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This paper considers Schrödinger operators, and presents a probabilistic interpretation of the variation (or shape derivative) of the Dirichlet groundstate energy when the associated domain is perturbed. This interpretation relies on the distribution on the boundary of a stopped random process with Feynman-Kac weights. Practical computations require in addition the explicit approximation of the normal derivative of the groundstate on the boundary. We then propose to use this formulation in the...

On a q-deformed harmonic oscillator with variable linear momentum.

Harold Exton (1992)

Collectanea Mathematica

On a theorem of Jörgens and Chernoff concerning essential self-adjointness of Dirac operators.

P.A. Rejto, J.J. Landgren (1981)

Journal für die reine und angewandte Mathematik

On absorbing boundary conditions for quantum transport equations

A. Arnold (1994)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

On an example of phase-space tunneling

Shu Nakamura (1995)

Annales de l'I.H.P. Physique théorique

On coupling constant thresholds and related eigenvalue properties of Dirac operators.

M. Klaus (1985)

Journal für die reine und angewandte Mathematik

On diamagnetism and de Haas-van Alphen effect

B. Helffer, J. Sjöstrand (1990)

Annales de l'I.H.P. Physique théorique

On essential selfadjointness of the Weyl quantized relativistic Hamiltonian.

Takashi Ichinose, Tetsuo Tsuchida (1993)

Forum mathematicum

On higher order estimates in quantum electrodynamics.

Matte, Oliver (2010)

Documenta Mathematica

On $m$ -accretive Schrödinger-type operators with singular potentials on manifolds of bounded geometry.

Milatovic, Ognjen (2003)

International Journal of Mathematics and Mathematical Sciences

On $m$ -sectorial Schrödinger-type operators with singular potentials on manifolds of bounded geometry

Ognjen Milatovic (2004)

Commentationes Mathematicae Universitatis Carolinae

We consider a Schrödinger-type differential expression $H_{V} = \nabla^{*} \nabla + V$ , where $\nabla$ is a $C^{\infty}$ -bounded Hermitian connection on a Hermitian vector bundle $E$ of bounded geometry over a manifold of bounded geometry $(M, g)$ with metric $g$ and positive $C^{\infty}$ -bounded measure $d μ$ , and $V$ is a locally integrable section of the bundle of endomorphisms of $E$ . We give a sufficient condition for $m$ -sectoriality of a realization of $H_{V}$ in $L^{2} (E)$ . In the proof we use generalized Kato’s inequality as well as a result on the positivity of $u \in L^{2} (M)$ satisfying the...

On nodal sets and nodal domains on $S^{2}$ and $ℝ^{2}$

Alexandre Eremenko, Dmitry Jakobson, Nikolai Nadirashvili (2007)

Annales de l’institut Fourier

We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on $S^{2}$ . We also construct a solution of the equation $Δ u = u$ in $ℝ^{2}$ that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.

On nonlinear Schrödinger equations in exterior domains

N Burq, P Gérard, N Tzvetkov (2004)

Annales de l'I.H.P. Analyse non linéaire

On orthogonality relations for dual discrete $q$ -ultraspherical polynomials.

Groza, Valentyna A., Kachuryk, Ivan I. (2006)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

On phase segregation in nonlocal two-particle Hartree systems

Walter Aschbacher, Marco Squassina (2009)

Open Mathematics

We prove the phase segregation phenomenon to occur in the ground state solutions of an interacting system of two self-coupled repulsive Hartree equations for large nonlinear and nonlocal interactions. A self-consistent numerical investigation visualizes the approach to this segregated regime.

On quantum twist maps and spectral properties of Floquet operators

Gunther Karner (1998)

Annales de l'I.H.P. Physique théorique

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