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  • 1. Alberts, Tom
    et al.
    Binder, Ilia
    Viklund, Fredrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A Dimension Spectrum for SLE Boundary Collisions2016In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 343, no 1, p. 273-298Article in journal (Refereed)
    Abstract [en]

    We consider chordal SLE curves for , where the intersection of the curve with the boundary is a random fractal of almost sure Hausdorff dimension . We study the random sets of points at which the curve collides with the real line at a specified "angle" and compute an almost sure dimension spectrum describing the metric size of these sets. We work with the forward SLE flow and a key tool in the analysis is Girsanov's theorem, which is used to study events on which moments concentrate. The two-point correlation estimates are proved using the direct method.

  • 2. Andersson, John
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Weiss, Georg S.
    Uniform Regularity Close to Cross Singularities in an Unstable Free Boundary Problem2010In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 296, no 1, p. 251-270Article in journal (Refereed)
    Abstract [en]

    We introduce a new method for the analysis of singularities in the unstable problem Delta u = chi{u> 0}, which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of "supercharacteristic" growth of the solution, i.e. points at which the solution grows faster than the characteristic/invariant scaling of the equation would suggest. At such points the classical theory is doomed to fail, due to incompatibility of the invariant scaling of the equation and the scaling of the solution. In the case of two dimensions our result shows that in a neighborhood of the set at which the second derivatives of u are unbounded, the level set {u = 0} consists of two C-1-curves meeting at right angles. It is important that our result is not confined to the minimal solution of the equation but holds for all solutions.

  • 3.
    Arnlind, Joakim
    et al.
    Inst Hautes Etud Sci.
    Bordemann, Martin
    Univ Haute Alsace, Lab MIA.
    Hofer, Laurent
    Univ Luxembourg.
    Hoppe, Jens
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Shimada, Hidehiko
    Max Planck Inst Gravitat Phys.
    Noncommutative Riemann Surfaces by Embeddings in R-32009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 288, no 2, p. 403-429Article in journal (Refereed)
    Abstract [en]

    We introduce C-Algebras of compact Riemann surfaces ∑ as non-commutative analogues of the Poisson algebra of smooth functions on ∑. Representations of these algebrasgive rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as N → ∞. For a particular class of surfaces, interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.

  • 4.
    Aspenberg, Magnus
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Yampolsky, Michael
    Mating Non-Renormalizable Quadratic Polynomials2009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 287, no 1, p. 1-40Article in journal (Refereed)
    Abstract [en]

    In this paper we prove the existence and uniqueness of matings of the basilica with any quadratic polynomial which lies outside of the 1/2-limb of M, is non-renormalizable, and does not have any non-repelling periodic orbits.

  • 5. Bao, Z.
    et al.
    Erdős, L.
    Schnelli, Kevin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Local Law of Addition of Random Matrices on Optimal Scale2017In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 349, no 3, p. 947-990Article in journal (Refereed)
    Abstract [en]

    The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.

  • 6. Bjerklöv, Kristian
    Dynamics of the quasi-periodic Schrodinger cocycle at the lowest energy in the spectrum2007In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 272, no 2, p. 397-442Article in journal (Refereed)
    Abstract [en]

    In this paper we consider the quasi-periodic Schrodinger cocycle over T-d (d >= 1) and, in particular, its projectivization. In the regime of large coupling constants and Diophantine frequencies, we give an affirmative answer to a question posed by M. Herman [21, p.482] concerning the geometric structure of certain Strange Nonchaotic Attractors which appear in the projective dynamical system. We also show that for some phase, the lowest energy in the spectrum of the associated Schrodinger operator is an eigenvalue with an exponentially decaying eigenfunction. This generalizes [39] to the multi-frequency case (d > 1).

