Displaying similar documents to “Localisation for non-monotone Schrödinger operators”

On the existence of viable solutions for a class of second order differential inclusions

Aurelian Cernea (2002)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We prove the existence of viable solutions to the Cauchy problem x” ∈ F(x,x’), x(0) = x₀, x’(0) = y₀, where F is a set-valued map defined on a locally compact set M R 2 n , contained in the Fréchet subdifferential of a ϕ-convex function of order two.

Monotone extenders for bounded c-valued functions

Kaori Yamazaki (2010)

Studia Mathematica

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Let c be the Banach space consisting of all convergent sequences of reals with the sup-norm, C ( A , c ) the set of all bounded continuous functions f: A → c, and C A ( X , c ) the set of all functions f: X → c which are continuous at each point of A ⊂ X. We show that a Tikhonov subspace A of a topological space X is strong Choquet in X if there exists a monotone extender u : C ( A , c ) C A ( X , c ) . This shows that the monotone extension property for bounded c-valued functions can fail in GO-spaces, which provides a negative answer...

Can we assign the Borel hulls in a monotone way?

Márton Elekes, András Máthé (2009)

Fundamenta Mathematicae

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A hull of A ⊆ [0,1] is a set H containing A such that λ*(H) = λ*(A). We investigate all four versions of the following problem. Does there exist a monotone (with respect to inclusion) map that assigns a Borel/ G δ hull to every negligible/measurable subset of [0,1]? Three versions turn out to be independent of ZFC, while in the fourth case we only prove that the nonexistence of a monotone G δ hull operation for all measurable sets is consistent. It remains open whether existence here is also...

Monotone Hurwitz Numbers and the HCIZ Integral

I. P. Goulden, Mathieu Guay-Paquet, Jonathan Novak (2014)

Annales mathématiques Blaise Pascal

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In this article, we prove that the complex convergence of the HCIZ free energy is equivalent to the non-vanishing of the HCIZ integral in a neighbourhood of z = 0 . Our approach is based on a combinatorial model for the Maclaurin coefficients of the HCIZ integral together with classical complex-analytic techniques.

Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval

Kamil Kaleta (2012)

Studia Mathematica

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We prove a uniform lower bound for the difference λ₂ - λ₁ between the first two eigenvalues of the fractional Schrödinger operator ( - Δ ) α / 2 + V , α ∈ (1,2), with a symmetric single-well potential V in a bounded interval (a,b), which is related to the Feynman-Kac semigroup of the symmetric α-stable process killed upon leaving (a,b). “Uniform” means that the positive constant C α appearing in our estimate λ - λ C α ( b - a ) - α is independent of the potential V. In the general case of α ∈ (0,2), we also find a uniform lower...

What is a monotone Lagrangian cobordism?

François Charette (2012-2014)

Séminaire de théorie spectrale et géométrie

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We explain the notion of Lagrangian cobordism. A flexibility/rigidity dichotomy is illustrated by considering Lagrangian tori in 2 . Towards the end, we present a recent construction by Cornea and the author [8], of monotone cobordisms that are not trivial in a suitable sense.

The converse problem for a generalized Dhombres functional equation

L. Reich, Jaroslav Smítal, M. Štefánková (2005)

Mathematica Bohemica

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We consider the functional equation f ( x f ( x ) ) = ϕ ( f ( x ) ) where ϕ J J is a given homeomorphism of an open interval J ( 0 , ) and f ( 0 , ) J is an unknown continuous function. A characterization of the class 𝒮 ( J , ϕ ) of continuous solutions f is given in a series of papers by Kahlig and Smítal 1998–2002, and in a recent paper by Reich et al. 2004, in the case when ϕ is increasing. In the present paper we solve the converse problem, for which continuous maps f ( 0 , ) J , where J is an interval, there is an increasing homeomorphism ϕ of J such...

Semiconjugacy to a map of a constant slope

Jozef Bobok (2012)

Studia Mathematica

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It is well known that any continuous piecewise monotone interval map f with positive topological entropy h t o p ( f ) is semiconjugate to some piecewise affine map with constant slope e h t o p ( f ) . We prove this result for a class of Markov countably piecewise monotone continuous interval maps.

On the equivalence of Green functions for general Schrödinger operators on a half-space

Abdoul Ifra, Lotfi Riahi (2004)

Annales Polonici Mathematici

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We consider the general Schrödinger operator L = d i v ( A ( x ) x ) - μ on a half-space in ℝⁿ, n ≥ 3. We prove that the L-Green function G exists and is comparable to the Laplace-Green function G Δ provided that μ is in some class of signed Radon measures. The result extends the one proved on the half-plane in [9] and covers the case of Schrödinger operators with potentials in the Kato class at infinity K considered by Zhao and Pinchover. As an application we study the cone L ( ) of all positive L-solutions continuously...

Nonlinear multivalued boundary value problems

Ralf Bader, Nikolaos S. Papageorgiou (2001)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, we study nonlinear second order differential inclusions with a multivalued maximal monotone term and nonlinear boundary conditions. We prove existence theorems for both the convex and nonconvex problems, when d o m A N and d o m A = N , with A being the maximal monotone term. Our formulation incorporates as special cases the Dirichlet, Neumann and periodic problems. Our tools come from multivalued analysis and the theory of nonlinear monotone operators.

Optimal potentials for Schrödinger operators

Giuseppe Buttazzo, Augusto Gerolin, Berardo Ruffini, Bozhidar Velichkov (2014)

Journal de l’École polytechnique — Mathématiques

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We consider the Schrödinger operator - Δ + V ( x ) on H 0 1 ( Ω ) , where Ω is a given domain of d . Our goal is to study some optimization problems where an optimal potential V 0 has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

Sharp trace asymptotics for a class of 2 D -magnetic operators

Horia D. Cornean, Søren Fournais, Rupert L. Frank, Bernard Helffer (2013)

Annales de l’institut Fourier

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In this paper we prove a two-term asymptotic formula for the spectral counting function for a 2 D magnetic Schrödinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a 2 D Fermi gas submitted to a constant external magnetic field. The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for...

Time-dependent Schrödinger perturbations of transition densities

Krzysztof Bogdan, Wolfhard Hansen, Tomasz Jakubowski (2008)

Studia Mathematica

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We construct the fundamental solution of t - Δ y - q ( t , y ) for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure...

L p estimates for Schrödinger operators with certain potentials

Zhongwei Shen (1995)

Annales de l'institut Fourier

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We consider the Schrödinger operators - Δ + V ( x ) in n where the nonnegative potential V ( x ) belongs to the reverse Hölder class B q for some q n / 2 . We obtain the optimal L p estimates for the operators ( - Δ + V ) i γ , 2 ( - Δ + V ) - 1 , ( - Δ + V ) - 1 / 2 and ( - Δ + V ) - 1 where γ . In particular we show that ( - Δ + V ) i γ is a Calderón-Zygmund operator if V B n / 2 and ( - Δ + V ) - 1 / 2 , ( - Δ + V ) - 1 are Calderón-Zygmund operators if V B n .

Spectral statistics for random Schrödinger operators in the localized regime

François Germinet, Frédéric Klopp (2014)

Journal of the European Mathematical Society

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We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy E in the localized phase. Assume the density of states function is not too flat near E . Restrict it to some large cube Λ . Consider now I Λ , a small energy interval centered at E that asymptotically contains infintely many eigenvalues when the volume of the cube Λ grows to infinity. We prove that, with probability one in the...