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We consider the Schrödinger operator in a bounded domain $ \mathbf{R}^n (n=2,3) $ R n (n = 2 , 3) with Neumann boundary condition. We suppose that this domain contains small deformable inclusions, i.e. regions where the potentials do not have the same values as the exterior medium. Our goal is to construct an asymptotic formula for the case of multiple and simple eigenvalue problems. We find an expansion that highlights the relation between the deformation parameters and the eigenvalues. The problem is devoted to a domain containing a finite number of inhomogeneities. Also, we focus on the interface of a small inclusion. [ABSTRACT FROM AUTHOR]

In this paper we study global nonlinear stability for the Dirac–Klein–Gordon system in two and three space dimensions for small and regular initial data. In the case of two space dimensions, we consider the Dirac–Klein–Gordon system with a massless Dirac field and a massive scalar field, and prove global existence, sharp time decay estimates and linear scattering for the solutions. In the case of three space dimensions, we consider the Dirac–Klein–Gordon system with a mass parameter in the Dirac equation, and prove uniform (in the mass parameter) global existence, unified time decay estimates and linear scattering in the top order energy space. [ABSTRACT FROM AUTHOR]

We investigate the following nonlinear bosonic equation on Euclidean space arising in string theory and cosmology: P − Δ e − c Δ u + m u = f (x , u) , x ∈ R n , where n ≥ 3 , m > 0 , c > 0 and f (x , u) u tends to a positive function h (x) independent of u as u → + ∞ , e − c Δ is given by a power series with Δ is the Euclidean Laplace operator. Here, the nonlinear term f (x , u) does not satisfy the usual condition: AR 0 ≤ F (x , u) : = ∫ 0 u f (x , t) d t ≤ 1 2 + θ f (x , u) u , for θ > 0 and | u | is large, which is important in using the mountain pass theorem, see Alves et al. (J. Differ. Equ. 323:229-252, 2022) and Corrêa et al. (J. Differ. Equ. 363:491-517, 2023). This paper is devoted to discuss how to use the mountain pass theorem to obtain the existence of nontrivial solution to problem (P) without the (AR) condition. [ABSTRACT FROM AUTHOR]

We consider semiclassically scaled, weakly nonlinear Schrödinger equations with external confining potentials and additional angular-momentum rotation term. This type of model arises in the Gross–Pitaevskii theory of trapped, rotating quantum gases. We construct asymptotic solutions in the form of semiclassical wave packets, which are concentrated in both space and in frequency around an classical Hamiltonian phase-space flow. The rotation term is thereby seen to alter this flow, but not the corresponding classical action. [ABSTRACT FROM AUTHOR]

We consider the classical vacua of the Weinberg–Salam (WS) model of electroweak forces. These are no-particle, static solutions to the WS equations minimizing the WS energy locally. We study the WS vacuum solutions exhibiting a non-vanishing average magnetic field of strength b and prove that (i) there is a magnetic field threshold b ∗ such that for b < b ∗ , the vacua are translationally invariant (and the magnetic field is constant), while, for b > b ∗ , they are not, (ii) for b > b ∗ , there are non-translationally invariant solutions with lower energy per unit volume and with the discrete translational symmetry of a 2D lattice in the plane transversal to b, and (iii) the lattice minimizing the energy per unit volume approaches the hexagonal one as the magnetic field strength approaches the threshold b ∗ . In the absence of particles, the Weinberg–Salam model reduces to the Yang–Mills–Higgs (YMH) equations for the gauge group U(2). Thus, our results can be rephrased as the corresponding statements about the U(2)-YMH equations. [ABSTRACT FROM AUTHOR]