  • 7.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    SNA's in the Quasi-Periodic Quadratic Family2009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 286, no 1, p. 137-161Article in journal (Refereed)
    Abstract [en]

    We rigorously show that there can exist Strange Nonchaotic Attractors (SNA) in the quasi-periodically forced quadratic ( or logistic) map (theta, x) -> (theta + omega, c(theta)x(1 - x)) for certain choices of c : T bar right arrow [3/2, 4] and Diophantine omega.

  • 8. Breuer, Jonathan
    et al.
    Duits, Maurice
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Universality of Mesoscopic Fluctuations for Orthogonal Polynomial Ensembles2016In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 342, no 2, p. 491-531Article in journal (Refereed)
    Abstract [en]

    We prove that the fluctuations of mesoscopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green's function for the associated Jacobi matrices. As a particular consequence we obtain a central limit theorem for the modified Jacobi Unitary Ensembles on all mesoscopic scales.

  • 9.
    Dahl, Mattias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Gicquaud, Romain
    Sakovich, Anna
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Asymptotically Hyperbolic Manifolds with Small Mass2014In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 325, no 2, p. 757-801Article in journal (Refereed)
    Abstract [en]

    For asymptotically hyperbolic manifolds of dimension n with scalar curvature at least equal to -n(n - 1) the conjectured positive mass theorem states that the mass is non-negative, and vanishes only if the manifold is isometric to hyperbolic space. In this paper we study asymptotically hyperbolic manifolds which are also conformally hyperbolic outside a ball of fixed radius, and for which the positive mass theorem holds. For such manifolds we show that the conformal factor tends to one as the mass tends to zero.

  • 10.
    de Woul, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.
    Langmann, Edwin
    KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.
    Exact Solution of a 2D Interacting Fermion Model2012In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 314, no 1, p. 1-56Article in journal (Refereed)
    Abstract [en]

    We study an exactly solvable quantum field theory (QFT) model describing interacting fermions in 2+1 dimensions. This model is motivated by physical arguments suggesting that it provides an effective description of spinless fermions on a square lattice with local hopping and density-density interactions if, close to half filling, the system develops a partial energy gap. The necessary regularization of the QFT model is based on this proposed relation to lattice fermions. We use bosonization methods to diagonalize the Hamiltonian and to compute all correlation functions. We also discuss how, after appropriate multiplicative renormalizations, all short- and long distance cutoffs can be removed. In particular, we prove that the renormalized two-point functions have algebraic decay with non-trivial exponents depending on the interaction strengths, which is a hallmark of Luttinger-liquid behavior.

  • 11.
    Ekholm, Tomas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Frank, Rupert
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On Lieb-Thirring inequalities for Schrödinger operators with virtual level2006In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 264, no 3, p. 725-740Article in journal (Refereed)
    Abstract [en]

    We consider the operator H = - Delta- V in L-2(R-d), d >= 3. For the moments of its negative eigenvalues we prove the estimate

    tr H--(gamma) <= C-gamma,C-d integral(Rd) (V(x) - (d-2)(2)/4\x\(2))(gamma+d/2) dx, gamma > 0.

    Similar estimates hold for the one-dimensional operator with a Dirichlet condition at the origin and for the two-dimensional Aharonov-Bohm operator.

  • 12.
    Frank, Rupert L.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Siedentop, Heinz
    Warzel, Simone
    The ground state energy of heavy atoms: Relativistic lowering of the leading energy correction2008In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 278, no 2, p. 549-566Article in journal (Refereed)
    Abstract [en]

    We describe atoms by a pseudo-relativistic model that has its origin in the work of Chandrasekhar. We prove that the leading energy correction for heavy atoms, the Scott correction, exists. It turns out to be lower than in the non-relativistic description of atoms. Our proof is valid up to and including the critical coupling constant. It is based on a renormalization of the energy whose zero level we adjust to be the ground-state energy of the corresponding non-relativistic problem. This allows us to roll the proof back to results for the Schrodinger operator.

