringassumption and the Born approximation of the wave equation. Limitations of these techniquesdue to lack of computer capacity in processing the large data sets involved, in particular invertinglarge matrices, then leads us to techniques of “backward and forward migration”,cross-correlations and Beylkin’s theorem. Migration means that, given an assumption of cand data d measured at the surface, one calculates the corresponding function u with numericmethods, e.g. by finite difference. Since one does not know c, an initial guess is usually updatedby repeating this calculation many times. It is thus crucial to be able to do it fast, whichstill is a bottleneck for applications. This then lead us to Fourier Integral Operators (FIO’s)and ”wave-packets”. We introduce the main i**de**as of wave-packet **de**composition algorithms,which is a very active area of research with applications far beyond seismic imaging, e.g. datacompression, signal processing etc. This finally leads us to the problem of approximatinga function by a sum of exponential functions, which is the topic of my other talk at thisconference.

Spectral Days, Santiago **de** Chile, September 20 – 24, 2010 2Mini-coursesStability of MatterRafael Benguria, **Pontificia** **Universidad** Católica **de** ChileE-mail: rbenguri@fis.puc.clI will give a short course on stability of matter: During this course I will present the mainmathematical and physical i**de**as behind the proof of stability of matter for various operators.I will start with the now classical proof of Lieb and Thirring, I will give a short review aboutdifferent techniques, including Lieb-Thirring inequalities, Electrostatic Inequalities, etc. I willalso briefly review more recent results.Localization for the Random Displacement Mo**de**lGünter Stolz, University of Alabama at BirminghamE-mail: stolz.gunter@gmail.comThe random displacement mo**de**l (RDM) is a random Schrödinger operator in L 2 (R d ) ofthe formH ω = −∆ + ∑ n∈Z d q(x − n − ω n ),where the single-site terms q are displaced from the sites n of the lattice Z d by an arrayω = (ω n ) of in**de**pen**de**nt, i**de**ntically distributed random displacement vectors.I will report on recent joint work with Frédéric Klopp, Michael Loss and Shu Nakamura(arXiv:1007.2483), proving that, un**de**r suitable assumptions and in any dimension d, the RDMexhibits localization, in the spectral as well as dynamical sense, at energies close to the bottomof its spectrum. This corresponds to what is known for other random Schrödinger operatorssuch as the An**de**rson and Poisson mo**de**ls. The proof for the RDM was complicated by thelack of monotonicity properties which had frequently been exploited for other mo**de**ls. As aconsequence, localization for the multi-dimensional RDM was a longstanding open problem.My lectures will focus on three particular results which provi**de**d the main steps towardsa localization proof: (i) A spectral geometric result for a related Neumann problem dubbed“bubbles tend to the corners”, (ii) a proof of Lifshitz tails for the integrated **de**nsity of states,extending previous work of Klopp and Nakamura, (iii) a proof of a new Wegner estimate forthe RDM. With these ingredients established, localization is a consequence of the method ofmultiscale analysis.In or**de**r to make my lectures accessible for non-specialists, I will try to review the basici**de**as un**de**rlying this strategy.

Spectral Days, Santiago **de** Chile, September 20 – 24, 2010 3On semiclassical limiting eigenvalue distribution theorems forclusters of eigenvaluesCarlos Villegas Blas, UNAM Cuernavaca, MexicoE-mail: villegas@matcuer.unam.mxIn this course we will **de**scribe distributions of eigenvalues in clusters originated by untroducinga suitable perturbation of a Hamiltonian operator H 0 having a discrete spectrumwith **de**generated eigenvalues. Namely, H 0 can be the Spherical Laplacian on the n-sphere,the hydrogen atom Hamiltonian or the Landau Hamiltonian. We will elaborate our study inthe semiclassical regime. We will explain some techniques involved in the problem like theuse of coherent states and the averaging method. The research work presented in this courseis in collaboration with Lawrence Thomas, Alejandro Uribe, Peter Hislop, Georgi Raikov andAlexan**de**r Pushnitski.Invited TalksApproximations by sums of exponential functionsMarcus CarlssonE-mail: mcarlsso@math.purdue.eduWe consi**de**r the problem of approximating a (C-valued) function f (on e.g. an interval,half-axis or rectangle), by sums of few exponential functions. To be more precise, let f be**de**fined on an interval, say. Then given ɛ > 0 we want to find a minimal n, coefficientsc 1 , . . . , c n ∈ C and no**de**s ζ 1 , . . . , ζ n ∈ C such thatn∑∥ f − c k e ζ kx∥ < ɛ,k=1where x is the in**de**pen**de**nt variable of the interval. This problem turns out to have strongconnections to the structure of certain Hankel-type operators, which we introduce and presentbasic theorems. If time allows, we also discuss various algorithms.

