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This paper concerned a fractional Schrödinger equation in whole space, for ɛ > 0 is a small parameter, ɛ2s(−Δ)su + V(x)u = |u|p−2u, where 1 2 < s < 1 , N > 1 and 2 < p < 2 N N − 2 s . We prove the non-degeneracy and uniqueness of bubble solutions by using local Pohozaev identity and finite dimensional reduction, which are the cornerstones to construct different type solutions. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the Sobolev norms of solutions for the linear wave equation with unbounded perturbations of order one remain bounded for all time. The main proof is based on the KAM reducibility of the linear wave equation. To the best of our knowledge, this is the first reducibility result for the linear wave equation with general quasi-periodic unbounded perturbations on the one-dimensional torus. [ABSTRACT FROM AUTHOR]

The flat band of edge states that occur in the simple tight-binding lattice model of graphene with a zigzag edge have long been conjectured to take up a ferromagnetic configuration. In this work, we prove that, for a large class of interaction Hamiltonians that can be added to the tight-binding model, and at the first order in perturbation theory, the degeneracy of edge states is resolved in such a way that the ground state is in the maximal, spin j = N/2 representation of the spin symmetry, where N is the number of edge states. [ABSTRACT FROM AUTHOR]

We have constructed a perturbation theory to treat interactions that can include the Coulomb interaction, describing a physical problem that is often encountered in nuclear physics. The Coulomb part is not treated perturbatively; the exact solutions are employed. The method is an extension of the results presented in the work of Hoffmann [J. Math. Phys. 62, 032105 (2021)]. It is designed to calculate phase shifts directly rather than the full form of the wavefunctions in position space. We present formulas that allow for the calculation of the phase shifts to second order in the perturbation. The phase shift results to second order, for a short-range potential, were compared with the exact solution, where we found an error of third order in the coupling strength. A different model, meant as a simple approximation of nuclear scattering of a proton on helium-4 and including a Coulomb potential and a spherical well, was constructed to test the theory. The wavepacket scattering formalism of Hoffmann [J. Phys. B: At., Mol. Opt. Phys. 50, 215302 (2017)], known to give everywhere finite results, was employed. We found physically acceptable results and a cross section of the correct order of magnitude. [ABSTRACT FROM AUTHOR]

In this paper, the local iterative Lie–Schwinger block-diagonalization method, introduced and developed in our previous work for quantum chains, is extended to higher-dimensional quantum lattice systems with Hamiltonians that can be written as the sum of an unperturbed gapped operator, consisting of a sum of on-site terms, and a perturbation, consisting of bounded interaction potentials of short range multiplied by a real coupling constant t. Our goal is to prove that the spectral gap above the ground-state energy of such Hamiltonians persists for sufficiently small values of |t|, independently of the size of the lattice. New ideas and concepts are necessary to extend our method to systems in dimension d > 1: As in our earlier work, a sequence of local block-diagonalization steps based on judiciously chosen unitary conjugations of the original Hamiltonian is introduced. The supports of effective interaction potentials generated in the course of these block-diagonalization steps can be identified with what we call minimal rectangles contained in the lattice, a concept that serves to tackle combinatorial problems that arise in the course of iterating the block-diagonalization steps. For a given minimal rectangle, control of the effective interaction potentials generated in each block-diagonalization step with support in the given rectangle is achieved by exploiting a variety of rather subtle mechanisms, which include, for example, the use of weighted sums of paths consisting of overlapping rectangles and of large denominators, expressed in terms of sums of orthogonal projections, which serve to control analogous sums of projections in the numerators resulting from the unitary conjugations of the interaction potential terms involved in the local block-diagonalization step. [ABSTRACT FROM AUTHOR]

Heun differential equations are the most general second order Fuchsian equations with four regular singularities. An explicit integral series representation of Heun functions involving only elementary integrands has hitherto been unknown and noted as an important open problem in a recent review. We provide such representations of the solutions of all equations of the Heun class: general, confluent, bi-confluent, doubly confluent, and triconfluent. All the series are illustrated with concrete examples of use, and Python implementations are available for download. We demonstrate the utility of the integral series by providing the first representation of the solution to the Teukolsky radial equation governing the metric perturbations of rotating black holes that is convergent everywhere from the black hole horizon up to spatial infinity. [ABSTRACT FROM AUTHOR]