  • 13.
    Frank, Rupert
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lieb, Elliott H.
    Departments of Mathematics and Physics, Princeton University.
    Seiringer, Robert
    Department of Physics, Princeton University.
    Stability of relativistic matter with magnetic fields for nuclear charges up to the critical value2007In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 275, no 2, p. 479-489Article in journal (Refereed)
    Abstract [en]

    We give a proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge Z alpha = 2/pi.

  • 14. Froehlich, Juerg
    et al.
    Jonsson, B. Lars G.
    KTH, School of Electrical Engineering (EES), Electromagnetic Engineering.
    Lenzmann, Enno
    Boson stars as solitary waves2007In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 274, no 1, p. 1-30Article in journal (Refereed)
    Abstract [en]

    We study the nonlinear equation i theta t psi = (root-Delta+m(2) - m) psi - (vertical bar x vertical bar(-1) * vertical bar psi vertical bar(2)) psi on R-3, which is known to describe the dynamics of pseudo-relativistic boson stars in the meanfield limit. For positive mass parameters, m > 0, we prove existence of travelling solitary waves, psi(t, x) = ei t mu phi(v)(x - vt), for some mu is an element of R and with speed vertical bar v vertical bar < 1, where c = 1 corresponds to the speed of light in our units. Due to the lack of Lorentz covariance, such travelling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with v = 0). To overcome this difficulty, we introduce and study an appropriate variational problem that yields the functions phi(v) H-1/2(R-3) as minimizers, which we call boosted ground states. Our existence proof makes extensive use of concentration-compactness-type arguments. In addition to their existence, we prove orbital stability of travelling solitary waves psi(t, x) = e i t mu(v)(x - vt) and pointwise exponential decay of phi(v)(x) in x.

  • 15. Frohlich, J.
    et al.
    Gustafson, S.
    Jonsson, B. Lars G.
    KTH, Superseded Departments, Electromagnetic Theory.
    Sigal, I. M.
    Solitary wave dynamics in an external potential2004In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 250, no 3, p. 613-642Article in journal (Refereed)
    Abstract [en]

    We study the behavior of solitary-wave solutions of some generalized nonlinear Schrodinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We consider solutions of the equations with a non-vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newton's equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping.

  • 16. Gicquaud, Romain
    et al.
    Sakovich, Anna
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A Large Class of Non-Constant Mean Curvature Solutions of the Einstein Constraint Equations on an Asymptotically Hyperbolic Manifold2012In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 310, no 3, p. 705-763Article in journal (Refereed)
    Abstract [en]

    We construct solutions of the constraint equation with non constant mean curvature on an asymptotically hyperbolic manifold by the conformal method. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then in letting the exponent tend to its true value. We prove that the solutions of the sub-critical equations remain bounded which yields solutions of the constraint equation unless a certain limit equation admits a non-trivial solution. Finally, we give conditions which ensure that the limit equation admits no non-trivial solution.

  • 17.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Tkachev, Vladimir
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    The Resultant on Compact Riemann Surfaces2009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 286, no 1, p. 313-358Article in journal (Refereed)
    Abstract [en]

    We introduce a notion of the resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the Schottky double of the domain.

  • 18.
    Hekmati, Pedram
    et al.
    KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.
    Mickelsson, Jouko
    KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.
    Fractional Loop Group and Twisted K-theory2010In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 299, no 3, p. 741-763Article in journal (Refereed)
    Abstract [en]

    We study the structure of abelian extensions of the group L (q) G of q-differentiable loops (in the Sobolev sense), generalizing from the case of the central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of the supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on G is discussed.

