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Toru Sasaki - One of the best experts on this subject based on the ideXlab platform.
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Gap Condition AND SELF-DUALIZED ${\mathcal N}=4$ SUPER-YANG–MILLS THEORY FOR ADE GAUGE GROUP ON K3
Modern Physics Letters A, 2004Co-Authors: Toru SasakiAbstract:We try to determine the partition function of [Formula: see text] super-Yang–Mills theory for ADE gauge group on K3 by self-dualizing our previous ADE partition function. The resulting partition function satisfies Gap Condition.
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Gap Condition AND SELF-DUALIZED ${\mathcal N}=4$ SUPER-YANG–MILLS THEORY FOR ADE GAUGE GROUP ON K3
Modern Physics Letters A, 2004Co-Authors: Toru SasakiAbstract:We try to determine the partition function of ${\mathcal N}=4$ super-Yang–Mills theory for ADE gauge group on K3 by self-dualizing our previous ADE partition function. The resulting partition function satisfies Gap Condition.
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Gap Condition and self dualized mathcal n 4 super yang mills theory for ade gauge group on k3
Modern Physics Letters A, 2004Co-Authors: Toru SasakiAbstract:We try to determine the partition function of ${\mathcal N}=4$ super-Yang–Mills theory for ADE gauge group on K3 by self-dualizing our previous ADE partition function. The resulting partition function satisfies Gap Condition.
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Hecke operator andS-duality of Script N = 4 super Yang-Mills for ADE gauge group onK3
Journal of High Energy Physics, 2003Co-Authors: Toru SasakiAbstract:We determine the partition functions of = 4 super Yang-Mills gauge theory for some ADE gauge groups on K3, under the assumption that they are holomorphic. Our partition functions satisfy the Gap Condition and Montonen-Olive duality at the same time, like the SU(N) partition functions of Vafa and Witten. As a result we find a close relation between Hecke operator and S-duality of = 4 super Yang-Mills for ADE gauge group on K3.
Paola Loreti - One of the best experts on this subject based on the ideXlab platform.
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Ingham type inequalities in lattices
arXiv: Classical Analysis and ODEs, 2015Co-Authors: Vilmos Komornik, Anna Chiara Lai, Paola LoretiAbstract:A classical theorem of Ingham extended Parseval's formula of the trigonometrical system to arbitrary families of exponentials satisfying a uniform Gap Condition. Later his result was extended to several dimensions, but the optimal integration domains have only been determined in very few cases. The purpose of this paper is to determine the optimal connected integration domains for all regular two-dimensional lattices.
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Discrete Ingham type inequalities with a weakened Gap Condition
arXiv: Classical Analysis and ODEs, 2007Co-Authors: Paola Loreti, Vilmos KomornikAbstract:We establish discrete Ingham type and Haraux type inequalities for exponential sums satisfying a weakened Gap Condition. They enable us to obtain discrete simultaneous observability theorems for systems of vibrating strings or beams.
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Dirichlet series and simultaneous observability: two problems solved by the same approach
Systems & Control Letters, 2002Co-Authors: Vilmos Komornik, Paola LoretiAbstract:Abstract In a classical paper, Ingham gave a simple proof of an important theorem of Polya on singular points of Dirichlet series under a uniform Gap assumption on the exponents. Bernstein generalized Polya's theorem by weakening this Gap Condition. We give a simpler proof of Bernstein's theorem by applying a recent generalization of Ingham's theorem. Furthermore, we also solve a simultaneous observability problem by using this theory.
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Ingham-Beurling type theorems with weakened Gap Conditions
Acta Mathematica Hungarica, 2002Co-Authors: Claudio Baiocchi, Vilmos Komornik, Paola LoretiAbstract:Completing a series of works begun by Wiener [34], Paley and Wiener [28] and Ingham [9], a far-reaching generalization of Parseval"s identity was obtained by Beurling [4] for nonharmonic Fourier series whose exponents satisfy a uniform Gap Condition. Later this Gap Condition was weakened by Ullrich [33], Castro and Zuazua [5], Jaffard, Tucsnak and Zuazua [11] and then in [2] in some particular cases. In this paper we prove a general theorem which contains all previous results. Furthermore, applying a different method, we prove a variant of this theorem for nonharmonic Fourier series with vector coefficients. This result, partly motivated by control-theoretical applications, extends several earlier results obtained in [15] and [2]. Finally, applying these results we obtain an optimal simultaneous observability theorem concerning a system of vibrating strings.
