Homoclinics for singular strong force Lagrangian systems in $${\mathbb {R}}^N$$

We will be concerned with the existence of homoclinics for Lagrangian systems in $${\mathbb {R}}^N$$ R N ($$N\ge 3 $$ N ≥ 3 ) of the form $$\frac{d}{dt}\left( \nabla \Phi (\dot{u}(t))\right) +\nabla _{u}V(t,u(t))=0$$ d dt ∇ Φ ( u ˙ ( t ) ) + ∇ u V ( t , u ( t ) ) = 0 , where $$t\in {\mathbb {R}}$$ t ∈ R , $$\Phi {:}\,{\mathbb {R}}^N\rightarrow [0,\infty )$$ Φ : R N → [ 0 , ∞ ) is a G-function in the sense of Trudinger, $$V{:}\,{\mathbb {R}}\times \left( {\mathbb {R}}^N{\setminus }\{\xi \} \right) \rightarrow {\mathbb {R}}$$ V : R × R N \ { ξ } → R is a $$C^2$$ C 2 -smooth potential with a single well of infinite depth at a point $$\xi \in {\mathbb {R}}^N{\setminus }\{0\}$$ ξ ∈ R N \ { 0 } and a unique strict global maximum 0 at the origin. Under a strong force type condition around the singular point $$\xi $$ ξ , we prove the existence of a homoclinic solution $$u{:}\,{\mathbb {R}}\rightarrow {\mathbb {R}}^N{\setminus }\{\xi \}$$ u : R → R N \ { ξ } via minimization of an action integral.

Thermal Ionization for Short-Range Potentials

We study a concrete model of a confined particle in form of a Schrödinger operator with a compactly supported smooth potential coupled to a bosonic field at positive temperature. We show, that the model exhibits thermal ionization for any positive temperature, provided the coupling is sufficiently small. Mathematically, one has to rule out that zero is an eigenvalue of the self-adjoint generator of time evolution—the Liouvillian. This will be done by using positive commutator methods with dilations in the space of scattering functions. Our proof relies on a spatial cutoff in the coupling but does otherwise not require any unnatural restrictions.

Bound states of a Klein–Gordon particle in the presence of a smooth potential well

We solve the one-dimensional time-independent Klein–Gordon equation in the presence of a smooth potential well. The bound state solutions are given in terms of the Whittaker [Formula: see text] function, and the antiparticle bound state is discussed in terms of potential parameters.

Homoclinics for singular strong force Lagrangian systems

We study the existence of homoclinic solutions for a class of Lagrangian systems $\begin{array}{} \frac{d}{dt} \end{array} $(∇Φ(u̇(t))) + ∇uV(t, u(t)) = 0, where t ∈ ℝ, Φ : ℝ2 → [0, ∞) is a G-function in the sense of Trudinger, V : ℝ × (ℝ2 ∖ {ξ}) → ℝ is a C1-smooth potential with a single well of infinite depth at a point ξ ∈ ℝ2 ∖ {0} and a unique strict global maximum 0 at the origin. Under a strong force condition around the singular point ξ, via minimization of an action integral, we will prove the existence of at least two geometrically distinct homoclinic solutions u± : ℝ → ℝ2 ∖ {ξ}.

Cooling phonon modes of a Bose condensate with uniform few body losses

We present a general analysis of the cooling produced by losses on condensates or quasi-condensates. We study how the occupations of the collective phonon modes evolve in time, assuming that the loss process is slow enough so that each mode adiabatically follows the decrease of the mean density. The theory is valid for any loss process whose rate is proportional to the jjth power of the density, but otherwise spatially uniform. We cover both homogeneous gases and systems confined in a smooth potential. For a low-dimensional gas, we can take into account the modified equation of state due to the broadening of the cloud width along the tightly confined directions, which occurs for large interactions. We find that at large times, the temperature decreases proportionally to the energy scale mc^2mc2, where mm is the mass of the particles and cc the sound velocity. We compute the asymptotic ratio of these two quantities for different limiting cases: a homogeneous gas in any dimension and a one-dimensional gas in a harmonic trap.