  • 19.
    Johansson, Kurt
    KTH, Superseded Departments, Mathematics.
    Determinantal processes with number variance saturation2004In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 252, no 3-Jan, p. 111-148Article in journal (Refereed)
  • 20.
    Johansson, Kurt
    KTH, Superseded Departments, Mathematics.
    Discrete polynuclear growth and determinantal processes2003In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 242, no 2-Jan, p. 277-329Article in journal (Refereed)
  • 21.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Non-colliding Brownian Motions and the Extended Tacnode Process2013In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 319, no 1, p. 231-267Article in journal (Refereed)
    Abstract [en]

    We consider non-colliding Brownian motions with two starting points and two endpoints. The points are chosen so that the two groups of Brownian motions just touch each other, a situation that is referred to as a tacnode. The extended kernel for the determinantal point process at the tacnode point is computed using new methods and given in a different form from that obtained for a single time in previous work by Delvaux, Kuijlaars and Zhang. The form of the extended kernel is also different from that obtained for the extended tacnode kernel in another model by Adler, Ferrari and van Moerbeke. We also obtain the correlation kernel for a finite number of non-colliding Brownian motions starting at two points and ending at arbitrary points.

  • 22.
    Johansson, Kurt
    KTH, Superseded Departments, Mathematics.
    Shape fluctuations and random matrices2000In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 209, no 2, p. 437-476Article in journal (Refereed)
  • 23.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Two Time Distribution in Brownian Directed Percolation2016In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, p. 1-52Article in journal (Refereed)
    Abstract [en]

    In the zero temperature Brownian semi-discrete directed polymer we study the joint distribution of two last-passage times at positions ordered in the time-like direction. This is the situation when we have the slow de-correlation phenomenon. We compute the limiting joint distribution function in a scaling limit. This limiting distribution is given by an expansion in determinants that is not a Fredholm expansion. A somewhat similar looking formula was derived non-rigorously in a related model by Dotsenko.

  • 24.
    Johansson, Kurt
    KTH, Superseded Departments, Mathematics.
    Universality of the local spacing distribution in certain ensembles of hermitian Wigner matrices2001In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 215, no 3, p. 683-705Article in journal (Refereed)
  • 25. Kaloshin, Vadim
    et al.
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    An Example of a Nearly Integrable Hamiltonian System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension2012In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 315, no 3, p. 643-697Article in journal (Refereed)
    Abstract [en]

    The famous ergodic hypothesis suggests that for a typical Hamiltonian on a typical energy surface nearly all trajectories are dense. KAM theory disproves it. Ehrenfest (The Conceptual Foundations of the Statistical Approach in Mechanics. Ithaca, NY: Cornell University Press, 1959) and Birkhoff (Collected Math Papers. Vol 2, New York: Dover, pp 462-465, 1968) stated the quasi-ergodic hypothesis claiming that a typical Hamiltonian on a typical energy surface has a dense orbit. This question is wide open. Herman (Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998). Doc Math 1998, Extra Vol II, Berlin: Int Math Union, pp 797-808, 1998) proposed to look for an example of a Hamiltonian near with a dense orbit on the unit energy surface. In this paper we construct a Hamiltonian which has an orbit dense in a set of maximal Hausdorff dimension equal to 5 on the unit energy surface.

  • 26.
    Kreiss, Heinz -Otto
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Reula, O.
    Sarbach, O.
    Winicour, J.
    Boundary Conditions for Coupled Quasilinear Wave Equations with Application to Isolated Systems2009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 289, no 3, p. 1099-1129Article in journal (Refereed)
    Abstract [en]

    We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form [0, T] x I pound, where I pound is a compact manifold with smooth boundaries a,I pound. By using an appropriate reduction to a first order symmetric hyperbolic system with maximal dissipative boundary conditions, well posedness of such problems is established for a large class of boundary conditions on a,I pound. We show that our class of boundary conditions is sufficiently general to allow for a well posed formulation for different wave problems in the presence of constraints and artificial, nonreflecting boundaries, including Maxwell's equations in the Lorentz gauge and Einstein's gravitational equations in harmonic coordinates. Our results should also be useful for obtaining stable finite-difference discretizations for such problems.