We explore the convergence of the global attractors for a class of perturbed severely damped wave equations with the Dirichlet boundary condition in the 3D bounded domain. With respect to the perturbation parameter, it is shown that the global attractors are both upper and lower semicontinuous. [ABSTRACT FROM AUTHOR]

In our prior studies, we synthesized special circuits with evolution matrices featuring degenerate eigenfrequencies and nontrivial Jordan blocks. The degeneracy of this type is sometimes referred to as exceptional point of degeneracy (EPD). Our focus here is on the simplest of our circuits featuring EPDs that are composed of only two LC-loops coupled by a gyrator. These circuits, when near an EPD state, can be used for enhanced sensitivity applications. With that in mind, we develop here a comprehensive perturbation theory for these simple circuits near an EPD. Using this theory, we propose an approach to sensing, allowing one to benefit from the proximity to an EPD on the one hand when providing for stable operation on the other hand. We also address a broader scope of problems related to perturbations of Jordan blocks and their numerical treatment that allow us to effectively detect proximity to Jordan blocks. [ABSTRACT FROM AUTHOR]

In nonrelativistic quantum mechanics, we require that the free and interacting Hamiltonians be related by a unitary transformation, as has been done by other authors. We then derive a stationary perturbation theory for the radial Schrödinger equation for scattering from a spherically symmetric potential. A resulting advantage over the more commonly used Green function method is that the expression for the interacting state vector is normalized to each order in the coupling constant, unlike, in general, the result of the Green function method. Other authors have applied the unitary transformation concept to time-dependent perturbation theory to give unitarity of the time evolution operator to each order in perturbation theory, with results that show improvement over the standard perturbation theory. In this paper, general formulas are obtained for the phase shifts at the first and second order in the coupling constant. We test the method on a simple system with a known exact solution and find complete agreement between our first- and second-order contributions to the s-wave phase shifts and the corresponding expansion to the second order of the exact solution. [ABSTRACT FROM AUTHOR]

We present analytic solutions to the Teukolsky equation for massless perturbations of any spin in the four-dimensional de Sitter background. The angular part of the equation fixes the separation constant to a discrete set, and its solution is given by hypergeometric polynomials. For the radial part, we derive an analytic power series solution that is regular at the poles and determine a transcendental function whose zeros give the characteristic values of the wave frequency. We study the existence of explicit polynomial solutions to the radial equation and obtain two classes of singular closed-form solutions, one with discrete wave frequencies and the other with continuous frequency spectra. [ABSTRACT FROM AUTHOR]

We analytically evaluate the generating integral K n l (β , β ′ ) = ∫ 0 ∞ ∫ 0 ∞ e − β r − β ′ r ′ G n l (r , r ′ ) r q r ′ q ′ d r d r ′ and integral moments J n l (β , r) = ∫ 0 ∞ d r ′ G n l (r , r ′ ) r ′ q e − β r ′ for the reduced Coulomb Green's function Gnl(r, r′) for all values of the parameters q and q′, when the integrals are convergent. These results can be used in second-order perturbation theory to analytically obtain the complete energy spectra and local physical characteristics such as electronic densities of multi-electron atoms or ions. [ABSTRACT FROM AUTHOR]

This paper deals with the well-known problem of constructing Lyapunov functions for a nonlinear system and the approximation of the basin of attraction associated with a given attractive equilibrium point. Following a paper by Spelberg-Korspeter et al., the problem is studied by means of perturbative methods, with particular focus on the time-reversed Van Der Pol model. As a difference, the theory is reformulated in terms of the Lie transform method, introduced by Giorgilli et al., which, remarkably, does not require any inverse function arguments to produce the inverse transformations during the normalization process. This will be shown to be, also in this case, a key feature in terms of concrete applications. The nonautonomous perturbation theory developed by the authors in previous works allows an effortless extension of such a construction to the (aperiodically) time-dependent case. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of both a free particle and an isotropic harmonic oscillator constrained to move on a spheroidal surface using two consecutive projections: a projection onto a sphere surface followed by the gnomonic projection onto a tangent plane to the spheroid. We obtain the Hamiltonian of the aforementioned systems in terms of the Cartesian coordinates of the tangent plane and then quantize it in the standard way. It is shown that the effect of nonsphericity of the surface can be treated as the appearance of an effective potential. By using the perturbation theory up to the first order in the second eccentricity of the spheroid, we approximately calculate the eigenfunctions and eigenvalues of the free particle as well as the isotropic harmonic oscillator on the spheroidal surface. We find that the deviation from the sphericity plays an important role in splitting the energy levels of the isotropic oscillator on a sphere and lifting the degeneracy. [ABSTRACT FROM AUTHOR]