Spectral Days, Santiago **de** Chile, September 20 – 24, 2010 4The mutually unbiased bases revisitedMonique CombescureE-mail: m.combescure@ipnl.in2p3.frThe mutually unbiased bases is an important issue in quantum information theory. Oneconsi**de**r the d-dimensional Hilbert space C d in which live the states called qdits. Two basesB; B ′ of C d are said to me mutually unbiased if for any pair b ∈ B; b ′ ∈ B ′ one has :|b · b ′ | = 1 √d(where the dot **de**notes the hermitian scalar product in C d ). An important question is toconstruct these bases and to find what is the number of such bases (which is majorated byd + 1). The problem has been completely solved when d is a prime number or a power ofa prime number. In this case one finds exactly d + 1 mutually unbiased bases. Note thatrepresenting the bases with respect to the natural base amounts to consi**de**r d × d unitaryHadamard matrices (all their entries have modulus 1/ √ d). The purpose of our work is torevisit this problem using a minimal toolbox which consists of the Discrete Fourier transform,the generalized Pauli matrices and one particular matrix which is circulant (all its rows aresuccessive circular permutations of the first). One knows that the Discrete Fourier Transformdiagonalizes all circulant matrices. Furthermore the circulant matrix we exhibit has all itspowers wich are also circulant Hadamard matrices. This solves the problem for prime dimensiond.In prime power dimension d = p n what replaces the single circulant matrix C is a groupof block-circulant with circulant blocks matrices which are also diagonalized by the DiscreteFourier Transform in this case.As a by-product of our construction we recover properties of quadratic Gauss sums (in theprime dimension case) and of Weil sums (in the prime power dimension).On the Lipschitz continuity of spectral bands of Harper-like andmagnetic Schrödinger operatorsHoria Cornean, Aalborg UniversityE-mail: cornean@math.aau.dkWe show for a large class of discrete Harper-like and continuous magnetic Schrödingeroperators that their band edges are Lipschitz continuous with respect to the intensity of theexternal constant magnetic field. We generalize a result obtained by J. Bellissard in 1994,and give examples in favor of a recent conjecture of G. Nenciu. A preprint version of thiswork may be found here: http://arxiv.org/abs/0912.0153