We consider the physical setup of a three-dimensional fluid–structure interaction problem. A viscous compressible gas or liquid interacts with a nonlinear, visco-elastic, three-dimensional bulk solid. The latter is described by an evolution with inertia, a non-linear dissipation term and a term that relates to a non-convex elastic energy functional. The fluid is modelled by the compressible Navier–Stokes equations with a barotropic pressure law. Due to the motion of the solid, the fluid domain is time-changing. Our main result is the long-time existence of a weak solution to the coupled system until the time of a collision. The nonlinear coupling between the motions of the two different matters is established via the method of minimising movements. The motion of both the solid and the fluid is chosen via an incrimental minimization with respect to dissipative and static potentials. These variational choices together with a careful construction of an underlying flow map for our approximation then directly result in the pressure gradient and the material time derivatives. [ABSTRACT FROM AUTHOR]

We investigate the Cauchy problem of the one dimensional Maxwell-Schrödinger (MS) system under the Lorenz gauge condition. Different from the classical case, we consider the electromagnetic and electrostatic potentials which are growing at space infinity. More precisely, the electrostatic potential is allowed to grow linearly, while for the electromagnetic potential the growth is sublinear. Based on the energy estimates and the gauge transformation, we prove the global existence and the uniqueness of the weak solutions to this system. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a class of planar Schrödinger–Poisson systems, namely, - Δ u + V (x) u + γ 2 π (log (| · |) ∗ | u | 2) u = b | u | p - 2 u in R 2 , where γ > 0 , b ≥ 0 , p > 2 and V ∈ C (R 2 , R) is an unbounded potential function with inf R 2 V > 0 . Suppose moreover that the potential V satisfies { x ∈ R 2 : V (x) ≤ M } < ∞ for every M > 0 , we establish the existence of ground state solutions for this system via variational methods. Furthermore, we also explore the minimax characterization of ground state solutions. Our main results can be viewed as a counterpart of the results from Molle and Sardilli (Proc Edinb Soc 65:1133–1146, 2022), where the authors studied the existence of ground state solutions for the above planar Schrödinger–Poisson system in the case where b > 0 and p > 4 . [ABSTRACT FROM AUTHOR]

We consider a Dirac operator with constant magnetic field defined on a half-plane with boundary conditions that interpolate between infinite mass and zigzag. By a detailed study of the energy dispersion curves, we show that the infinite mass case generically captures the profile of these curves, which undergoes a continuous pointwise deformation into the topologically different zigzag profile. Moreover, these results are applied to the bulk-edge correspondence. In particular, by means of a counterexample, we show that this correspondence does not always hold true in the zigzag case. [ABSTRACT FROM AUTHOR]

This work addresses the resolvent convergence of generalized MIT bag operators to Dirac operators with zigzag type boundary conditions. We prove that the convergence holds in strong but not in norm resolvent sense. Moreover, we show that the only obstruction for having norm resolvent convergence is the existence of an eigenvalue of infinite multiplicity for the limiting operator. More precisely, we prove the convergence of the resolvents in operator norm once projected into the orthogonal of the corresponding eigenspace. [ABSTRACT FROM AUTHOR]

This paper is concerned with a planar Schrödinger–Poisson system involving Stein–Weiss nonlinearity 0.1 - Δ u + V (x) u + ϕ u = 1 | x | β ∫ R 2 F (u (y)) | x - y | μ | y | β d y f (u) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , and its degenerate case 0.2 - Δ u + ϕ u = ∫ R 2 F (u (y)) | x - y | μ d y f (u) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where β ≥ 0 , 0 < μ < 2 , 2 β + μ < 2 , V ∈ C (R 2 , R) and f is of exponential critical growth. By combining variational methods, Stein–Weiss inequality and some delicate analysis, we derive the existence of ground state solution for the first system. Under some mild assumptions, we introduce the Pohozaev identity of the equivalent equation of the second system and use Jeanjean's monotonicity method to achieve the existence of nontrivial solution for the second system. [ABSTRACT FROM AUTHOR]

