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Journal of Physics A Mathematical and Theoretical

Spectral and transport properties of quantum systems: in memory of Pierre Duclos (1948–2010)

J. M. Combes, Pavel Exner, Valentin Anatol'evich Zagrebnov

Published Nov 9, 2010• 1 citation
Publisher

Abstract

This issue is devoted to our colleague and friend Pierre Duclos who passed away suddenly and prematurely in Prague on 12 January this year. We want to honour his memory in the way he would have liked, by collecting fresh and original work from his area of interest. Pierre Duclos was born on 8 January 1948 in Paris. He started as an engineering student but also attended graduate courses at the Centre de Physique Théorique (CPT) in Marseille, which inspired him to change his path and pursue the professional career of a researcher. He joined the Mathematical Physics team at CPT and obtained a position at the University of Toulon, where he later became a full professor. He was never solitary; always being full of energy and a smart and sociable person, he started and maintained many international collaborations and organized numerous conferences and seminars. In the early eighties he had strong ties with the Free and Technical Universities of Berlin. From the beginning of the nineties, he collaborated with coleagues in Prague, Bucharest, Santiago de Chile, and more recently also in Aalborg and Dublin. His scientific interests were wide, with a focus on mathematical methods of quantum theory. He made important contributions to our understanding of multiple-well Schrödinger operators, geometrically-induced properties of quantum waveguides, spectra of Wannier–Stark systems, dynamics with time-periodic perturbations, and transport in mesoscopic systems, to name his most significant results. We choose for this issue the title `Spectral and transport properties of quantum systems' which cover the subjects of most papers to which his colleagues, and often coauthors, contributed. We have also included a few other papers with topics related to Pierre's work. We are glad we were able to gather a numerous collection of papers which in our view represent interesting new developments. A few of them are works which bear Pierre's signature and have been completed by his collaborators. Moreover, we are aware of several other works which their authors dedicated to Pierre's memory but which for some reason or another did not make it to this issue. A list of these papers can be found at the end of the preface. This multitude of memorial papers shows that Pierre was popular not only as a colleague, coauthor, and teacher, but also as a person. Those who had the good fortune to work with him will always recall his blend of hard-working habits, strong views, and human warmth, which made him so unique. He will be remembered with gratitude and admiration by all who knew him well. We will miss him a lot. Jean-Michel Combes , Université du Sud Toulon-Var and Centre de Physique Théorique, CNRS Marseille, France Pavel Exner , Doppler Institute and Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences Valentin A Zagrebnov , Université de la Mediterranée and Centre de Physique Théorique, CNRS Marseille, France Guest Editors Other works dedicated to Pierre's memory: Aschbacher W, Barbaroux J-M, Faupin J and Guillot J-C 2010 Spectral theory for a mathematical model of weak interactions: the decay of the intermediate bosons W ± . II Annales Henri Poincaré at press Bellissard J and Palmer I 2009 The Jewett–Krieger construction for tilings arXiv:0906.2997 Gesztesy F and Zinchenko M 2010 Symmetrized perturbation determinants and applications to boundary data maps and Krein-type resolvent formulas arXiv:1007.4605 Kostrykin V, Potthoff J and Schrader R 2010 Brownian motions on metric graphs: Feller Brownian motions on intervals revisited arXiv:1008.3761 Stollmann P 2010 From uncertainty principles to Wegner estimates Math. Phys. Anal. Geom. 13 145–57