We extend the theory of perturbations of KMS states to a class of unbounded perturbations using noncommutative Lp-spaces. We also prove certain stability of the domain of the modular operator associated with a ∥·∥p-continuous state. This allows us to define an analytic multiple-time KMS condition and to obtain its analyticity together with some bounds to its norm. [ABSTRACT FROM AUTHOR]

We determine explicitly a boundary triple for the Dirac operator H ≔ − i α ⋅ ∇ + m β + V (x) in R 3 , for m ∈ R and V (x) = | x | − 1 (ν I 4 + μ β − i λ α ⋅ x / | x | β) , with ν , μ , λ ∈ R. Consequently, we determine all the self-adjoint realizations of H in terms of the behavior of the functions of their domain in the origin. When sup x | x | | V (x) | ≤ 1 , we discuss the problem of selecting the distinguished extension requiring that its domain is included in the domain of the appropriate quadratic form. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to propose an (α, q)-analogue of the Poisson operators on the Fock space of type B in the sense of Bożejko, Ejsmont, and Hasebe [J. Funct. Anal. 269, 1769–1795 (2015)] and to find a probability law of this operator. We shall show that the probability law is expressed by the q-Meixner distribution in the sense of Definition 3.2. Our results contain not only symmetric distributions as in Bożejko-Ejsmont-Hasebe but also the non-symmetric ones such as free Poisson, q and q2-deformations of Poisson, Pascal, Gamma, and Meixner distributions. [ABSTRACT FROM AUTHOR]

The number of quantum systems for which the stationary Schrodinger equation is exactly solvable is very limited. These systems constitute the basic elements of the quantum theory of perturbation. The exact polynomial solutions for real quantum potential systems provided by the use of Lagrange interpolation allows further development of the quantum perturbation theory. In fact, the first order of correction for the value of the energy appears to be sufficient since the chosen perturbation Hamiltonian is very small or even negligible compared to the main Hamiltonian. Here, we use the perturbation theory to derive polynomial solutions, and we then find that our approximated results agree very well with previous published or numerically achieved ones. We believe that this study is an operational tool for the verification and improvement of numerical and approximate methods. [ABSTRACT FROM AUTHOR]

We demonstrate the flux-approximation problem of the isothermal relativistic Euler equations describing a perfect fluid flow in special relativity. First, the Riemann problem of the isothermal relativistic Euler equations under flux perturbation is discussed, and four kinds of solutions are obtained. Second, we rigorously prove that, as the flux perturbation vanishes, any two-shock Riemann solution tends to a delta-shock solution to the pressureless relativistic Euler equations and the intermediate density between the two shocks tends to a weighted δ-measure which forms a delta shock wave. Correspondingly, any two-rarefaction Riemann solution tends to a two-contact-discontinuity solution to the pressureless relativistic Euler equations and the nonvacuum intermediate state in between tends to a vacuum state. [ABSTRACT FROM AUTHOR]

The paper deals with a problem of asymptotic soliton-like solutions to the Benjamin-Bona-Mahony (BBM) equation with a small parameter at the highest derivative and variable coefficients depending on the variables x and t, as well as a small parameter. An algorithm for constructing solutions to the BBM equation has been proposed, and theorems on accuracy of such solutions have been proved. [ABSTRACT FROM AUTHOR]

In this paper, we study the reducibility of time quasiperiodic perturbations of the quantum harmonic or anharmonic oscillator in one space dimension. We modify known algorithms obtaining a reducibility result which allows us to deal with perturbations of order strictly larger than the ones considered in all the previous papers. [ABSTRACT FROM AUTHOR]

We construct a minitwistor action for Yang–Mills–Higgs (YMH) theory in three dimensions. The Feynman diagrams of this action will construct perturbation theory around solutions of the Bogomolny equations in much the same way that MHV (maximally helicity violating) diagrams describe perturbation theory around the self-dual Yang Mills equations in four dimensions. We also provide a new formula for all tree amplitudes in YMH theory (and its maximally supersymmetric extension) in terms of degree d maps to minitwistor space. We demonstrate its relationship to the Roiban-Spradlin-Volovich-Witten (RSVW) formula in four dimensions and show that it generates the correct MHV amplitudes at d = 1 and factorizes correctly in all channels for all degrees. [ABSTRACT FROM AUTHOR]