  • 27.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Rosenzweig, Lior
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Superscars for arithmetic toral point scatterers2016In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 349, no 1, p. 329-360Article in journal (Refereed)
    Abstract [en]

    We investigate eigenfunctions of the Laplacian perturbed by a delta potential on the standard tori in dimensions . Despite quantum ergodicity holding for the set of "new" eigenfunctions we show that superscars occur-there is phase space localization along families of closed orbits, in the sense that some semiclassical measures contain a finite number of Lagrangian components of the form , for uniformly bounded from below. In particular, for both and , eigenfunctions fail to equidistribute in phase space along an infinite subsequence of new eigenvalues. For , we also show that some semiclassical measures have both strongly localized momentum marginals and non-uniform quantum limits (i.e., the position marginals are non-uniform). For , superscarred eigenstates are quite rare, but for we show that the phenomenon is quite common-with denoting the counting function for the new eigenvalues below x, there are eigenvalues with the property that any semiclassical limit along these eigenvalues exhibits superscarring.

  • 28. Kurlberg, Pär
    et al.
    Rudnick, Z.
    On quantum ergodicity for linear maps of the torus2001In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 222, no 1, p. 201-227Article in journal (Refereed)
    Abstract [en]

    We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (cat maps). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence., all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly distributed. A key step in the argument is to show that for a hyperbolic matrix in the modular group. there is a density one sequence of integers N for which its order (or period) modulo N is somewhat larger than rootN.

  • 29.
    Langmann, Edwin
    KTH, Superseded Departments, Physics.
    Second quantization of the elliptic Calogero-Sutherland model2004In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 247, no 2, p. 321-351Article in journal (Refereed)
    Abstract [en]

    We construct a quantum field theory model of anyons on a circle and at finite temperature. We find an anyon Hamiltonian providing a second quantization of the elliptic Calogero-Sutherland model. This allows us to prove a remarkable identity which is a starting point for an algorithm to construct eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland Hamiltonian.

  • 30.
    Langmann, Edwin
    et al.
    KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.
    Lebowitz, Joel L.
    Departments of Mathematics and Physics, Rutgers University.
    Mastropietro, Vieri
    Dipartimento di Matematica, Università degli Studi di Milano.
    Moosavi, Per
    KTH, School of Engineering Sciences (SCI), Theoretical Physics.
    Steady states and universal conductance in a quenched Luttinger model2016In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, p. 1-32Article in journal (Refereed)
    Abstract [en]

    We obtain exact analytical results for the evolution of a 1+1-dimensional Luttinger model prepared in a domain wall initial state, i.e., a state with different densities on its left and right sides. Such an initial state is modeled as the ground state of a translation invariant Luttinger Hamiltonian (Formula presented.) with short range non-local interaction and different chemical potentials to the left and right of the origin. The system evolves for time t > 0 via a Hamiltonian (Formula presented.) which differs from (Formula presented.) by the strength of the interaction. Asymptotically in time, as (Formula presented.), after taking the thermodynamic limit, the system approaches a translation invariant steady state. This final steady state carries a current I and has an effective chemical potential difference (Formula presented.) between right- (+) and left- (−) moving fermions obtained from the two-point correlation function. Both I and (Formula presented.) depend on (Formula presented.) and (Formula presented.). Only for the case (Formula presented.) does (Formula presented.) equal the difference in the initial left and right chemical potentials. Nevertheless, the Landauer conductance for the final state, (Formula presented.), has a universal value equal to the conductance quantum (Formula presented.) for the spinless case.

  • 31.
    Laptev, Ari
    et al.
    KTH, Superseded Departments, Mathematics.
    Naboko, S.
    Safronov, O.
    On new relations between spectral properties of Jacobi matrices and their coefficients2003In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 241, no 1, p. 91-110Article in journal (Refereed)
    Abstract [en]

    We study the spectral properties of Jacobi matrices. By using ``higher order'' trace formulae we obtain a result relating the properties of the elements of Jacobi matrices and the corresponding spectral measures. Complicated expressions for traces of some operators can be magically simplified allowing us to apply induction arguments. Our theorems are generalizations of a recent result of R. Killip and B. Simon [17].