DOI
10.1088/1751-8121/43/47/470301

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This issue is devoted to our colleague and friend Pierre Duclos who passed away suddenly and prematurely in Prague on 12 January this year. We want to honour his memory in the way he would have liked, by collecting fresh and original work from his area of interest. Pierre Duclos was born on 8 January 1948 in Paris. He started as an engineering student but also attended graduate courses at the Centre de Physique Théorique (CPT) in Marseille, which inspired him to change his path and pursue the professional career of a researcher. He joined the Mathematical Physics team at CPT and obtained a position at the University of Toulon, where he later became a full professor. He was never solitary; always being full of energy and a smart and sociable person, he started and maintained many international collaborations and organized numerous conferences and seminars. In the early eighties he had strong ties with the Free and Technical Universities of Berlin. From the beginning of the nineties, he collaborated with coleagues in Prague, Bucharest, Santiago de Chile, and more recently also in Aalborg and Dublin. His scientific interests were wide, with a focus on mathematical methods of quantum theory. He made important contributions to our understanding of multiple-well Schrödinger operators, geometrically-induced properties of quantum waveguides, spectra of Wannier–Stark systems, dynamics with time-periodic perturbations, and transport in mesoscopic systems, to name his most significant results. We choose for this issue the title `Spectral and transport properties of quantum systems' which cover the subjects of most papers to which his colleagues, and often coauthors, contributed. We have also included a few other papers with topics related to Pierre's work. We are glad we were able to gather a numerous collection of papers which in our view represent interesting new developments. A few of them are works which bear Pierre's signature and have been completed by his collaborators. Moreover, we are aware of several other works which their authors dedicated to Pierre's memory but which for some reason or another did not make it to this issue. A list of these papers can be found at the end of the preface. This multitude of memorial papers shows that Pierre was popular not only as a colleague, coauthor, and teacher, but also as a person. Those who had the good fortune to work with him will always recall his blend of hard-working habits, strong views, and human warmth, which made him so unique. He will be remembered with gratitude and admiration by all who knew him well. We will miss him a lot. Jean-Michel Combes , Université du Sud Toulon-Var and Centre de Physique Théorique, CNRS Marseille, France Pavel Exner , Doppler Institute and Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences Valentin A Zagrebnov , Université de la Mediterranée and Centre de Physique Théorique, CNRS Marseille, France Guest Editors Other works dedicated to Pierre's memory: Aschbacher W, Barbaroux J-M, Faupin J and Guillot J-C 2010 Spectral theory for a mathematical model of weak interactions: the decay of the intermediate bosons W ± . II Annales Henri Poincaré at press Bellissard J and Palmer I 2009 The Jewett–Krieger construction for tilings arXiv:0906.2997 Gesztesy F and Zinchenko M 2010 Symmetrized perturbation determinants and applications to boundary data maps and Krein-type resolvent formulas arXiv:1007.4605 Kostrykin V, Potthoff J and Schrader R 2010 Brownian motions on metric graphs: Feller Brownian motions on intervals revisited arXiv:1008.3761 Stollmann P 2010 From uncertainty principles to Wegner estimates Math. Phys. Anal. Geom. 13 145–57
On the emptiness formation probability in quasi-free states by W. H. Aschbacher Wegner-Stollmann type estimates for some quantum lattice systems by V. Chulaevsky The mutually unbiased bases revisited by M. Combescure Pertubative vs. variational methods in the study of carbon nanotubes by H. D. Cornean, T. G. Pedersen, and B. Ricaud Normal transport at positive temperatures in classical Hamiltonian open systems by S. De Bievre, P. Lafitte, and P. E. Parris Equivalence of resolvent and scattering resonances on quantum graphs by P. Exner and J. Lipovsky Optimal uniform elliptic estimates for the Ginzburg-Landau system by S. Fournais and B. Helffer Localization for a continuum Cantor-Anderson Hamiltonian by F. Germinet and A. Klein Generalized fractal dimensions on the negative axis for non compactly supported measures by F. Germinet and S. Tcheremchantsev Localization for Schrodinger operators with random vector potentials by F. Ghribi, P. D. Hislop, and F. Klopp Vibrational levels associated with hydrogen bonds and semiclassical Hamiltonian normal forms by G. A. Hagedorn and A. Joye On the strict positivity of entropy production by V. Jaksic and C.-A. Pillet Uniqueness results for transient dynamics of quantum systems by A. Jensen and G. Nenciu Heat kernels on metric graphs and a trace formula by V. Kostrykin, J. Potthoff, and R. Schrader On a random matrix model of quantum relaxation by J. L. Lebowitz, A. Lytova, and L. Pastur Revivals of wave packets and Bohr-Sommerfeld quantization rules by D. Robert On a linear stochastic wave equation modeling heat flow by L. E. Thomas and Y. Wang Exponential decay of eigenfunctions of first order systems by D. R. Yafaev.