We consider the initial-boundary value problem of the nonlinear Schrödinger equation on the half plane with a nonlinear Neumann boundary condition. After establishing the boundary Strichartz estimate in L 2 (R + 2) and H s (R + 2) , we consider the time local well-posedness of the problem in L 2 (R + 2) and H s (R + 2) . [ABSTRACT FROM AUTHOR]

It is well-known that, due to the interaction between the spin and the magnetic field, the two-dimensional Pauli operator has an eigenvalue 0 at the threshold of its essential spectrum. We show that when perturbed by an effectively positive perturbation, V, coupled with a small parameter ε , these eigenvalues become resonances. Moreover, we derive explicit expressions for the leading terms of their imaginary parts in the limit ε ↘ 0 . These show, in particular, that the dependence of the imaginary part of the resonances on ε is determined by the flux of the magnetic field. The cases of non-degenerate and degenerate zero eigenvalue are treated separately. We also discuss applications of our main results to particles with anomalous magnetic moments. [ABSTRACT FROM AUTHOR]

A Landau–Zener-type formula for a degenerate avoided-crossing is studied in the non-coupled regime. More precisely, a 2 × 2 system of first-order h-differential operator with O (ε) off-diagonal part is considered in 1D. Asymptotic behavior as ε h m / (m + 1) → 0 + of the local scattering matrix near an avoided-crossing is given, where m stands for the contact order of two curves of the characteristic set. A generalization including the cases with vanishing off-diagonals and non-Hermitian symbols is also given. [ABSTRACT FROM AUTHOR]

We consider the regularized Landau-Pekar equations with positive speed of sound and prove the existence of subsonic traveling waves. We provide a definition of the effective mass for the regularized Landau-Pekar equations based on the energy-velocity expansion of subsonic traveling waves. Moreover we show that this definition of the effective mass agrees with the definition based on an energy-momentum expansion of low energy states. [ABSTRACT FROM AUTHOR]

In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (−Δ)su+μu=(Iα*F(u))F′(u) inRN, ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), Iα∼1|x|N−α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u∈Hs(RN) $u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$ , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫RNu2=m>0 ${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ >}0$ prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C 1-regularity.

This article is concerned with the following Hamiltonian elliptic system: −ε2Δu+εb→⋅∇u+u+V(x)v=Hv(u,v)inRN,−ε2Δv−εb→⋅∇v+v+V(x)u=Hu(u,v)inRN,\left\{\begin{array}{l}-{\varepsilon }^{2}\Delta u+\varepsilon \overrightarrow{b}\cdot \nabla u+u+V\left(x)v={H}_{v}\left(u,v)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ -{\varepsilon }^{2}\Delta v-\varepsilon \overrightarrow{b}\cdot \nabla v+v+V\left(x)u={H}_{u}\left(u,v)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array}\right. where ε>0\varepsilon \gt 0 is a small parameter, VV is a potential function, and HH is a super-quadratic sub-critical Hamiltonian. Applying suitable variational arguments and refined analysis techniques, we construct a new multiplicity result of semiclassical solutions which depends on the number of global minimum points of VV. This result indicates how the shape of the graph of VV affects the number of semiclassical solutions.

In this paper, we combine the arguments of [L. Fanelli, J. Zhang and J. Zheng, Uniform resolvent estimates for Schrödinger operators in critical magnetic fields, Int. Math. Res. Not. IMRN 2023), 10.1093/imrn/rnac362] and [Y. Sire, C. D. Sogge, C. Wang and J. Zhang, Reversed Strichartz estimates for wave on non-trapping asymptotically hyperbolic manifolds and applications, Comm. Partial Differential Equations 47 2022, 6, 1124–1132] to prove the maximal estimates for fractional Schrödinger equations (i ⁢ ∂ t + ℒ 퐀 α 2 ) ⁢ u = 0 in the purely magnetic fields which includes the Aharonov–Bohm fields. The proof is based on the cluster spectral measure estimates. In particular, for α = 1 , the maximal estimate for wave equation is sharp up to the endpoint. [ABSTRACT FROM AUTHOR]

