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163
CurvatureInduced Bound States In Quantum Waveguides In Two And Three Dimensions
 Math. Phys
, 1995
"... Dirichlet Laplacian on curved tubes of a constant cross section in two and three dimensions is investigated. It is shown that if the tube is nonstraight and its curvature vanishes asymptotically, there is always a bound state below the bottom of the essential spectrum. An upper bound to the number ..."
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Cited by 108 (12 self)
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Dirichlet Laplacian on curved tubes of a constant cross section in two and three dimensions is investigated. It is shown that if the tube is nonstraight and its curvature vanishes asymptotically, there is always a bound state below the bottom of the essential spectrum. An upper bound to the number of these bound states in thin tubes is derived. Furthermore, if the tube is only slightly bent, there is just one bound state; we derive its behaviour with respect to the bending angle. Finally, perturbation theory of these eigenvalues in any thin tube with respect to the tube radius is constructed and some open questions are formulated. October 1994 CPT94/P.3023 anonymous ftp or gopher: cpt.univmrs.fr Unit'e Propre de Recherche 7061 1 and PHYMAT, Universit'e de Toulon et du Var, 83130 Lagarde, France duclos@naxos.unice.fr 2 Nuclear Physics Institute, AS CR, 25068 Rez near Prague and Doppler Institute, Czech Technical University, Brehov'a 7, 11519 Prague, Czech Republic exner@uj...
Scale invariance of the PNG droplet and the Airy process
 J. Stat. Phys
"... We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process, A(y). The Airy process is stationary, it has continuous sample paths, its single “time ” (fixed y) distribution is the T ..."
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Cited by 93 (13 self)
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We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process, A(y). The Airy process is stationary, it has continuous sample paths, its single “time ” (fixed y) distribution is the Tracy–Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y−2. Roughly the Airy process describes the last line of Dyson’s Brownian motion model for random matrices. Our construction uses a multi–layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation. 1 The PNG droplet The polynuclear growth (PNG) model is a simplified model for layer by layer growth [1, 2]. Initially one has a perfectly flat crystal in contact with its supersaturated vapor. Once in a while a supercritical seed is formed, which then spreads laterally by further attachment of particles at its perimeter sites. Such islands coalesce if they are in the same layer and further islands may be nucleated upon already existing ones. The PNG model ignores the lateral lattice
Localization at Weak Disorder: Some Elementary Bounds
, 1993
"... An elementary proof is given of localization for linear operators H=H o +lV, with H o translation invariant, or periodic, and V( . ) a random potential, in energy regimes which for weak disorder (l®0) are close to the unperturbed spectrum s(H o ). The analysis is within the approach introduced in ..."
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Cited by 75 (6 self)
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An elementary proof is given of localization for linear operators H=H o +lV, with H o translation invariant, or periodic, and V( . ) a random potential, in energy regimes which for weak disorder (l®0) are close to the unperturbed spectrum s(H o ). The analysis is within the approach introduced in the recent study of localization at high disorder by Aizenman and Molchanov [AM]; the localization regimes discussed in the two works being supplementary. Included also are some general auxiliary results enhancing the method, which now yields uniform exponential decay for the matrix elements <0P [a,b] e itH x> of the spectrally filtered unitary time evolution operators, with [a,b] in the relevant energy range. corrected 7/12/93 Localization at Weak Disorder 2 1. Introduction This work presents an elementary derivation of localization for time evolutions generated by linear operators consisting of a translation invariant, or periodic, part and an added random potential, at energy rang...
Almost Periodic Schrödinger Operators III. The Absolutely Continuous Spectrum in One Dimension
, 1983
"... We discuss the absolutely continuous spectrum of H = — d 2 /dx 2 + V(x) with F almost periodic and its discrete analog (hu)(n) = u(n +1) + u(n — 1) + V(ri)u(ri). Especial attention is paid to the set, A, of energies where the Lyaponov exponent vanishes. This set is known to be the essential supp ..."
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Cited by 43 (11 self)
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We discuss the absolutely continuous spectrum of H = — d 2 /dx 2 + V(x) with F almost periodic and its discrete analog (hu)(n) = u(n +1) + u(n — 1) + V(ri)u(ri). Especial attention is paid to the set, A, of energies where the Lyaponov exponent vanishes. This set is known to be the essential support of the a.c. part of the spectral measure. We prove for a.e. Fin the hull and a.e. E in A, H and h have continuum eigenfunctions, u9 with \u \ almost periodic. In the discrete case, we prove that ^4^4 with equality only if V = const. If k is the integrated density of states, we prove that on A, 2kdk/dE^π ~ 2 in the continuum case and that 2πsmπkdk/dE^.l in the discrete case. We also provide a new proof of the PasturIshii theorem and that the multiplicity of the absolutely continuous spectrum is 2.
Internal Lifshits tails for random perturbations of periodic Schrödinger operators
 Duke Math. J
, 1999
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Ground State of the Massless Nelson Model Without Infrared Cutoff in a NonFock Representation
 Rev. Math. Phys
, 2000
"... We consider a model of quantum particles coupled to a massless quantum scalar field, called the massless Nelson model, in a nonFock representation of the timezero fields which satisfy the canonical commutation relations. We show that the model has a ground state for all values of the coupling const ..."
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Cited by 26 (1 self)
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We consider a model of quantum particles coupled to a massless quantum scalar field, called the massless Nelson model, in a nonFock representation of the timezero fields which satisfy the canonical commutation relations. We show that the model has a ground state for all values of the coupling constant even in the case where no infrared cutoff is made. The nonFock representation used is inequivalent to the Fock one if no infrared cutoff is made. Key words: Nelson's model, massless quantum field, infrared divergence, ground state, canonical commutation relations, nonFock representation 1 Introduction We consider a system of N quantum particles (N 2 IN) moving on the ddimensional Euclidean space IR d (d 2 IN) under the influence of an external potential V : IR dN ! IR (Borel measurable) and coupled to a massless quantum scalar field. The model we discuss here is the socalled massless Nelson model [10]. The problem to which we address ourselves in this paper is that of existen...
The Essential Spectrum of Neumann Laplacians on Some Bounded Singular Domains
 JOURNAL OF FUNCTIONAL ANALYSIS 102, 448483 (1991)
, 1991
"... In the present paper we consider Neumann Laplacians on singular domains of the type “rooms and passages” or “combs” and we show that, in typical situations, the essential spectrum can be determined from the geometric data. Moreover, given an arbitrary closed subset S of the nonnegative reals, we co ..."
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Cited by 26 (3 self)
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In the present paper we consider Neumann Laplacians on singular domains of the type “rooms and passages” or “combs” and we show that, in typical situations, the essential spectrum can be determined from the geometric data. Moreover, given an arbitrary closed subset S of the nonnegative reals, we construct domains Q = Q(S) such that the essential spectrum of the Neumann Laplacian on R is just this set S.
Trace Class Perturbations and the Absence of Absolutely Continuous Spectra
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1989
"... We show that various Hamiltonians and Jacobi matrices have no absolutely continuous spectrum by showing that under a trace class perturbation they become a direct sum of finite matrices. ..."
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Cited by 25 (9 self)
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We show that various Hamiltonians and Jacobi matrices have no absolutely continuous spectrum by showing that under a trace class perturbation they become a direct sum of finite matrices.