Tensor field theory is the quantum field theoretic counterpart of tensor models and enhanced tensor field theory enlarges this theory space to accommodate "enhanced tensor interactions." These interactions were introduced to explore new large N limits and to probe different phases for tensor models. We undertake the multi-scale renormalization analysis for two types of enhanced theories with rank d tensor fields ϕ : (U (1) D ) d → C and with the so-called quartic "melonic" interactions of the form p2aϕ4 reminiscent of derivative couplings expressed in momentum space. Scrutinizing the degree of divergence of both theories, we identify generic conditions for their renormalizability at all orders of perturbation at high momenta, i.e., the ultraviolet regime. For the first type of theory, we identify a 2-parameter space of just-renormalizable models for generic (d, D). These models have dominant non-melonic four-point functions. Finally, by specifying the parameters, we detail the renormalization analysis of the second type of model which is more exotic: it exhibits an infinite family of logarithmically divergent two-point amplitudes and all four-point amplitudes are convergent. [ABSTRACT FROM AUTHOR]

We provide regularity of solutions to a large class of evolution equations on Banach spaces where the generator is composed of a static principal part plus a non-autonomous perturbation. Regularity is examined with respect to the graph norm of the iterations of the principal part. The results are applied to the Schrödinger equation and conditions on a time-dependent scalar potential for the regularity of the solution in higher Sobolev spaces are derived. [ABSTRACT FROM AUTHOR]

For bounded right linear operators, in a right quaternionic Hilbert space with a left multiplication defined on it, we study the approximate S-point spectrum. In the same Hilbert space, then we study the Fredholm operators and the Fredholm index. In particular, we prove the invariance of the Fredholm index under small norm operator and compact operator perturbations. Finally, in association with the Fredholm operators, we develop the theory of essential S-spectrum. We also characterize the S-spectrum in terms of the essential S-spectrum and Fredholm operators. In the sequel, we study left and right S-spectra as needed for the development of the theory presented in this note. [ABSTRACT FROM AUTHOR]

Integrable couplings are associated with non-semisimple Lie algebras. In this paper, we propose a new method to generate new integrable systems through making perturbation in matrix spectral problems for integrable couplings, which is called the "completion process of integrable couplings." As an example, the idea of construction is applied to the Ablowitz-Kaup-Newell-Segur integrable coupling. Each equation in the resulting hierarchy has a bi-Hamiltonian structure furnished by the component-trace identity. [ABSTRACT FROM AUTHOR]

Using three different approaches Perturbation Theory (PT), the Lagrange Mesh Method (LMM), and the variational method, we study the low-lying states of the Yukawa potential. First orders of PT, in powers of the screening parameter, are calculated in the framework of the non-linearization procedure. It is found that the Padé approximants to PT series together with the LMM provide highly accurate values of energy and the positions of the radial nodes of the wave function. The most accurate results, at present, of the critical screening parameters for some low-lying states and the first coefficients in the expansion of the energy at the critical parameter are presented. A locally accurate and compact trial function for the eigenfunctions of the low-lying states is discovered. This function used as a zeroth order entry in PT leads to energies as precise as those of Padé approximants and LMM. Finally, a compact analytical expression for the energy, that reproduces at least 6 decimal digits in the entire physical range of the screening parameter, is found. [ABSTRACT FROM AUTHOR]

We study discrete spectral quantities associated with Schrödinger operators of the form − Δ R d + V N , d odd. The potential VN models a highly disordered crystal; it varies randomly at scale N−1 ≪ 1. We use perturbation analysis to obtain almost sure convergence of the eigenvalues and scattering resonances of − Δ R d + V N as N → ∞. We identify a stochastic and a deterministic regime for the speed of convergence. The type of regime depends whether the low frequency effects due to large deviations overcome the (deterministic) constructive interference between highly oscillatory terms. [ABSTRACT FROM AUTHOR]