We consider the solutions of the Klein–Gordon equation in the 2 + 1 -dimensional space-time which gravity is analyzed, i.e., the manifold ℝ t × ℝ + × ℝ / 2 ⁢ π ⁢ α ⁢ ℤ created by a massive point particle. Using the Schwartz kernel of resolvent and spectral measure for Schrödinger operator on the spinless cone, we prove the dispersive estimates and Strichartz estimates for the Klein–Gordon equation. In a future paper, we will consider the problem on the spinning cone. [ABSTRACT FROM AUTHOR]

In this paper, the fractional nonlinear Schrödinger equation (NLSE) has been studied through conformable fraction space-time derivatives sense. Namely, we introduce some vital solutions for the fractional NLSE by using robust solver approach based on the Jacobian elliptic function method. This solver is easy to use, reliable, practical, and sturdy. The fractional properties structures that obtained from the equation are given in form of hyperbolic, soliton, shocks, explosive, superperiodic and trigonometric structures. It was noticed that raising the fractal factors causes the nonlinear wave to propagate with a different phase and wave frequency. The physical models describe the tidal energy generations play the important roles in the modern green power technologies. The solutions of nonlinear equations produce the parametric description for wave features in these processes. The solutions developed can be used in novel communications, energy applications, fractional quantum modes, and complicated astrophysical phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Schrödinger–Pauli problem on a finite square domain. Because of the boundary conditions we have proposed as a solution a suitable trigonometric series. We have shown that these coefficients exist and can be univocally determined by solving a simple algebraic system. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonlinear boundary value problem D D ψ - a (x) ψ = f (x , ψ) + ϵ h (x , ψ) on M B CHI ψ = 0 on ∂ M where M is a compact Riemannian spin manifold of dimension m ≥ 2 and the boundary ∂ M has non-negative mean curvature, and D is the Dirac-Atiyah-Singer operator. Let S(M) denote the spinor bundle on M and ψ : M → S (M) be a section. a(x) is a scalar field on M, ϵ ∈ R . f (x , ψ) , h (x , ψ) : M × S m → S m are two nonlinear map, and B CHI is the chirality boundary operator. Under some mild assumptions on a, f, and h, we obtain the ground state solution of (D) with ϵ = 0 and infinitely many large norm solutions of (D) with ϵ ∈ R and infinitely many small energy solutions of (D) with ϵ > 0 by using variational methods. [ABSTRACT FROM AUTHOR]

We study the focusing intercritical NLS NLS i ∂ t u + Δ x , y u = - | u | α u on the semiperiodic waveguide manifold R x d × T y with d ≥ 5 and α ∈ (4 d , 4 d - 1) . In the case d ≤ 4 , with the aid of the semivirial vanishing theory (Luo in Normalized ground states and threshold scattering for focusing NLS on Rd × T via semivirial-free geometry, 2022), the author was able to construct a sharp threshold, which being uniquely characterized by the ground state solutions, that sharply determines the bifurcation of global scattering and finite time blow-up solutions in dependence of the sign of the semivirial functional. As the derivative of the nonlinear potential is no longer Lipschitz in d ≥ 5 and the underlying domain possesses an anisotropic nature, the proof in Luo (Normalized ground states and threshold scattering for focusing NLS on Rd × T via semivirial-free geometry, 2022), which makes use of the concentration compactness principle, can not be extended to higher dimensional models. In this paper, we exploit a well-tailored adaptation of the interaction Morawetz–Dodson–Murphy (IMDM) estimates, which were only known to be applicable on Euclidean spaces, into the waveguide setting, in order to prove that the large data scattering result formulated in Luo (Normalized ground states and threshold scattering for focusing NLS on Rd × T via semivirial-free geometry, 2022) continues to hold for all d ≥ 5 . Together with Tzvetkov–Visciglia (Rev. Mat. Iberoam. 32:1163–1188, 2016) and the author (Luo in Normalized ground states and threshold scattering for focusing NLS on Rd × T via semivirial-free geometry, 2022), we thus give a complete characterization of the large data scattering for (NLS) in both defocusing and focusing case and in arbitrary dimension. [ABSTRACT FROM AUTHOR]