Spectral Days, Santiago **de** Chile, September 20 – 24, 2010 5Self-adjoint extensions and SUSY breaking in a mo**de**l ofsupersymmetric quantum mechanicsHoracio Falomir, **Universidad** Nacional **de** La PlataE-mail: falomir@fisica.unlp.edu.arWe consi**de**r the self-adjoint extensions (SAE) of the symmetric supercharges and Hamiltonianfor a mo**de**l of SUSY quantum mechanics in the half-line with a singular superpotential,and consi**de**r the realization of the gra**de**d superalgebra in the domain of the Hamiltonian.We show that only for two particular SAE, whose domains are scale invariant, the algebraof N = 2 SUSY is realized, one with manifest SUSY and the other with spontaneously brokenSUSY. Otherwise, only the N = 1 SUSY algebra is obtained, with spontaneously brokenSUSY and non-**de**generate energy spectrum.Exponential **de**cay and resonances in a driven systemClaudio Fernán**de**z, **Pontificia** **Universidad** Católica **de** ChileE-mail: cfernand@mat.puc.clWe study resonances for a time periodic family H(t) of quantum Hamiltonians on the Hilbertspace H = L 2 (R n ). We assume thatH ω (t) = H + ωW (x, t),where W (x, t) is a time periodic potential, W (x, t + T ) = W (x, t) and H = ∆ + V (x). We assumethat the operator H has a bound state ϕ, corresponding to an simple isolated eigenvalue.We use the Floquet structure, combined with an application of the Mourre theory to studyresonances, to characterize the resonant behavior for the family family H ω (t), in terms of anaverage exponential **de**cay of the survival probability,P (t) ≡ |〈ϕ, U(t, 0)ϕ〉| 2 ,where U(t, s) is the corresponding propagator.Above is a joint work with Philippe Briet (CPT, Marseille).

Spectral Days, Santiago **de** Chile, September 20 – 24, 2010 6Perturbation of Near Threshold EigenvaluesArne Jensen, Aalborg UniversityE-mail: matarne@math.aau.dkFor a two-channel mo**de**l of the form[ ] [ ]Hop 0 0 W12H ε =+ ε0 E 0 W 21 0on H = H op ⊕ C,appearing in the study of Feshbach resonances, I will give results on the **de**cay laws forresonances produced by perturbation of an unstable bound state close to a threshold. Theoperator H op is assumed to have the properties of a Schrödinger operator in odd dimensions,with a threshold at zero. We consi**de**r for ε > 0 and small the survival probability|〈Ψ 0 , e −itHε Ψ 0 〉| 2 ,where Ψ 0 is the eigenfunction corresponding to E 0 for ε = 0. Depending on how close theeigenvalue E 0 is to the threshold zero, one may get bound state behavior, an exponential**de**cay law, or a non-exponential **de**cay law, or even a combination of these.The results presented are joint work with V. Dinu and G. Nenciu, Bucharest, Romania.Localization Properties of the Chalker-Coddington Mo**de**lAlain Joye, University of GrenobleE-mail: Alain.Joye@ujf-grenoble.frThe Chalker Coddington mo**de**l is an effective random unitary mo**de**l **de**signed to un**de**rstandthe **de**localization transition of the quantum Hall effect. Despite its popularity intheoretical and computational physics, no rigorous analysis of its properties had been un**de**rtaken.After a **de**scription of the mo**de**l, recent mathematical results about the localization propertiesof the Chalker Coddington o**de**l restricted to a cylin**de**r of perimeter 2M will be presented:The Lyapunov spectrum is first proven to be simple, which, in particular, yields finitenessof the localization length. It is then shown that this implies spectral localization. Finally,a Thouless formula is proven and the **de**nsity of states is shown to be flat. This makes itpossible to compute the mean Lyapunov exponent which is in**de**pen**de**nt of M.This is joint work with Joachim Asch and Olivier Bourget.