It is well known that the equations governing the evolution of scalar, electromagnetic, and gravitational perturbations of the background geometry of a Schwarzschild black hole can be reduced to a single master equation. We use Kovacic’s algorithm to obtain all Liouvillian solutions, i.e., essentially all solutions in terms of quadratures, of this master equation. We prove that the algebraically special Liouvillian solutions χ and χ ∫ d r * χ 2 , initially found by Chandrasekhar in the gravitational case, are the only Liouvillian solutions to the master equation. We show that the Liouvillian solution χ ∫ d r * χ 2 is a product of elementary functions, one of them being a polynomial solution P to an associated confluent Heun equation. P admits a finite expansion both in terms of truncated confluent hypergeometric functions of the first kind, and also in terms of associated Laguerre polynomials. Remarkably both expansions entail not constant coefficients but appropriate function coefficients instead. We highlight the relation of these results with inspiring new developments. Our results set the stage for deriving similar results in other black hole geometries 4-dim and higher. [ABSTRACT FROM AUTHOR]

For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in the work of Higuchi et al. (e-print arXiv:1506.06457) [see also E. Segawa and A. Suzuki, Quantum Stud.: Math. Found. 3, 11 (2016)]. In this paper, as an application of the spectral mapping theorem, we investigate the spectrum of a one-dimensional split-step quantum walk. We give a criterion for when there are no eigenvalues around ±1 in terms of a discriminant operator. We also provide a criterion for when eigenvalues ±1 exist in terms of birth eigenspaces. Moreover, we prove that eigenvectors from the birth eigenspaces decay exponentially at spatial infinity and that the birth eigenspaces are robust against perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and multiplicity of solutions for the following quasilinear Choquard equation: − Δ u + V (x) u − u Δ ( u 2 ) = (| x | − μ * | u | p ) | u | p − 2 u , x ∈ R N , where N ≥ 3, μ ∈ (0 , N + 2 2 ) , p ∈ (2 , 4 N − 4 μ N − 2 ). Under some assumptions on V, we obtain the existence of positive solutions, negative solutions, and high-energy solutions via perturbation method. [ABSTRACT FROM AUTHOR]

The higher order discrete rogue waves (RWs) of the integrable discrete Ablowitz-Ladik equation are reported using a novel discrete version of generalized perturbation Darboux transformation. The dynamical behaviors of strong and weak interactions of these RWs are analytically and numerically discussed, which exhibit the abundant wave structures. We numerically show that a small noise has the weaker effect on strong-interaction RWs than weak-interaction RWs, whose main reason may be related to main energy distributions of RWs. The interaction of two first-order RWs is shown to be non-elastic. Moreover, we find that the maximal number (Smax) of the possibly split first-order ones of higher order RWs is related to the number (Pmax) of peak points of their strongest-interaction cases, that is, Smax = (Pmax + 1)/2. The results will excite to further understand the discrete RW phenomena in nonlinear optics and relevant fields. [ABSTRACT FROM AUTHOR]

Using the Batalin-Vilkovisky-formalism of mathematical physics, an explicit construction for the minimal model of a quantum L∞-algebra is given as a formal super integral. The approach taken herein to these formal integrals is axiomatic, and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the Appendix. Additionally, there exists a canonical differential graded Lie algebra morphism mapping formal functions on homology to formal functions on the whole space. An inverse L∞-algebra morphism to this differential graded Lie algebra morphism is constructed as a formal super integral. [ABSTRACT FROM AUTHOR]

We study functional and spectral properties of perturbations of the operator − ( ∂ s + i a ) 2 in L 2 ( S 1 ). This operator appears when considering the restriction to the unit circle of a two-dimensional Schrödinger operator with the Bohm-Aharonov vector potential. We prove a Hardy-type inequality on R 2 and, on S 1 , a sharp interpolation inequality and a sharp Keller-Lieb-Thirring inequality. [ABSTRACT FROM AUTHOR]

We study XXZ spin systems on general graphs. In particular, we describe the formation of droplet states near the bottom of the spectrum in the Ising phase of the model, where the Z-term dominates the XX-term. As key tools, we use particle number conservation of XXZ systems and symmetric products of graphs with their associated adjacency matrices and Laplacians. Of particular interest to us are strips and multi-dimensional Euclidean lattices, for which we discuss the existence of spectral gaps above the droplet regime. We also prove a Combes-Thomas bound which shows that the eigenstates in the droplet regime are exponentially small perturbations of strict (classical) droplets. [ABSTRACT FROM AUTHOR]

The scalar-tensor theories have become popular recently in particular in connection with attempts to explain present accelerated expansion of the universe, but they have been considered as a natural extension of general relativity long time ago. The Horndeski scalar-tensor theory involving four invariantly defined Lagrangians is a natural choice since it implies field equations involving at most second derivatives. Following the formalisms of defining covariant global quantities and conservation laws for perturbations of spacetimes in standard general relativity, we extend these methods to the general Horndeski theory and find the covariant conserved currents for all four Lagrangians. The current is also constructed in the case of linear perturbations involving both metric and scalar fields. As a specific illustration, we derive a superpotential that leads to the covariantly conserved current in the Branse-Dicke theory. [ABSTRACT FROM AUTHOR]

