Narrow Results

We establish some embedding results of weighted Sobolev spaces of radially symmetric functions. The results are then used to obtain ground state solutions of nonlinear Schrödinger equations with unbounded and decaying radial potentials. Our work unifies and generalizes many existing partial results in the literature.

We review the mathematical formalism of the equilibrium quantum statistical mechanics of lattice models with an infinite degree of freedom. The equivalence of the KMS boundary condition, the Gibbs-Araki condition and the variational principle is established for a class of long range interactions. Our technical tool is one-parameter semigroups of completely positive maps on UHF C*-algebras, which we call Feller semigroups. Uniqueness of Gibbs and ground sates is re-examined from the viewpoint of unique ergodicity of Feller semigroups.

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.

In this work, we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms of the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.

Monte Carlo simulation techniques, like simulated annealing and parallel tempering, are often used to evaluate low-temperature properties and find ground states of disordered systems. Here we compare these methods using direct calculations of ground states for three-dimensional Ising diluted antiferromagnets in a field (DAFF) and three-dimensional Ising spin glasses (ISG). For the DAFF, we find that, with respect to obtaining ground states, parallel tempering is superior to simple Monte Carlo and to simulated annealing. However, equilibration becomes more difficult with increasing magnitude of the externally applied field. For the ISG with bimodal couplings, which exhibits a high degeneracy, we conclude that finding true ground states is easy for small systems, as is already known. But finding each of the degenerate ground states with the same probability (or frequency), as required by Boltzmann statistics, is considerably harder and becomes almost impossible for larger systems.

In this paper, we consider the generalized inverse iteration for computing ground states of the Gross–Pitaevskii eigenvector (GPE) problem. For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross–Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [W. Bao and Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput.25 (2004) 1674–1697] or the damped inverse iteration suggested in [P. Henning and D. Peterseim, Sobolev gradient flow for the Gross–Pitaevskii eigenvalue problem: Global convergence and computational efficiency, SIAM J. Numer. Anal.58 (2020) 1744–1772]. Our analysis also reveals why the inverse iteration for the GPE does not react favorably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments.

We study the nonlinear coupled Kirchhoff system with purely Sobolev critical exponent. By using appropriate transformation, we get one equivalent system involving a critical Schrödinger system and an algebraic system. Through solving the critical Schrödinger system with a corresponding algebraic system, under suitable conditions we obtain the existence and classification of positive ground states for the Kirchhoff system in dimensions 3 and 4. Furthermore, for the degenerate case, we give a complete classification of positive ground states for the Kirchhoff system in any dimension. To the best of our knowledge, this paper is the first to give classification results for the ground states of Kirchhoff systems. The results in this paper partially extend and complement the main results established by Lü and Peng [Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differ. Equ. 263 (2017) 8947–8978] considering the linearly coupled Kirchhoff system with subcritical exponent and some partial results established by Chen and Zou [Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012) 515–551; Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differ. Equ. 52 (2015) 423–467], where the authors considered the coupled purely critical Schrödinger system.

We investigate the soliton dynamics for the Schrödinger–Newton system by proving suitable modulational stability estimates in the spirit of those obtained by Weinstein for local equations.

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 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.

In this first part, we study the existence and uniqueness of solutions of a general nonlinear Schrödinger system in the presence of diamagnetic field, local and nonlocal nonlinearities. This kind of systems models many important phenomena in nonlinear optics; multimodal optical fibers, optical pulse propagation, ferromagnetic film and optical pulse propagation in the birefringent fibers. They also govern the interaction of electron and nucleii through Coulombic potential and under the action of external magnetic field in quantum mechanics.

We study the positive solutions of the Lane–Emden problem -Δ_pu = λ_p|u|^q-2u in Ω, u = 0 on ∂Ω, where Ω ⊂ ℝ^N is a bounded and smooth domain, N ≥ 2, λ_p is the first eigenvalue of the p-Laplacian operator Δ_p, p > 1, and q is close to p. We prove that any family of positive solutions of this problem converges in to the function θ_pe_p when q → p, where e_p is the positive and L^∞-normalized first eigenfunction of the p-Laplacian and . A consequence of this result is that the best constant of the immersion is differentiable at q = p. Previous results on the asymptotic behavior (as q → p) of the positive solutions of the nonresonant Lane–Emden problem (i.e. with λ_p replaced by a positive λ ≠ λ_p) are also generalized to the space and to arbitrary families of these solutions. Moreover, if u_λ,q denotes a solution of the nonresonant problem for an arbitrarily fixed λ > 0, we show how to obtain the first eigenpair of the p-Laplacian as the limit in , when q → p, of a suitable scaling of the pair (λ, u_λ,q). For computational purposes the advantage of this approach is that λ does not need to be close to λ_p. Finally, an explicit estimate involving L^∞- and L¹-norms of u_λ,q is also derived using set level techniques. It is applied to any ground state family {v_q} in order to produce an explicit upper bound for ‖v_q‖_∞ which is valid for q ∈ [1, p + ϵ] where .

We consider a class of Lévy-type processes derived via a Doob transform from Lévy processes conditioned by a control function called potential. These ground state transformed processes (also called $P{(\varphi )}_1$-processes) have position-dependent and generally unbounded components, with stationary distributions given by the ground states of the Lévy generators perturbed by the potential. We derive precise upper envelopes for the almost sure long-time behavior of these ground state-transformed Lévy processes, characterized through escape rates and integral tests. We also highlight the role of the parameters by specific examples.