This issue comprises two dozen research papers which are all in one sense or another devoted to models in which the interaction is singular and sharply localized; a typical example is a quantum particle interacting with a family of δ-type potentials. Such an idealization usually makes analysis of their properties considerably easier, sometimes allowing us to reduce it to a simple algebraic problem—this is why one speaks about solvable models. The subject can be traced back to the early days of quantum mechanics; however, the progress in this field was slow and uneven until the 1960s, mostly because singular interactions are often difficult to deal with mathematically and intuitive arguments do not work. After overcoming the initial difficulties the `classical' theory of point interactions was developed, and finally summarized in 1988 in a monograph by Albeverio, Gesztesy, Høegh-Krohn, and Holden, which you will find quoted in numerous places within this issue. A reliable way to judge theories is to observe the progress they make within one or two decades. In this case there is no doubt that the field has witnessed a continuous development and covered areas which nobody had thought of when the subject first emerged. The reader may see it in the second edition of the aforementioned book which was published by AMS Chelsea only recently and contained a brief survey of these new achievements. It is no coincidence that this topical issue appears at the same time; it has been conceived as its counterpart and a forum at which fresh results in the field can demonstrated. Let us briefly survey the contents of the issue. While the papers included have in common the basic subject, they represent a broad spectrum philosophically as well as technically, and any attempt to classify them is somewhat futile. Nevertheless, we will divide them into a few groups. The first comprises contributions directly related to the usual point-interaction ideology. M Correggi and one of the editors study a toy model of a decay under the influence of a time-periodic δ potential. E Demiralp describes the spectrum of a spherical harmonic oscillator amended with a concentric family of δ-shell interactions. Another of the editors presents an isoperimetric problem for point interactions arranged at vertices of a polygon. W Huddell and R Hughes show how singular perturbations of a one-dimensional Dirac operator can be approximated by regular potentials, and J Brasche constructs a family of Hamiltonians in which the singular interaction has a more complicated support, namely a Brownian path. Finally, B Pavlov and I Antoniou apply the singular perturbation technique to another classical Hamiltonian, that of a generalized Friedrichs model; no matter that the unperturbed observable is called momentum in their paper. The three papers in the following group are distinguished by the fact that they consider systems which are fully or partially periodic. F Bentosela and M Tater analyse scattering on a crystalline `slab' modelled by point interactions distributed periodically on a finite number of parallel plates. E de Prunelé studies evolution of wavepackets in crystal models of different geometries, and M Avdonin et al discuss a simple model of a spin-dependent scattering on a one-dimensional array of quantum dots. The next group of papers is devoted to a topic which was untouched at the time of the aforementioned first edition, namely quantum graphs, which became a subject of interest after numerous applications of such systems to semiconductor, carbon and other nanostructures. Most contributions here deal with the `usual' model in which the Hamiltonian is a Schrödinger operator supported by the graph. P Kuchment describes spectral properties of such graphs, in particular periodic ones and those with decorations. S Albeverio and K Pankrashkin present a modification of Krein's formula which is suitable for constructing Hamiltonians of quantum graphs using boundary conditions at vertices directly. Two papers are devoted to inverse problems in this context: M Harmer studies inverse scattering for the matrix Schrödinger operator on the halfline with applications to star graphs, while P Kurasov and M Nowaczyk give a mathematically rigorous version of the known Gutkin–Smilansky result on the inverse spectral problem. The paper by O Post contributes to the question of how graphs can be approximated