Spectral Days, Santiago **de** Chile, September 20 – 24, 2010 7Absolute continuity of the spectrum of a Landau Hamiltonianperturbed by a generic periodic potentialFrédéric Klopp, University Paris 13E-mail: klopp@math.univ-paris13.frConsi**de**r Γ, a non-**de**generate lattice in R 2 and a constant magnetic field B with a fluxthough a cell of Γ that is a rational multiple of 2π. We prove that for a generic Γ-periodicpotential V , the spectrum of the Landau Hamiltonian with magnetic field B and periodicpotential V is purely absolutely continuous.Mathematical and Experimental Aharonov-Bohm BoundaryConditionsCésar R. **de** Oliveira, Fe**de**ral University of São Carlos, SP, BrazilE-mail: oliveira@pq.cnpq.brIn the framework of nonrelativistic quantum mechanics, we present a study of three topicsrelated to the Aharonov-Bohm (AB) effect. We always consi**de**r a cylindrical solenoid of radiusgreater than zero and mainly in the plane. First we present a classification of all selfadjointSchrödinger operators (i.e., the possible boundary conditions on the solenoid bor**de**r) thatmathematically could characterize the AB operator, whose domains are contained in thenatural space of twice differentiable functions (i.e., in a Sobolev space H 2 ).We then consi**de**r the traditional Dirichlet, Neumann and Robin boundary conditions onthe solenoid bor**de**r and calculate and compare their scattering operators and cross sections.Hopefully, a self-adjoint extension could be experimentally selected after comparing the experimentalresults with the theoretical previsions.Finally, we discuss a theoretical mechanism to select and so justify the usual AB hamiltonianwith Dirichlet boundary conditions on the solenoid. This is obtained by way of increasingsequences of finitely long solenoids together with a natural impermeability procedure; furthermore,it is shown that both limits commute. Such rigorous limits are in the strong resolventsense.This is based on works in collaboration with Marciano Pereira from State University ofPonta Grossa, PR, Brazil.

Spectral Days, Santiago **de** Chile, September 20 – 24, 2010 8Hid**de**n spectral properties in the Quantum Hall effect:conductance and Chern numbersGianluca Panati, Universitá di RomaE-mail: panati@mat.uniroma1.itAiming to a mathematical un**de**rstanding of the Quantum Hall effect (QHE), one investigatesthe dynamics of non-interacting electrons in a 2-dimensional periodic background,un**de**r the additional influence of a uniform transverse magnetic field. Since a direct study ofthe Hamiltonian is a formidable task, one usually refers to effective mo**de**ls which correctly**de**scribe the phenomenology in suitable regimes. In particular, it is known that in the limitof weak (resp. strong) magnetic field, the original Hamiltonian is isospectral to a simpler one,the so-called Hofstadter Hamiltonian (resp. Harper mo**de**l).Beyond the spectrum, the most relevant physical property is the conductance, which isnot invariant un**de**r isospectrality. In view of that, we re**de**rive the effective mo**de**ls showingthat they are asymptotically unitarily equivalent to the original Hamiltonian. More importantly,we study the geometric structure hid**de**n in such effective mo**de**ls, and its relation withtopological quantum numbers. The duality between the geometric structures becomes thekey-i**de**a to obtain a rigorous proof of the TKNN formula, which relates the value of theconductance in the limit of weak and strong magnetic fields.The seminar is based on joint works with G. De Nittis and F. Faure.Integrated **de**nsity of states of Schroedinger operators withperiodic or almost-periodic potentialsLeonid Parnovski, University College LondonE-mail: leonid@math.ucl.ac.ukI will discuss recent results on the asymptotic behaviour of the integrated **de**nsity of states.In particular, we have proved the existence of a complete polynomial asymptotics of the**de**nsity of states when the potential is either smooth periodic, or generic quasi-periodic (finitelinear combination of exponentials), or belongs to a wi**de** class of almost-periodic functions.This is a joint work with Roman Shterenberg (Birmingham Alabama).

Spectral Days, Santiago **de** Chile, September 20 – 24, 2010 9Analysis of the equilibrium of a perturbed fermion systemRolando Rebolledo, **Pontificia** **Universidad** Católica **de** ChileE-mail: rrebolle@uc.clA fermion system immersed in a boson reservoir provi**de**s an interesting example of an opensystem dynamics. The conference will consi**de**r the construction of its semigroup together withits large time behavior. In particular, conditions on the existence of steady states, ergodicityand **de**coherence will be addressed. In addition, the connection between equilibrium (zeroentropy production) and **de**tailed balance conditions will be consi**de**red if time permits.References[1] F. Fagnola and R. Rebolledo, Algebraic conditions for convergence of a quantum Markovsemigroup to a steady state, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11(2008), 467–474.[2] F. Fagnola and R. Rebolledo, On the existence of stationary states for quantum dynamicalsemigroups, J. Math. Phys. 42(3) (2001), 1296–1308.[3] F. Fagnola and R. Rebolledo, Subharmonic projections for a quantum Markov semigroup.,J. Math. Phys. 43(2) (2002), 1074–1082.[4] F. Fagnola and R. Rebolledo, Transience and recurrence of quantum Markov semigroups.,Probab. Theory Related Fields 126(2) (2003), 289–306.[5] R. Rebolledo, Decoherence of quantum Markov semigroups., Ann. Inst. H. PoincaréProbab. Statist. 41(3) (2005), 349–373.[6] R. Rebolledo, A view on **de**coherence via master equations., Open Syst. Inf. Dyn. 12(1)(2005), 37–54.[7] R. Rebolledo, Unraveling Open Quantum Systems: Classical Reductions and ClassicalDilations of Quantum Markov Semigroups., Confluentes Mathematici 1(1) (2009), 123-167.