  • 32.
    Laptev, Ari
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Naboko, S.
    Safronov, Oleg
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Absolutely continuous spectrum of Schrödinger operators with slowly decaying and oscillating potentials2005In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 253, no 3, p. 611-631Article in journal (Refereed)
    Abstract [en]

    The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional Schrodinger operator is essentially supported by [0, infinity). Our main theorem states that this property is preserved for slowly decaying potentials provided that there are some oscillations with respect to one of the variables.

  • 33. Laptev, Ari
    et al.
    Safronov, Oleg
    Eigenvalue Estimates for Schrodinger Operators with Complex Potentials2009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 292, no 1, p. 29-54Article in journal (Refereed)
    Abstract [en]

    We discuss properties of eigenvalues of non-self-adjoint Schrodinger operators with complex-valued potential V. Among our results are estimates of the sum of powers of imaginary parts of eigenvalues by the L-p-norm of JV.

  • 34.
    Lenells, Jonatan
    Baylor University, United States .
    Boundary Value Problems for the Stationary Axisymmetric Einstein Equations: A Disk Rotating Around a Black Hole2011In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 304, no 3, p. 585-635Article in journal (Refereed)
    Abstract [en]

    We solve a class of boundary value problems for the stationary axisymmetric Einstein equations involving a disk rotating around a central black hole. The solutions are given explicitly in terms of theta functions on a family of hyperelliptic Riemann surfaces of genus 4. In the absence of a disk, they reduce to the Kerr black hole. In the absence of a black hole, they reduce to the Neugebauer-Meinel disk.

  • 35.
    Lenells, Jonatan
    et al.
    Leibniz Universität Hannover, Germany .
    Misiołek, G.
    Tiǧlay, F.
    Integrable evolution equations on spaces of tensor densities and their peakon solutions2010In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 299, no 1, p. 129-161Article in journal (Refereed)
    Abstract [en]

    We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct "peakon" and "multi-peakon" solutions for all λ ≠ 0, 1, and "shock-peakons" for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold's approach to Euler equations on Lie groups.

  • 36.
    Lester, Stephen
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Rudnick, Zeev
    Small Scale Equidistribution of Eigenfunctions on the Torus2017In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 350, no 1, p. 279-300Article in journal (Refereed)
    Abstract [en]

    We study the small scale distribution of the L-2 mass of eigenfunctions of the Laplacian on the flat torus T-d. Given an orthonormal basis of eigenfunctions, we show the existence of a density one subsequence whose L-2 mass equidistributes at small scales. In dimension two our result holds all the way down to the Planck scale. For dimensions d = 3, 4 we can restrict to individual eigenspaces and show small scale equidistribution in that context. We also study irregularities of quantum equidistribution: We construct eigenfunctions whose L-2 mass does not equidistribute at all scales above the Planck scale. Additionally, in dimension d = 4 we show the existence of eigenfunctions for which the proportion of L-2 mass in small balls blows up at certain scales.

  • 37.
    Lundholm, Douglas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Portmann, Fabian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Solovej, Jan Philip
    University of Copenhagen, Denmark.
    Lieb-Thirring Bounds for Interacting Bose Gases2015In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 335, no 2, p. 1019-1056Article in journal (Refereed)
    Abstract [en]

    We study interacting Bose gases and prove lower bounds for the kinetic plus interaction energy of a many-body wave function in terms of its particle density. These general estimates are then applied to various types of interactions, including hard sphere (in 3D) and hard disk (in 2D) as well as a general class of homogeneous potentials.

  • 38.
    Lundholm, Douglas
    et al.
    University of Copenhagen.
    Solovej, Jan Philip
    University of Copenhagen.
    Hardy and Lieb-Thirring Inequalities for Anyons2013In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 322, no 3, p. 883-908Article in journal (Refereed)
    Abstract [en]

    We consider the many-particle quantum mechanics of anyons, i.e. identical particles in two space dimensions with a continuous statistics parameter α∈[0,1] ranging from bosons (α = 0) to fermions (α = 1). We prove a (magnetic) Hardy inequality for anyons, which in the case that α is an odd numerator fraction implies a local exclusion principle for the kinetic energy of such anyons. From this result, and motivated by Dyson and Lenard’s original approach to the stability of fermionic matter in three dimensions, we prove a Lieb-Thirring inequality for these types of anyons.

