Narrow Results

One electron system minimally coupled to a quantized radiation field is considered. It is assumed that the quantized radiation field is massless, and no infrared cutoff is imposed. The Hamiltonian, H, of this system is defined as a self-adjoint operator acting on L² (ℝ³) ⊗ ℱ ≅ L² (ℝ³; ℱ), where ℱ is the Boson Fock space over L² (ℝ³ × {1, 2}). It is shown that the ground state, ψ_g, of H belongs to , where N denotes the number operator of ℱ. Moreover, it is shown that for almost every electron position variable x ∈ ℝ³ and for arbitrary k ≥ 0, ‖(1 ⊗ N^k/2) ψ_g (x)‖_ℱ ≤ D_k e^-δ|x|m+1 with some constants m ≥ 0, D_k > 0, and δ > 0 independent of k. In particular for 0 < β < δ/2 is obtained.

We consider a quantum system described by a concrete C*-algebra acting on a Hilbert space ℋ with a vector state ω induced by a cyclic vector Ω and a unitary evolution U_t such that U_tΩ = Ω, ∀t ∈ ℝ. It is proved that this vector state is a ground state if and only if it is non-faithful and completely passive. This version of a result of Pusz and Woronowicz is reviewed, emphasizing other related aspects: passivity from the point of view of moving observers and stability with respect to local perturbations of the dynamics.

Generalized Pauli–Fierz Hamiltonian with Kato-class potential K_PF in nonrelativistic quantum electrodynamics is defined and studied by a path measure. K_PF is defined as the self-adjoint generator of a strongly continuous one-parameter symmetric semigroup and it is shown that its bound states spatially exponentially decay pointwise and the ground state is unique.

We consider a 3D quantum system of $N$ identical bosons in a trapping potential $|x{|}^p$, with $p\ge 0$, interacting via a Newton potential with an attractive interaction strength $a_N$. For a fixed large $N$ and the coupling constant $a_N$ smaller than a critical value $a_{\ast }$ (Chandrasekhar limit mass), in an appropriate sense, the many-body system admits a ground state. We investigate the blow-up behavior of the ground state energy as well as the ground states when $a_N$ approaches $a_{\ast }$ sufficiently slowly in the limit $N\to \infty$. The blow-up profile is given by the Gagliardo–Nirenberg solutions.

In this work, we obtain the integrated density of states for the Schrödinger operators with decaying random potentials acting on ${\ell }^2({\mathbb{Z}}^d)$. We also study the asymptotic of the largest and smallest eigenvalues of its finite volume approximation.

We prove the existence, uniqueness, and strict positivity of ground states of the possibly massless renormalized Nelson operator under an infrared regularity condition and for Kato decomposable electrostatic potentials fulfilling a binding condition. If the infrared regularity condition is violated, then we show non-existence of ground states of the massless renormalized Nelson operator with an arbitrary Kato decomposable potential. Furthermore, we prove the existence, uniqueness, and strict positivity of ground states of the massless renormalized Nelson operator in a non-Fock representation where the infrared condition is unnecessary. Exponential and superexponential estimates on the pointwise spatial decay and the decay with respect to the boson number for elements of spectral subspaces below localization thresholds are provided. Moreover, some continuity properties of ground state eigenvectors are discussed. Byproducts of our analysis are a hypercontractivity bound for the semigroup and a new remark on Nelson’s operator theoretic renormalization procedure. Finally, we construct path measures associated with ground states of the renormalized Nelson operator. Their analysis entails improved boson number decay estimates for ground state eigenvectors, as well as upper and lower bounds on the Gaussian localization with respect to the field variables in the ground state. As our results on uniqueness, positivity, and path measures exploit the ergodicity of the semigroup, we restrict our attention to one matter particle. All results are non-perturbative.