Vilmos Komornik - One of the best experts on this subject based on the ideXlab platform.
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Ingham type inequalities in lattices
arXiv: Classical Analysis and ODEs, 2015Co-Authors: Vilmos Komornik, Anna Chiara Lai, Paola LoretiAbstract:A classical theorem of Ingham extended Parseval's formula of the trigonometrical system to arbitrary families of exponentials satisfying a uniform Gap Condition. Later his result was extended to several dimensions, but the optimal integration domains have only been determined in very few cases. The purpose of this paper is to determine the optimal connected integration domains for all regular two-dimensional lattices.
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Discrete Ingham type inequalities with a weakened Gap Condition
arXiv: Classical Analysis and ODEs, 2007Co-Authors: Paola Loreti, Vilmos KomornikAbstract:We establish discrete Ingham type and Haraux type inequalities for exponential sums satisfying a weakened Gap Condition. They enable us to obtain discrete simultaneous observability theorems for systems of vibrating strings or beams.
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Dirichlet series and simultaneous observability: two problems solved by the same approach
Systems & Control Letters, 2002Co-Authors: Vilmos Komornik, Paola LoretiAbstract:Abstract In a classical paper, Ingham gave a simple proof of an important theorem of Polya on singular points of Dirichlet series under a uniform Gap assumption on the exponents. Bernstein generalized Polya's theorem by weakening this Gap Condition. We give a simpler proof of Bernstein's theorem by applying a recent generalization of Ingham's theorem. Furthermore, we also solve a simultaneous observability problem by using this theory.
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Ingham-Beurling type theorems with weakened Gap Conditions
Acta Mathematica Hungarica, 2002Co-Authors: Claudio Baiocchi, Vilmos Komornik, Paola LoretiAbstract:Completing a series of works begun by Wiener [34], Paley and Wiener [28] and Ingham [9], a far-reaching generalization of Parseval"s identity was obtained by Beurling [4] for nonharmonic Fourier series whose exponents satisfy a uniform Gap Condition. Later this Gap Condition was weakened by Ullrich [33], Castro and Zuazua [5], Jaffard, Tucsnak and Zuazua [11] and then in [2] in some particular cases. In this paper we prove a general theorem which contains all previous results. Furthermore, applying a different method, we prove a variant of this theorem for nonharmonic Fourier series with vector coefficients. This result, partly motivated by control-theoretical applications, extends several earlier results obtained in [15] and [2]. Finally, applying these results we obtain an optimal simultaneous observability theorem concerning a system of vibrating strings.
Alexander Elgart - One of the best experts on this subject based on the ideXlab platform.
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Smooth adiabatic evolutions with leaky power tails
Journal of Physics A: Mathematical and General, 1999Co-Authors: Joseph E. Avron, Alexander ElgartAbstract:Adiabatic evolutions with a Gap Condition have, under a range of circumstances, exponentially small tails that describe the leaking out of the spectral subspace. In general, adiabatic evolutions without a Gap Condition do not seem to have this feature. This is a known fact for eigenvalue crossing. We show that this is also the case for eigenvalues at the threshold of the continuous spectrum by considering the Friedrichs model.
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adiabatic theorem without a Gap Condition
Communications in Mathematical Physics, 1999Co-Authors: J E Avron, Alexander ElgartAbstract:We prove the adiabatic theorem for quantum evolution without the traditional Gap Condition. All that this adiabatic theorem needs is a (piecewise) twice differentiable finite dimensional spectral projection. The result implies that the adiabatic theorem holds for the ground state of atoms in quantized radiation field. The general result we prove gives no information on the rate at which the adiabatic limit is approached. With additional spectral information one can also estimate this rate.
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an adiabatic theorem without a Gap Condition
arXiv: Mathematical Physics, 1998Co-Authors: J E Avron, Alexander ElgartAbstract:The basic adiabatic theorems of classical and quantum mechanics are over-viewed and an adiabatic theorem in quantum mechanics without a Gap Condition is described.