  • 39. Martens, Marco
    et al.
    Winckler, Björn
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On the Hyperbolicity of Lorenz Renormalization2014In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 325, no 1, p. 185-257Article in journal (Refereed)
    Abstract [en]

    We consider infinitely renormalizable Lorenz maps with real critical exponent alpha > 1 of certain monotone combinatorial types. We prove the existence of periodic points of the renormalization operator, and that each map in the limit set of renormalization has an associated two-dimensional strong unstable manifold. For monotone families of Lorenz maps we prove that each infinitely renormalizable combinatorial type has a unique representative within the family. We also prove that each infinitely renormalizable map has no wandering intervals, is ergodic, and has a uniquely ergodic minimal Cantor attractor of measure zero.

  • 40.
    Mickelsson, Jouko
    et al.
    KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.
    Pellonpää, Juha-Pekka
    Families index theorem in supersymmetric WZW model and twisted K-theory: The SU(2) case2007In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 271, no 3, p. 775-789Article in journal (Refereed)
    Abstract [en]

    The construction of twisted K-theory classes on a compact Lie group is reviewed using the supersymmetric Wess-Zumino-Wittenmodel on a cylinder. The Quillen superconnection is introduced for a family of supercharges parametrized by a compact Lie group and the Chern character is explicitly computed in the case of SU( 2). For large euclidean time, the character form is localized on a D-brane.

  • 41.
    Olofsson, Rikard
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Large Supremum Norms and Small Shannon Entropy for Hecke Eigenfunctions of Quantized Cat Maps2009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 286, no 3, p. 1051-1072Article in journal (Refereed)
    Abstract [en]

    This paper concerns the behavior of eigenfunctions of quantized cat maps and in particular their supremum norm. We observe that for composite integer values of N, the inverse of Planck's constant, some of the desymmetrized eigenfunctions have very small support and hence very large supremum norm. We also prove an entropy estimate and show that our functions satisfy equality in this estimate. In the case when N is a prime power with even exponent we calculate the supremum norm for a large proportion of all desymmetrized eigenfunctions and we find that for a given N there is essentially at most four different values these assume.

  • 42.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Instability of Spatially Homogeneous Solutions in the Class of T-2-Symmetric Solutions to Einstein's Vacuum Equations2015In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 334, no 3, p. 1299-1375Article in journal (Refereed)
    Abstract [en]

    In the subject of cosmology, spatially homogeneous solutions are often used to model the universe. It is therefore of interest to ask what happens when perturbing into the spatially inhomogeneous regime. To this end, we, in the present paper, study the future asymptotics of solutions to Einstein's vacuum equations in the case of T-2-symmetry. It turns out that in this setting, whether the solution is spatially homogeneous or not can be characterized in terms of the asymptotics of one variable appearing in the equations; there is a monotonic function such that if its limit is finite, then the solution is spatially homogeneous and if the limit is infinite, then the solution is spatially inhomogeneous. In particular, regardless of how small the initial perturbation away from spatial homogeneity is, the resulting asymptotics are very different. Using spatially homogeneous solutions as models is therefore, in this class, hard to justify.

  • 43. Ringström, Hans
    On curvature decay in expanding cosmological models2006In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 264, no 3, p. 613-630Article in journal (Refereed)
    Abstract [en]

    Consider a globally hyperbolic cosmological spacetime. Topologically, the spacetime is then a compact 3-manifold in cartesian product with an interval. Assuming that there is an expanding direction, is there any relation between the topology of the 3-manifold and the asymptotics? In fact, there is a result by Michael Anderson, where he obtains relations between the long-time evolution in General Relativity and the geometrization of 3-manifolds. In order to obtain conclusions however, he makes assumptions concerning the rate of decay of the curvature as proper time tends to infinity. It is thus of interest to find out if such curvature decay conditions are always fulfilled. We consider here the Gowdy spacetimes, for which we prove that the decay condition holds. However, we observe that for general Bianchi VIII spacetimes, the curvature decay condition does not hold, but that some aspects of the expected asymptotic behaviour are still true.