This paper discusses the mean-field limit for the quantum dynamics of N identical bosons in R 3 interacting via a binary potential with Coulomb-type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in Golse and Paul (Commun Math Phys 369:1021–1053, 2019). Our first main result is a definition of the interaction nonlinearity in the equation governing the dynamics of quantum Klimontovich solutions for a class of interaction potentials slightly less general than those considered in Kato (Trans Am Math Soc 70:195–211, 1951). Our second main result is a new operator inequality satisfied by the quantum Klimontovich solution in the case of an interaction potential with Coulomb-type singularity. When evaluated on an initial bosonic pure state, this operator inequality reduces to a Gronwall inequality for a functional introduced in Pickl (Lett Math Phys 97:151-164, 2011), resulting in a convergence rate estimate for the quantum mean-field limit leading to the time-dependent Hartree equation. [ABSTRACT FROM AUTHOR]

We present the self-consistent Pauli equation, a semi-relativistic model for charged spin- 1 / 2 particles with self-interaction with the electromagnetic field. The Pauli equation arises as the O ⁢ (1 / c) approximation of the relativistic Dirac equation. The fully relativistic self-consistent model is the Dirac–Maxwell equation where the description of spin and the magnetic field arises naturally. In the non-relativistic setting, the correct self-consistent equation is the Schrödinger–Poisson equation which does not describe spin and the magnetic field and where the self-interaction is with the electric field only. The Schrödinger–Poisson equation also arises as the mean field limit of the 푁-body Schrödinger equation with Coulomb interaction. We propose that the Pauli–Poisson equation arises as the mean field limit N → ∞ of the linear 푁-body Pauli equation with Coulomb interaction where one has to pay extra attention to the fermionic nature of the Pauli equation. We present the semiclassical limit of the Pauli–Poisson equation by the Wigner method to the Vlasov equation with Lorentz force coupled to the Poisson equation which is also consistent with the hierarchy in 1 / c of the self-consistent Vlasov equation. This is a non-trivial extension of the groundbreaking works by Lions & Paul and Markowich & Mauser, where we need methods like magnetic Lieb–Thirring estimates. [ABSTRACT FROM AUTHOR]

We analyze a numerical method to solve the time-dependent linear Pauli equation in three space dimensions. The Pauli equation is a semi-relativistic generalization of the Schrödinger equation for 2-spinors which accounts both for magnetic fields and for spin, with the latter missing in preceding numerical work on the linear magnetic Schrödinger equation. We use a four term operator splitting in time, prove stability and convergence of the method and derive error estimates as well as meshing strategies for the case of given time-independent electromagnetic potentials, thus providing a generalization of previous results for the magnetic Schrödinger equation. [ABSTRACT FROM AUTHOR]

We investigate the following problem - div (v (x) | ∇ u | m - 2 ∇ u) + V (x) | u | m - 2 u = | x | - θ ∗ | u | b | x | α | u | b - 2 | x | α u + λ | x | - γ ∗ | u | c | x | β | u | c - 2 | x | β u in R N , where b , c , α , β > 0 , θ , γ ∈ (0 , N) , N ≥ 3 , 2 ≤ m < ∞ and λ ∈ R . Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively. [ABSTRACT FROM AUTHOR]

Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, we study the noncommutative Bloch transform of Fuchsian groups that we call the hyperbolic Bloch transform. First, we prove that the hyperbolic Bloch transform is injective and "asymptotically unitary" already in the simplest case—that is, when the Hilbert space is the regular representation of the Fuchsian group, Γ . Second, when Γ ⊂ PSU (1 , 1) acts isometrically on the hyperbolic plane H and the Hilbert space is L 2 H , we define a modified, geometric Bloch transform that sends wave functions to sections of irreducible, flat, Hermitian vector bundles over Σ = H / Γ and transforms the hyperbolic Laplacian into the covariant one. [ABSTRACT FROM AUTHOR]

We establish improved Hardy inequalities for the magnetic p -Laplacian due to adding nontrivial magnetic fields. We also prove that for Aharonov–Bohm magnetic fields the sharp constant in the Hardy inequality becomes strictly larger than in the case of a magnetic-free p -Laplacian. We also post some remarks with open problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the relationship between the Dirac–Fock model and the electron-positron Hartree–Fock model. We justify the Dirac–Fock model as a variational approximation of QED when the vacuum polarization is neglected and when the fine structure constant α is small and the velocity of light c is large. As a byproduct, we also prove, when α is small or c is large, the no-unfilled shells theory in the Dirac–Fock theory for atoms and molecules. The proof is based on some new properties of the Dirac–Fock model. [ABSTRACT FROM AUTHOR]

In this article, we study the following quasilinear equation with nonlocal nonlinearity −Δu−κuΔ(u2)+λu=(∣x∣−μ*F(u))f(u),inRN,-\Delta u-\kappa u\Delta \left({u}^{2})+\lambda u=\left({| x| }^{-\mu }* F\left(u))f\left(u),\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where κ\kappa is a parameter, N≥3N\ge 3, μ∈(0,N)\mu \in \left(0,N), F(t)=∫0tf(s)dsF\left(t)={\int }_{0}^{t}f\left(s){\rm{d}}s, and λ\lambda is a positive constant. We are going to analyze two cases: the L2{L}_{2}-norm of the solution is not confirmed and the L2{L}_{2}-norm of the solution is prescribed. Under the almost optimal assumptions on ff, we obtain the existence of a sequence of radial solutions for two cases.

This paper is concerned with the global well-posedness and relaxation-time limits for the solutions in the full quantum hydrodynamic model, which can be used to analyze the thermal and quantum influences on the transport of carriers in semiconductor devices. For the Cauchy problem in ℝ3, we prove the global existence, uniqueness and exponential decay estimate of smooth solutions, when the initial data are small perturbations of an equilibrium state. Moreover, we show that the solutions converge into that of the simplified quantum energy-transport model and the quantum drift-diffusion model for the moment relaxation limit, and the moment and energy relaxation limit, respectively. [ABSTRACT FROM AUTHOR]

This paper is concerned with the mass-energy threshold dynamics for the elliptic–elliptic Davey–Stewartson system in dimension three. By proving the compactness of minimizing sequence for the Weinstein functional, we show the long time behavior of solutions with data being at the mass-energy threshold. Our proof is based on a Gagliardo–Nirenberg type inequality, localized virial estimates and the concentration-compactness lemma. [ABSTRACT FROM AUTHOR]

We discuss strong local and global well–posedness for the three–dimensional NLS equation with nonlinearity concentrated on S 2 . Precisely, local well–posedness is proved for any C 2 power–nonlinearity, while global well–posedness is obtained either for small data or in the defocusing case under some growth assumptions. With respect to point–concentrated NLS models, widely studied in the literature, here the dimension of the support of the nonlinearity does not allow a direct extension of the known techniques and calls for new ideas. [ABSTRACT FROM AUTHOR]

We deal with the attractive Bose–Einstein condensates with sinusoidal potential in a circular region S : = { x : 0 ≤ | x | ≤ π } ⊂ R 2 . The existence, non-existence and mass concentration behavior (i.e., blow-up) of constrained minimizers for the related Gross-Pitaevskii energy functional are analyzed. Once the blow-up behavior arises, we prove that the mass of positive minimizers must concentrate at center of the circular region, rather than the points of circumference. [ABSTRACT FROM AUTHOR]