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adiabatic theorem without a Gap Condition two level system coupled to quantized radiation field
Physical Review A, 1998Co-Authors: J E Avron, Alexander ElgartAbstract:We prove an adiabatic theorem for the ground state of the Dicke model in a slowly rotating magnetic field and show that for weak electron-photon coupling, the adiabatic time scale is close to the time scale of the corresponding two level system--without the quantized radiation field. There is a correction to this time scale which is the Lamb shift of the model. The photon field affect the rate of approach to the adiabatic limit through a logarithmic correction originating from an infrared singularity characteristic of QED.
Jochen Schmid - One of the best experts on this subject based on the ideXlab platform.
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adiabatic theorems with and without spectral Gap Condition for non semisimple spectral values
QMath12 – Mathematical Results in Quantum Mechanics, 2014Co-Authors: Jochen SchmidAbstract:We present adiabatic theorems with and without spectral Gap Condition for operators A(t) : D(A(t)) ⊂ X → X in a Banach space X. In the case with spectral Gap, the considered spectral values λ(t) ∈ σ(A(t)) are not required to be semisimple and, in particular, need not be eigenvalues. In the case without spectral Gap, the considered eigenvalues λ(t) ∈ ∂σ(A(t)) are not required to be weakly semisimple. We also discuss adiabatic theorems for operators A(t) with time-dependent domains and an application to operators defined by symmetric sesquilinear forms.
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adiabatic theorems with and without spectral Gap Condition for non semisimple spectral values
arXiv: Mathematical Physics, 2013Co-Authors: Jochen SchmidAbstract:We establish adiabatic theorems with and without spectral Gap Condition for general operators $A(t): D(A(t)) \subset X \to X$ with possibly time-dependent domains in a Banach space $X$. We first prove adiabatic theorems with uniform and non-uniform spectral Gap Condition (including a slightly extended adiabatic theorem of higher order). In these adiabatic theorems the considered spectral subsets $\sigma(t)$ have only to be compact -- in particular, they need not consist of eigenvalues. We then prove an adiabatic theorem without spectral Gap Condition for not necessarily (weakly) semisimple eigenvalues: in essence, it is only required there that the considered spectral subsets $\sigma(t) = \{ \lambda(t) \}$ consist of eigenvalues $\lambda(t) \in \partial \sigma(A(t))$ and that there exist projections $P(t)$ reducing $A(t)$ such that $A(t)|_{P(t)D(A(t))}-\lambda(t)$ is nilpotent and $A(t)|_{(1-P(t))D(A(t))}-\lambda(t)$ is injective with dense range in $(1-P(t))X$ for almost every~$t$. In all these theorems, the regularity Conditions imposed on $t \mapsto A(t)$, $\sigma(t)$, $P(t)$ are fairly mild. We explore the strength of the presented adiabatic theorems in numerous examples. And finally, we apply the adiabatic theorems for time-dependent domains to obtain -- in a very simple way -- adiabatic theorems for operators $A(t)$ defined by symmetric sesquilinear forms.
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Adiabatens\"atze mit und ohne Spektrall\"uckenbedingung
arXiv: Mathematical Physics, 2011Co-Authors: Jochen SchmidAbstract:In this work we generalize some of the previously known adiabatic theorems to situations with non-unitary evolutions in Banach spaces. We prove adiabatic theorems with uniform Gap Condition (generalizing a theorem of Abou Salem), adiabatic theorems with non-uniform Gap Condition (generalizing a theorem of Kato) and qualitative as well as quantitative adiabatic theorems without Gap Condition (generalizing theorems of Avron and Elgart, and Teufel). Additionally, we give a generalized version of an adiabatic theorem of higher order due to Nenciu. In all these adiabatic theorems the considered spectral values need not lie on the imaginary axis and in the adiabatic theorems with spectral Gap Condition and the adiabatic theorem of higher order compact subsets of the spectrum are sufficient (in particular, these subsets need not consist of eigenvalues). We explore the strength of the presented adiabatic theorems in numerous examples. In particular, we show that the theorems of the present work are more general than the previously known theorems. This work was finished in October 2010 and handed in as a diploma thesis at the mathematics department of the University of Stuttgart. The more recent results of Avron, Fraas, Graf und Grech which appeared in the meantime (in June 2011) are therefore not taken into consideration here. We point out, however, that the theorems of the present work are more general than the corresponding theorems of Avron, Fraas, Graf und Grech - with the exception of the adiabatic theorem of higher order which is in no logical relation to the adiabatic theorem of higher order of Avron, Fraas, Graf and Grech. We are about to gather the most important theorems of the present work in an article and will upload it to arXiv soon.
