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239
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 183 (21 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
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 159 (16 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...
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 116 (7 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 58 (10 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 Mathematical J
, 1999
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Periodic nonlinear Schrödinger equation with application to photonic crystals
 Milan J. Math
, 2005
"... We present basic results, known and new, on nontrivial solutions of periodic stationary nonlinear Schrödinger equations. We also sketch an application to nonlinear optics and discuss some open problems. 0 ..."
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Cited by 46 (3 self)
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We present basic results, known and new, on nontrivial solutions of periodic stationary nonlinear Schrödinger equations. We also sketch an application to nonlinear optics and discuss some open problems. 0
Wellposedness for semirelativistic Hartree equations of critical type
 Math. Phys. Anal. Geom
"... We prove local and global wellposedness for semirelativistic, nonlinear Schrödinger equations i∂tu = √ − ∆ + m 2 u+F(u) with initial data in H s (R 3), s ≥ 1/2. Here F(u) is a critical Hartree nonlinearity that corresponds to Coulomb or Yukawa type selfinteractions. For focusing F(u), which aris ..."
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Cited by 39 (9 self)
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We prove local and global wellposedness for semirelativistic, nonlinear Schrödinger equations i∂tu = √ − ∆ + m 2 u+F(u) with initial data in H s (R 3), s ≥ 1/2. Here F(u) is a critical Hartree nonlinearity that corresponds to Coulomb or Yukawa type selfinteractions. For focusing F(u), which arise in the quantum theory of boson stars, we derive a sufficient condition for globalintime existence in terms of a solitary wave ground state. Our proof of wellposedness does not rely on Strichartz type estimates, and it enables us to add external potentials of a general class. 1
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 37 (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.