In the present paper, we are concerned with the existence and multiplicity of normalized solutions of coupled Hartree–Fock system with upper critical exponent and lower power perturbation. This type system arises from the basic quantum chemistry model of small number of electrons interacting with static nuclei which can be approximated by Hartree or Hartree–Fock minimization problems. First, by constraint manifold methods and refined energy estimates, we prove the existence of ground and bound states normalized solutions for critical Hartree–Fock system. Then the precise asymptotic behaviors of two normalized solutions are also obtained. Finally, we discuss the orbital stability and finite time blow-up of the associated solitary wave solutions. [ABSTRACT FROM AUTHOR]

In Quantum Hydro-Dynamics the following problem is relevant: let (ρ , Λ) ∈ W 1 , 2 (R d , L d , R +) × L 2 (R d , L d , R d) be a finite energy hydrodynamics state, i.e. Λ = 0 when ρ = 0 and E = ∫ R d 1 2 | ∇ ρ | 2 + 1 2 Λ 2 L d < ∞. The question is under which conditions there exists a wave function ψ ∈ W 1 , 2 (R d , L d , C) such that ρ = | ψ | , J = ρ Λ = ℑ ( ψ ¯ ∇ ψ). The second equation gives for ψ = ρ w smooth, | w | = 1 , that i Λ = ρ w ¯ ∇ w . Interpreting ρ L d as a measure in the metric space R d , this question can be stated in generality as follows: given metric measure space (X , d , μ) and a cotangent vector field v ∈ L 2 (T ∗ X) , is there a function w ∈ W 1 , 2 (X , μ , S 1) such that d w = i w v. Under some assumptions on the metric measure space (X , d , μ) (conditions which are verified on Riemann manifolds with the measure μ = ρ Vol or more generally on non-branching MCP (K , N) ), we show that the necessary and sufficient conditions for the existence of w is that (in the case of differentiable manifold) ∫ v (γ (t)) · γ ˙ (t) d t ∈ 2 π Z for π -a.e. γ , where π is a test plan supported on closed curves. This condition generalizes the condition that the vorticity is quantized. We also give a representation of every possible solution. In particular, we deduce that the wave function ψ = ρ w is in W 1 , 2 (X , μ , C) whenever ρ ∈ W 1 , 2 (X , μ , R +) . [ABSTRACT FROM AUTHOR]

We investigate the stability of plane wave solutions of equations describing quantum particles interacting with a complex environment. The models take the form of PDE systems with a non local (in space or in space and time) self-consistent potential; such a coupling lead to challenging issues compared to the usual nonlinear Schrödinger equations. The analysis relies on the identification of suitable Hamiltonian structures and Lyapounov functionals. We point out analogies and differences between the original model, involving a coupling with a wave equation, and its asymptotic counterpart obtained in the large wave speed regime. In particular, while the analogies provide interesting intuitions, our analysis shows that it is illusory to obtain results on the former based on a perturbative analysis from the latter. [ABSTRACT FROM AUTHOR]

Fragmented Bose–Einstein condensates are large systems of identical bosons displaying multiple macroscopic occupations of one-body states, in a suitable sense. The quest for an effective dynamics of the fragmented condensate at the leading order in the number of particles, in analogy to the much more controlled scenario for complete condensation in one single state, is deceptive both because characterising fragmentation solely in terms of reduced density matrices is unsatisfactory and ambiguous, and because as soon as the time evolution starts the rank of the reduced marginals generically passes from finite to infinite, which is a signature of a transfer of occupations on infinitely many more one-body states. In this work we review these difficulties, we refine previous characterisations of fragmented condensates in terms of marginals, and we provide a quantitative rate of convergence to the leading effective dynamics in the double limit of infinitely many particles and infinite energy gap. [ABSTRACT FROM AUTHOR]