by more realistic `fat' graphs, and describes a class leading to disconnected quantum graphs. Finally, S Kondej and one of the editors study scattering in the context of `leaky' graphs which takes quantum tunnelling into account. While most results in this field describe one-particle Hamiltonians, more complicated systems have also been studied. In this issue we have three examples. C Cacciapuito, R Carlone, and R Figari discuss decoherence in a simple model of two particles, one heavy and one light, interacting through a δ potential; they give a rigorous meaning to a formula derived by Joos and Zeh. A related model by R Figari and A Teta is used to describe ionization. M Hallnäs, E Langmann, and C Paufler treat a true N -body situation, namely a model of one-dimensional gas of distinguishable particles interacting through generalized point interactions; they write the Bethe ansatz and present the solution of a particular case. The last group is a collection of contributions which in one sense or another are outside quantum mechanics, either modifying its postulates or applying it to a different physical situation. The latter applies to the paper of D Noja and A Posilicano in which they study nonlinear wave equations with point perturbations and show the existence of a solution to the Cauchy problem. F Coutinho et al discuss one-dimensional point interactions with energy-dependent coupling constant, S Albeverio and S Kuzhel examine a class of point interactions which are not symmetric but P-symmetric, where P is the parity operator, and M Znojil and V Jakubský consider a `double-well' PT-symmetric model with two δ interactions with an imaginary coupling. The last two papers present mathematical constructions. Yu Shondin demonstrates a way to define an interaction more singular than the usual δ potentials obtained by means of self-adjoint extensions, and V Koshmanenko presents a construction of strongly singular perturbations leading to rather unusual `Hamiltonians'.
This work is dedicated to Pavel Exner on the occasion of his anniversary.\nThis volume is dedicated to Pavel Exner on the occasion of his 70th anniversary. It collects contributions by numerous scientists with expertise in mathematical physics and in particular in problems arising from quantum mechanics. The questions addressed in the contributions cover a large range of topics. A lot of attention was paid to differential operators with zero range interactions, which are often used as models in quantum mechanics. Several authors considered problems related to systems with mixed-dimensions such as quantum waveguides, quantum layers and quantum graphs. Eigenvalues and eigenfunctions of Laplace and Schrödinger operators are discussed too, as well as systems with adiabatic time evolution. Although most of the problems treated in the book have a quantum mechanical background, some contributions deal with issues which go well beyond this framework, for example the Cayley–Hamilton theorem, approximation formulae for contraction semigroups or factorization of analytic operator-valued Fredholm functions. As for the mathematical tools involved, the book provides a wide variety of techniques from functional analysis and operator theory.\nAltogether the volume presents a collection of research papers which will be of interest to any active scientist working in one of the above mentioned fields.
The idea to organize a central-european workshop on quantum optics goes back to the activity of prof. V. Buzžek (Bratislava) and prof. J. Janszky (Budapest). The first meetings were held in Budapest and Bratislava and started a tradition bringing together people working on problems related to quantum optics. Later workshops have been held at Budmerice (Slovakia), Balaton (Hungary), Olomouc and Prague (Czech republic). The already eighth central-european workshop found his way back to Prague and was organized in cooperation with the Doppler Institute and the Department of Physics of the Faculty for Nuclear Science and Physical Engineering of the Czech Technical University in Prague. The contributions of the participants focus on problems in quantum communication, quantum information, quantum state reconstruction, atom optics and nonlinear quantum optics. The success of the workshop was enabled by the kind cooperation of all the participants during the conference. The present collection of contributions gives a good review of topics presented. We hope that the present issue will be able to transfer the constructive and stimulating atmosphere of the workshop to the readership. I would like to thank the editor prof. D. Lüst and co-editor prof. W. P. Schleich for opening up the possibility to publish the proceedings of the workshop in the Fortschritte der Physik. It surely helps to enlarge the readership of the proceedings and promotes the idea of the central-european workshop among colleagues around the world.