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  • 1. Balinsky, A.
    et al.
    Laptev, Ari
    KTH, Superseded Departments, Mathematics.
    Sobolev, A. V.
    Generalized Hardy inequality for the magnetic Dirichlet forms2004In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 116, no 4-Jan, p. 507-521Article in journal (Refereed)
    Abstract [en]

    We obtain lower bounds for the magnetic Dirichlet form in dimensions d greater than or equal to 2. For d = 2 the results generalize a well known lower bound by the magnetic field strength: we replace the actual magnetic field B by an non-vanishing effective field which decays outside the support of B as dist( x, supp B)(-2). In the case d greater than or equal to 3 we establish that the magnetic form is bounded from below by the magnetic field strength, if one assumes that the field does not vanish and its direction is slowly varying.

  • 2. Chanillo, S.
    et al.
    Helffer, B.
    Laptev, Ari
    KTH, Superseded Departments, Mathematics.
    Nonlinear eigenvalues and analytic hypoellipticity2004In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 209, no 2, p. 425-443Article in journal (Refereed)
    Abstract [en]

    Motivated by the problem of analytic hypoellipticity, we show that a special family of compact non-self-adjoint operators has a nonzero eigenvalue. We recover old results obtained by ordinary differential equations techniques and show how it can be applied to the higher dimensional case. This gives in particular a new class of hypoelliptic, but not analytic hypoelliptic operators.

  • 3. Dolbeault, Jean
    et al.
    Laptev, Ari
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Loss, Michael
    Lieb-Thirring inequalities with improved constants2008In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 10, no 4, p. 1121-1126Article in journal (Refereed)
    Abstract [en]

    Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one dimension. This allows us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multidimensional Schrodinger operators.

  • 4. Frank, Rupert L.
    et al.
    Laptev, Ari
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Inequalities between Dirichlet and Neumann Eigenvalues on the Heisenberg Group2010In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, no 15, p. 2889-2902Article in journal (Refereed)
    Abstract [en]

    We prove that for any domain in the Heisenberg group the (k+1)th Neumann eigenvalue of the sub-Laplacian is strictly less than the kth Dirichlet eigenvalue. As a byproduct, we obtain similar inequalities for the Euclidean Laplacian with a homogeneous magnetic field.

  • 5.
    Frank, Rupert L.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Princeton University, United States.
    Laptev, Ari
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Imperial College London, United Kingdom.
    Molchanov, Stanislav
    Eigenvalue estimates for magnetic Schrödinger operators in domains2008In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 136, no 12, p. 4245-4255Article in journal (Refereed)
    Abstract [en]

    Inequalities are derived for sums and quotients of eigenvalues of magnetic Schrodinger operators with non-negative electric potentials in domains. The bounds reflect the correct order of growth in the semi-classical limit.

  • 6.
    Frank, Rupert L.
    et al.
    Princeton Univ, Dept Math, Princeton, NJ 08544 USA..
    Laptev, Ari
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom.
    Seiringer, Robert
    Princeton Univ, Dept Phys, Princeton, NJ 08544 USA..
    A Sharp Bound on Eigenvalues of Schrodinger Operators on the Half-line with Complex-valued Potentials2011In: SPECTRAL THEORY AND ANALYSIS / [ed] Janas, J Kurasov, P Laptev, A Naboko, S Stolz, G, BIRKHAUSER VERLAG AG , 2011, p. 39-+Conference paper (Refereed)
    Abstract [en]

    We derive a sharp bound on the location of non-positive eigenvalues of Schrodinger operators on the half-line with complex-valued potentials.

  • 7.
    Frank, Rupert
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Laptev, Ari
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lieb, Elliott H.
    Department of Physics, Princeton University.
    Seiringer, Robert
    Department of Physics, Princeton University.
    Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials2006In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 77, no 3, p. 309-316Article in journal (Refereed)
    Abstract [en]

    Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schrodinger operator with a complex-valued potential.

  • 8. Gordon, A.
    et al.
    Holt, J.
    Laptev, Ari
    Molchanov, S.
    On the Simon-Spencer theorem2008Article in journal (Refereed)
    Abstract [en]

    This paper presents a generalization of the classical result by B. Simon and T. Spencer on the absence of absolutely continuous spectrum for the continuous one-dimensional Schrodinger operator with an unbounded potential.

  • 9.
    Hansson, Anders M
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Laptev, Ari
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Sharp spectral inequalitites for the Heisenberg Laplacian2008In: Groups and analysis: the legacy of Hermann Weyl / [ed] Tent, Katrin, Cambridge: Cambridge University Press , 2008, p. 13 sidor-Chapter in book (Other academic)
  • 10. Hoffmann-Ostenhof, M.
    et al.
    Hoffmann-Ostenhof, T.
    Laptev, Ari
    KTH, Superseded Departments, Mathematics.
    A geometrical version of Hardy's inequality2002In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 189, no 2, p. 539-548Article in journal (Refereed)
    Abstract [en]

    We prove a version of Hardy's type inequality in a domain Omega subset of R-n which involves the distance to the boundary and the volume of Omega. In particular, we obtain a result which gives a positive answer to a question asked by H. Brezis and M. Marcus.

  • 11. Hoffmann-Stenhof, M.
    et al.
    Hoffmann-Ostenhof, T.
    Laptev, Ari
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Tidblom, J.
    Many-particle hardy inequalities2008In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 77, p. 99-114Article in journal (Refereed)
    Abstract [en]

    In this paper we prove three different types of the so-called many-particle Hardy inequalities. One of them is a 'classical type' which is valid in any dimension d not equal 2. The second type deals with 2-dimensional magnetic Dirichlet forms, where every particle is supplied with a solenoid. Finally we show that Hardy inequalities for fermionic functions (fully anti-symmetric for d = 1, 2) hold true in all dimensions.