Spectral Days, Santiago **de** Chile, September 20 – 24, 2010 10On the generalized bosonic string equationEnrique Reyes, **Universidad** **de** Santiago **de** ChileE-mail: ereyes@fermat.usach.clIn this talk I review some recent work on the generalized bosonic string equation∆e −c∆ φ = U(x, φ) (1)on Eucli**de**an space and on more general Riemannian manifolds. I show how to interpret theleft hand si**de** of this equation using Fourier analysis on R n (in the Eucli**de**an context) andspectral properties of the Laplace operator (in the Riemannian manifold context), and I discussthe existence of regular solutions to the bosonic string equation assuming some technicalhypotheses on the growth of the non-linearity U.This work is based on several papers written in collaboration with Przemyslaw Górka (**Universidad****de** Talca and Warsaw University of Technology) and Humberto Prado (**Universidad****de** Santiago).The Asymptotics of the Ground State Energy: Functionals of theOne-Particle Ground Density MatrixHeinz Sie**de**ntop, University of MunichE-mail: h.s@lmu.**de**Large atoms are **de**scribed by the Schrödinger operator in high dimension. We are interestedin the infimum of the spectrum of this operator (ground state energy) as the dimensionof the un**de**rlying space (particle number) increases. We will show how a certain class ofnonlinear functionals can be used to **de**scribe the infimum. In particular, we are going todiscuss a functional introduced by Müller which shares several features with the Schrödingeroperator.

Spectral Days, Santiago **de** Chile, September 20 – 24, 2010 11Quasiclassical asymptotics for pseudo-differential operatorswith discontinuous symbolsAlex Sobolev, University College London, UKE-mail: asobolev@math.ucl.ac.ukRelying on the known two-term quasiclassical asymptotic formula for the trace of the functionf(A) of a truncated Wiener-Hopf type operator A in dimension one, in 1982 H. Widomconjectured a multi-dimensional generalization of that formula for a pseudo-differential operatorA with a symbol a(x, ξ) having jump discontinuities in both variables. In 1990 heproved the conjecture for the special case when the jump in any of the two variables occurson a hyperplane. I will present a proof of Widom’s Conjecture un**de**r the assumption that thesymbol has jumps in both variables on arbitrary smooth boun**de**d surfaces.Hydrogen-like atoms in relativistic QEDEdgardo Stockmeyer, University of MunichE-mail: stock@math.lmu.**de**In this talk I will consi**de**r two different mo**de**ls of a hydrogenic atom in a quantize**de**lectromagnetic field that treat the electron relativistically. The first one is a no-pair mo**de**l inthe free picture, the second one is given by the semi-relativistic Pauli-Fierz Hamiltonian. Forboth mo**de**ls I will discuss basic spectral theoretical results like semi-boun**de**dness, exponentiallocalization, and binding. Finally I will shortly comment on the proofs of the existence of aground state.(This is joint work with M. Könenberg and O. Matte)Spectral properties and averaging for discrete alloy type potentialsIvan Veselic, Chemnitz Technical UniversityE-mail: ivan.veselic@mathematik.tu-chemnitz.**de**The talk is **de**voted to discrete alloy type mo**de**ls, in particular those with non-monotoneparameter **de**pen**de**nce. We discuss recent results on these mo**de**ls, in particular Wegner estimates,averaging techniques, appropriate transformations on the probability space, fractionalmoment bounds on the resolvent and **de**coupling properties. This is related to the exponential**de**cay of the Greens function and localization properties.

