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  • 1. Deift, Percy
    et al.
    Östensson, Jörgen
    A Riemann-Hilbert approach to some theorems on Toeplitz operators and orthogonal polynomials2006Ingår i: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 139, s. 144-171Artikel i tidskrift (Refereegranskat)
  • 2. Foerster, Clemens
    et al.
    Östensson, Jörgen
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Lieb-Thirring inequalities for higher order differential operators2008Ingår i: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 281, nr 2, s. 199-213Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We derive Lieb-Thirring inequalities for the Riesz means of eigenvalues of order gamma >= 3/4 for a fourth order operator in arbitrary dimensions. We also consider some extensions to polyharmonic operators, and to systems of such operators, in dimensions greater than one. For the critical case gamma=1-1/(2l) in dimension d=1 with l >= 2 we prove the inequality L-l,r,d(o) < L-l,L-r,L-d, which holds in contrast to current conjectures.

  • 3. Hoppe, Jens
    et al.
    Laptev, Ari
    Östensson, Jörgen
    Solitons and the Removal of Eigenvalues for Fourth Order Differential Operators2006Ingår i: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247Artikel i tidskrift (Refereegranskat)
  • 4. Its, A. R.
    et al.
    Kuijlaars, A. B. J.
    Östensson, Jörgen
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Asymptotics for a special solution of the thirty fourth Painleve equation2009Ingår i: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 22, nr 7, s. 1523-1558Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In a previous paper we studied the double scaling limit of unitary random matrix ensembles of the form Z(n,N)(-1) vertical bar det M vertical bar(2 alpha)e(-NTrV(M)) dM with alpha > -1/2. The factor vertical bar det M vertical bar(2 alpha) induces critical eigenvalue behaviour near the origin. Under the assumption that the limiting mean eigenvalue density associated with V is regular, and that the origin is a right endpoint of its support, we computed the limiting eigenvalue correlation kernel in the double scaling limit as n, N -> infinity such that n(2/3)(n/N - 1) = O(1) by using the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials on the line orthogonal with respect to the weight vertical bar x vertical bar(2 alpha)e(-NV(x)). Our main attention was on the construction of a local parametrix near the origin by means of the psi-functions associated with a distinguished solution u(alpha) of the Painleve XXXIV equation. This solution is related to a particular solution of the Painleve II equation, which, however, is different from the usual Hastings-McLeod solution. In this paper we compute the asymptotic behaviour of u(alpha)(s) as s -> +/-infinity. We conjecture that this asymptotics characterizes u(alpha) and we present supporting arguments based on the asymptotic analysis of a one-parameter family of solutions of the Painleve XXXIV equation which includes u(alpha). We identify this family as the family of tronquee solutions of the thirty fourth Painleve equation.

  • 5. Its, Alexander R.
    et al.
    Kuijlaars, Arno B. J.
    Östensson, Jörgen
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Critical Edge Behavior in Unitary Random Matrix Ensembles and the Thirty-Fourth Painleve Transcendent2008Ingår i: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, s. rnn017-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We describe a new universality class for unitary invariant random matrix ensembles. It arises in the double scaling limit of ensembles Z(n,N)(-1)vertical bar det M vertical bar (2 alpha)e(-NTrV(M)) dM, with alpha > -1/2, defined on n x n Hermitian matrices M. Assuming that the limiting mean eigenvalue density is regular and that the origin is a right endpoint of its support, we compute the limiting eigenvalue correlation kernel in the double scaling limit as n, N -> infinity such that n(2/3)(n/N-1) = O(1). We use the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials orthogonal with respect to the weight vertical bar x vertical bar(2 alpha)e(-NV(x)). Our main attention is on the construction of a local parametrix near the origin by means of the psi-functions associated with a distinguished solution of the Painleve XXXIV equation.

  • 6. Laptev, Ari
    et al.
    Shterenberg, Roman
    Sukhanov, Vladimir
    Östensson, Jörgen
    Reflectionless potentials for an ordinary differential operator of order four2006Ingår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 22, s. 135-153Artikel i tidskrift (Refereegranskat)
  • 7.
    Östensson, Jörgen
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Yafaev, Dimitri R.
    A trace formula for differential operators of arbitrary order2012Ingår i: A Panorama of Modern Operator Theory and Related Topics: The Israel Gohberg Memorial Volume / [ed] Harry Dym, Marinus A. Kaashoek, Peter Lancaster, Heinz Langer & Leonid Lerer, Basel: Birkhäuser Verlag, 2012, Vol. 218, s. 541-570Kapitel i bok, del av antologi (Refereegranskat)
    Abstract [en]

    An operator H = H0 +V where H0 = i (N is arbitrary) and V is a differential operator of order N-1 with coefficients decaying sufficiently rapidly at infinity is considered in the space H2(R). The goal of the paper is to find an expression for the trace of the difference of the resolvents (H) -1 and (H0 - z) -1 in terms of the Wronskian of appropriate solutions to the differential equation Hu = zu. This also leads to a representation for the perturbation determinant of the pair H0H.

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