  • 44.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Power Law Inflation2009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 290, no 1, p. 155-218Article in journal (Refereed)
    Abstract [en]

    The subject of this paper is Einstein's equations coupled to a non-linear scalar field with an exponential potential. The problem we consider is that of proving future global non-linear stability of a class of spatially locally homogeneous solutions to the equations. There are solutions on R(+)xR(n) with accelerated expansion of power law type. We prove a result stating that if we have initial data that are close enough to those of such a solution on a ball of a certain radius, say B-4R0 (p), then all causal geodesics starting in B-R0 (p) are complete to the future in the maximal globally hyperbolic development of the data we started with. In other words, we only make local assumptions in space and obtain global conclusions in time. We also obtain asymptotic expansions in the region over which we have control. As a consequence of this result and the fact that one can analyze the asymptotic behaviour in most of the spatially homogeneous cases, we obtain quite a general stability statement in the spatially locally homogeneous setting.

  • 45.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The singular set for the composite membrane problem2007In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 271, no 1, p. 93-101Article in journal (Refereed)
    Abstract [en]

    In this paper we study the behavior of the singular set {u = vertical bar del u vertical bar = 0}, for solutions u to the free boundary problem Delta u = f chi{u >= 0} - g chi{u < 0}, with f > 0, f (x) + g(x) < 0, and f, g is an element of C-alpha. Such problems arise in an eigenvalue optimization for composite membranes. Here we show that if for a singular point z is an element of {u =del u = 0}, there are r(0) > 0, and c(0) > 0 such that the density assumption vertical bar{u < 0} boolean AND B-r (z)vertical bar >= c(0)r(2), for all r < r(0), holds, then z is isolated. The density assumption can be motivated by the following example: u = x(1)(2), f = 2, g < -2, and {u < 0} = theta.

  • 46.
    Szepessy, Anders
    KTH, Superseded Departments, Mathematics.
    Dynamics and stability of a weak detonation wave1999In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 202, no 3, p. 547-569Article in journal (Refereed)
    Abstract [en]

    One dimensional weak detonation waves of a basic reactive shock wave model are proved to be nonlinearly stable, i.e. initially perturbed waves tend asymptotically to translated weak detonation waves. This model system was derived as the low Math number limit of the one component reactive Navier-Stokes equations by Majda and Roytburd [SIAM J. Sci. Stat. Comput. 43, 1086-1118 (1983)], and its weak detonation waves have been numerically observed as stable. The analysis shows in particular the key role of the new nonlinear dynamics of the position of the shock wave, The shock translation solves a nonlinear integral equation, obtained by Green's function techniques, and its solution is estimated by observing that the kernel can be split into a dominating convolution operator and a remainder. The inverse operator of the convolution and detailed properties of the traveling wave reduce, by monotonicity, the remainder to a small L-1 perturbation.

  • 47.
    Wigman, Igor
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Fluctuations of the nodal length of random spherical harmonics2010In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 298, no 3, p. 787-831Article in journal (Refereed)
    Abstract [en]

    Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree n having Laplace eigenvalue E = n(n + 1). We study the length distribution of the nodal lines of random spherical harmonics. It is known that the expected length is of order n. It is natural to conjecture that the variance should be of order n, due to the natural scaling. Our principal result is that, due to an unexpected cancelation, the variance of the nodal length of random spherical harmonics is of order log n. This behaviour is consistent with the one predicted by Berry for nodal lines on chaotic billiards (Random Wave Model). In addition we find that a similar result is applicable for "generic" linear statistics of the nodal lines.

1 - 47 of 47
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