Spectral Days, Santiago **de** Chile, September 20 – 24, 2010 12The Dirac operator in the Kerr-Newman metricMonika Winklmeier, **Universidad** **de** Los An**de**s, BogotaE-mail: mwinklme@unian**de**s.edu.coThe Dirac equation in the Kerr-Newman metric **de**scribes a spin-1/2 particle in the spacetimegenerated by a rotating black hole. We will show that the Dirac equation in this casecan be written as an evolution equation with a selfadjoint operator in an appropriate Hilbertspace. In the second part of the talk we will give an estimate for the lowest eigenvalue inmodulus of the angular part of the Dirac operator in the Kerr-Newman metric.Stu**de**nt TalksEquivariant families of Magnetic Pseudodifferential Operatorsand some of their Classical LimitsFabián Belmonte, **Universidad** **de** ChileE-mail: fabianbelmonte@gmail.comRecently, a magnetic version of the Weyl pseudodifferential calculus has been **de**veloped.One can put this calculus in a C ∗ -algebraic framework. In this talk we show how we can useC ∗ -algebraic techniques in a more general context to obtain:1. a new calculus, which mo**de**ls equivariant families of magnetic pseudodifferential operators,2. a new example of Strict Deformation Quantization in the sense of Rieffel.The techniques we **de**velop could be used to prove results about the spectral behavior ofmagnetic Hamiltonians.Joint work with Max Lein and Marius Măntoiu.

Spectral Days, Santiago **de** Chile, September 20 – 24, 2010 13Discrete spectrum for a quantum Hall effect HamiltonianPablo Miranda, **Universidad** **de** ChileE-mail: pmirandar@ug.uchile.clLet H 0 = (−i∇ − A) 2 + W be a magnetic Schrödinger operator on L 2 (R 2 ) where b =curl A > 0 is a constant magnetic field and W is a monotone function which **de**pends onlyon the first variable x. Let V ∈ L ∞ (R 2 , R) **de**cay at infinity. We consi**de**r the operatorH = H 0 + V and find an effective Hamiltonian appropriate to the analysis of the discretespectrum of H near the edges of its essential spectrum. We use this result in or**de**r to establisha criterion for finiteness of the discrete spectrum in each gap of the essential spectrum of Hwhen V has a compact support.Joint work with V. Bruneau (Bor**de**aux I) and G. Raikov (PUC, Santiago **de** Chile).Characterization of the An**de**rson metal-insulator transition fornon ergodic random operatorsConstanza Rojas-Molina, University of Cergy-PontoiseE-mail: constanza777@yahoo.comWe investigate the An**de**rson metal-insulator transition for non ergodic random Schrödingeroperators in both annealed and quenched regimes, based on a dynamical approach of localization,improving known results for ergodic operators into this more general setting. In theprocedure, we reformulate the Bootstrap Multiscale Analysis of Germinet and Klein to fitthe non ergodic setting and we obtain ”uniform” Wegner Estimates, nee**de**d to perform thisadapted Multiscale Analysis. As an application we can study operators with An**de**rson-typepotentials mo**de**ling aperiodic solids, where the impurities lie on a Delone set rather than alattice or the impurities are not i**de**ntically distributed, yielding a break of ergodicity.Dynamical localization and its spectral propertiesAmal Taarabt, University of Cergy-PontoiseE-mail: Amal.Taarabt@u-cergy.frWe will discuss some spectral criteria for dynamical localization that we will classify intwo groups, the first does not involve the multiplicity while the second one does. We will alsoinvestigate the different relations between these properties.