1,720,997 research outputs found
A quantum hybrid with a thin antenna at the vertex of a wedge
No full textWe study the spectrum, resonances and scattering matrix of a quantum Hamiltonian on a “hybrid surface” consisting of a half-line attached by its endpoint to the vertex of a concave planar wedge. At the boundary of the wedge, outside the vertex, homogeneous Dirichlet conditions are imposed. The system is tunable by varying the measure of the angle at the vertex
Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one
Full text linkThe paper discusses nonlinear singular delta-type perturbations of the fractional Schrödinger equation ı∂ψ/∂t = (−∆)^s ψ, with s ∈ (1/2 , 1] , in dimension one. In particular, we investigate local and global well-posedness (in a strong
sense), conservation laws and the existence of blow-up solutions and standing waves
NLS Ground States on a Hybrid Plane
Full text linkWe study the existence, the nonexistence, and the shape of the ground states of a Nonlinear Schrödinger Equation on a manifold called hybrid plane, that consists of a half-line whose origin is connected to a plane. The nonlinearity is of power type, focusing and subcritical. The energy is the sum of the Nonlinear Schrödinger energies with a contact interaction on the half-line and on the plane with an additional quadratic term that couples the two components. By ground state we mean every minimizer of the energy at a fixed mass. As a first result, we single out the following rule: a ground state exists if and only if the confinement near the junction is energetically more convenient than escaping at infinity along the halfline, while escaping through the plane is shown to be never convenient. The problem of existence reduces then to a competition with the one-dimensional solitons. By this criterion, we prove existence of ground states for large and small values of the mass. Moreover, we show that at given mass a ground state exists if one of the following conditions is satisfied: the interaction at the origin of the half-line is not too repulsive; the interaction at the origin of the plane is sufficiently attractive; the coupling between the half-line and the plane is strong enough. On the other hand, nonexistence holds if the contact interactions on the half-line and on the plane are repulsive enough and the coupling is not too strong. Finally, we provide qualitative features of ground states. In particular, we show that in the presence of coupling every ground state is supported both on the half-line and on the plane and each component has the shape of a ground state at its mass for the related Nonlinear Schrödinger energy with a suitable contact interaction. These are the first results for the Nonlinear Schrödinger Equation on a manifold of mixed dimensionality
NLS ground states on the half-line with point interactions
Full text linkWe investigate the existence and the uniqueness of NLS ground states of fixed
mass on the half-line in the presence of a point interaction at the origin. The
nonlinearity is of power type, and the regime is either -subcritical or
-critical, while the point interaction is either attractive or
repulsive. In the -subcritical case, we prove that ground states exist
for every mass value if the interaction is attractive, while ground states
exist only for sufficiently large masses if the interaction is repulsive. In
the latter case, if the power is less or equal to four, ground states coincide
with the only bound state. If instead, the power is greater than four, then
there are values of the mass for which two bound states exist, and neither of
the two is a ground state, and values of the mass for which two bound states
exist, and one of them is a ground state. In the -critical case, we
prove that ground states exist for masses strictly below a critical mass value
in the attractive case, while ground states never exist in the repulsive case.Comment: 17 page
The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions
Full text linkFor kernels which are positive and integrable we show that the operator on a finite time interval enjoys a regularizing effect when applied to H"older continuous and Lebesgue functions and a ``contractive'' effect when applied to Sobolev functions. For H"older continuous functions, we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel, namely by the factor . For functions in Lebesgue spaces, we prove that an improvement always exists, and it can be expressed in terms of Orlicz integrability. Finally, for functions in Sobolev spaces, we show that the operator ``shrinks'' the norm of the argument by a factor that, as in the H"older case, depends on the function (whereas no regularization result can be obtained).
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These results can be applied, for instance, to Abel kernels and to the Volterra function , the latter being relevant for instance in the analysis of the Schr"odinger equation with concentrated nonlinearities in
Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit
Full text linkIn this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the L-2-subcritical case, they converge to the bound states of the nonlinear Schrodinger equation in the nonrelativistic limit
Costruzione di concetti e competenze matematiche nella modellizzazione di fenomeni fisici con l�uso di sistemi informatici.
No full textStability of the standing waves of the concentrated NLSE in dimension two
Full text linkIn this paper we will continue the analysis of two dimensional Schrödinger equation with a fixed, pointwise, nonlinearity started in [2, 13]. In this model, the occurrence of a blow-up phenomenon has two peculiar features: The energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves; there is no critical power nonlinearity, i.e., for every power there exist blow-up solutions. Here we study the stability properties of stationary states to verify whether the anomalies mentioned before have any counterpart on the stability features
The search for NLS ground states on a hybrid domain: Motivations, methods, and results
Full text linkWe discuss the problem of establishing the existence of the Ground States for the subcritical focusing Nonlinear Schrödinger energy on a domain made of a line and a plane intersecting at a point. The problem is physically motivated by the experimental realization of hybrid traps for Bose-Einstein Condensates, that are able to concentrate the system on structures close to the domain we consider. In fact, such a domain approximates the trap as the temperature approaches the absolute zero. The spirit of the paper is mainly pedagogical, so we focus on the formulation of the problem and on the explanation of the result, giving references for the technical points and for the proofs
Complete ionization for a non-autonomous point interaction model in d = 2
Full text linkWe consider the two dimensional Schr\"odinger equation with time dependent
delta potential, which represents a model for the dynamics of a quantum
particle subject to a point interaction whose strength varies in time. First,
we prove global well-posedness of the associated Cauchy problem under general
assumptions on the potential and on the initial datum. Then, for a
monochromatic periodic potential (which also satisfies a suitable no-resonance
condition) we investigate the asymptotic behavior of the survival probability
of a bound state of the time-independent problem. Such probability is shown to
have a time decay of order , up to lower order terms.Comment: 38 pages, 1 figure. Final version to appear on Comm. Math. Phy
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