  • 12.
    Hoppe, Jens
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Laptev, Ari
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Ostensson, J.
    Solitons and the removal of eigenvalues for fourth-order differential operators2006In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247Article in journal (Refereed)
    Abstract [en]

    A nonlinear functional Q[u, v] is given that governs the loss, respectively gain, of ( doubly degenerate) eigenvalues of fourth-order differential operators L = partial derivative(4) + partial derivative u partial derivative + v on the line. Apart from factorizing L as A*A + E-0, providing several explicit examples, and deriving various relations between u, v, and the eigenfunctions of L, we find u and v such that L is isospectral to the free operator L-0 = partial derivative(4) up to one (multiplicity 2) eigenvalue E-0 < 0. Not unexpectedly, this choice of u, v leads to exact solutions of the corresponding time-dependent PDE's. Removal of eigenvalues allows us to obtain a sharp Lieb-Thirring inequality for a class of operators L whose negative eigenvalues are of multiplicity two.

  • 13. Hundertmark, D.
    et al.
    Laptev, Ari
    KTH, Superseded Departments, Mathematics.
    Weidl, T.
    New bounds on the Lieb-Thirring constants2000In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 140, no 3, p. 693-704Article in journal (Refereed)
    Abstract [en]

    Improved estimates on the constants L(gamma,)d, foT 1/2 < gamma < 3/2, d epsilon N, in the inequalities for the eigenvalue moments of Schrodinger operators are established.

  • 14.
    Langmann, Edwin
    et al.
    KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.
    Laptev, Ari
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Paufler, Cornelius
    Singular factorizations, self-adjoint extensions and applications to quantum many-body physics2006In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 39, no 5, p. 1057-1071Article in journal (Refereed)
    Abstract [en]

    We study self-adjoint operators defined by factorizing second-order differential operators in first-order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum-mechanical models such as the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero-Sutherland type.

  • 15.
    Laptev, Ari
    KTH, Superseded Departments, Mathematics.
    The negative spectrum of a class of two-dimensional Schrodinger operators with potentials depending only on radius2000In: Functional analysis and its applications, ISSN 0016-2663, E-ISSN 1573-8485, Vol. 34, no 4, p. 305-307Article in journal (Refereed)
  • 16.
    Laptev, Ari
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Naboko, S.
    Safronov, O.
    A Szego condition for a multidimensional Schrodinger operator2005In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 219, no 2, p. 285-305Article in journal (Refereed)
    Abstract [en]

    We consider spectral properties of a Schrodinger operator perturbed by a potential vanishing at infinity and prove that the corresponding spectral measure satisfies a Szego-type condition.

  • 17.
    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].

  • 18.
    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.

  • 19. 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.

  • 20.
    Laptev, Ari
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Safronov, Oleg
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    The negative discrete spectrum of a class of two-dimensional Schrödinger operators with magnetic fields2005In: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 41, no 2, p. 107-117Article in journal (Refereed)
    Abstract [en]

    We obtain an asymptotic formula for the number of negative eigenvalues of a class of two-dimensional Schrodinger operators with small magnetic fields. This number increases as a coupling constant of the magnetic field tends to zero.

  • 21.
    Laptev, Ari
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shterenberg, R.
    Sukhanov, V.
    Ostensson, J.
    Reflectionless potentials for an ordinary differential operator of order four2006In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 22, no 1, p. 135-153Article in journal (Refereed)
    Abstract [en]

    The aim of this paper is to construct exact formulae for reflectionless potentials for ordinary differential operators of order four. They lead to soliton-type solutions which are well known for one-dimensional Schrodinger operators. Such solitons are solutions of some nonlinear integrable systems which appeared in Gelfand and Dikey 1976 Funct. Anal. Appl. 10 13-29 (in Russian) (see also Hoppe et al 2003 Preprint math-ph/0311011).

  • 22.
    Laptev, Ari
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Shterenberg, Roman
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Sukhanov, Vladimir
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Inverse spectral problems for Schrodinger operators with energy depending potentials2007In: Probability and Mathematical Physics: A Volume in Honor of Stanislav Molchanov / [ed] Dawson, DA; Jaksic, V; Vainberg, B, 2007, Vol. 42, p. 341-351Conference paper (Refereed)
    Abstract [en]

    We study an inverse problem for a class of Schrodinger operators with energy depending potentials. In particular, we show that introduction of the discrete spectrum generically does not lead to singularities of the corresponding soliton solutions. In our last chapter we derive some new trace formulas which could be considered as generalization of a standard trace formulas for Schrodinger operators.

  • 23.
    Laptev, Ari
    et al.
    KTH, Superseded Departments, Mathematics.
    Sigal, I. M.
    Global Fourier integral operators and semiclassical asymptotics2000In: Reviews in Mathematical Physics, ISSN 0129-055X, Vol. 12, no 5, p. 749-766Article, review/survey (Refereed)
    Abstract [en]

    In this paper we introduce a class of semiclassical Fourier integral operators with global complex phases approximating the fundamental solutions (propagators) for time-dependent Schrodinger equations. Our construction is elementary, it is inspired by the joint work of the first author with Yu. Safarov and D. Vasiliev. We consider several simple but basic examples.

  • 24.
    Laptev, Ari
    et al.
    KTH, Superseded Departments, Mathematics.
    Weidl, T.
    Sharp Lieb-Thirring inequalities in high dimensions2000In: Acta Mathematica, ISSN 0001-5962, E-ISSN 1871-2509, Vol. 184, no 1, p. 87-111Article in journal (Refereed)
1 - 24 of 24
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