We address the problem of characterising the compatible tuples of measurements that admit a unique joint measurement. We derive a uniqueness criterion based on the method of perturbations and apply it to show that extremal points of the set of compatible tuples admit a unique joint measurement, while all tuples that admit a unique joint measurement lie in the boundary of such a set. We also provide counter-examples showing that none of these properties are both necessary and sufficient, thus completely describing the relation between the joint measurement uniqueness and the structure of the compatible set. As a by-product of our investigations, we completely characterise the extremal and boundary points of the set of general tuples of measurements and of the subset of compatible tuples. [ABSTRACT FROM AUTHOR]

We derive a spontaneous emission rate and line shift for two-level atoms coupled to the radiation field using causal perturbation theory. In this approach, employing the theory of distribution splitting prevents the occurrence of divergent integrals. Our method confirms the result for an atomic decay rate but suggests that the cutoff frequency for the atomic line shift is determined by the atomic mass, rather than the Bohr radius or electron mass. [ABSTRACT FROM AUTHOR]

We study motion of a charged particle confined to a Dirichlet layer of a fixed width placed into a homogeneous magnetic field. If the layer is planar and the field is perpendicular to it, the spectrum consists of infinitely degenerate eigenvalues. We consider translationally invariant geometric perturbations and derive several sufficient conditions under which a magnetic transport is possible, that is, the spectrum, in its entirety or a part of it, becomes absolutely continuous. [ABSTRACT FROM AUTHOR]

In this paper, we consider the 3-dimensional Klein-Gordon-Schrödinger system under the dinucleon interactions. By introducing the atomic spaces and establishing local Strichartz estimates for the perturbed Schrödinger equation to overcome the lack of compactness in whole space, we prove the unique solvability in the energy space. Additionally, we derive a Lipschitz estimate in a weak topology for the linearized equations to obtain the continuous dependence on the initial data. [ABSTRACT FROM AUTHOR]

We study the quasilinear Schrödinger equation arising from the nonlinear dynamics of the superfluid condensate -Δu+λu+ K/2 (Δu2)u = β[1/α3 - (1/α+u2)3 ]u, x ϵ R3, where λ, k,α and β are positive constants. By developing perturbation arguments, we prove that for each λ, θ,M > 0 with α k = θ and θ α-3 k =M, there exists k 0 > 0 such that for k ϵ (0, k0), the equation has a positive classical radial solution uk satisfyingmaxxϵR3 ǀkμuk (x)ǀ→0 for any μ ⩾ 1/2 as k → 0+. Moreover, up to a subsequence, it follows that uk → u0 in H2(R3) ∩ C2(R3) as k → 0+, where u0 is the least energy solution of problem -Δu + λu = 3Mθ-1u3, x 2 R3. Our existence result generalizes the previous result in onedimensional space obtained by Brüll and Lange in 1986. [ABSTRACT FROM AUTHOR]

We study the quasilinear Schrödinger equation arising from the nonlinear dynamics of the superfluid condensate -Δu+λu+ K/2 (Δu2)u = β[1/α3 - (1/α+u2)3 ]u, x ϵ R3, where λ, k,α and β are positive constants. By developing perturbation arguments, we prove that for each λ, θ,M > 0 with α k = θ and θ α-3 k =M, there exists k 0 > 0 such that for k ϵ (0, k0), the equation has a positive classical radial solution uk satisfyingmaxxϵR3 ǀkμuk (x)ǀ→0 for any μ ⩾ 1/2 as k → 0+. Moreover, up to a subsequence, it follows that uk → u0 in H2(R3) ∩ C2(R3) as k → 0+, where u0 is the least energy solution of problem -Δu + λu = 3Mθ-1u3, x 2 R3. Our existence result generalizes the previous result in onedimensional space obtained by Brüll and Lange in 1986. [ABSTRACT FROM AUTHOR]

We study resonances generated by rank-one perturbations of self-adjoint operators with eigenvalues embedded in the continuous spectrum. Instability of these eigenvalues is analyzed and almost exponential decay for the associated resonant states is exhibited. We show how these results can be applied to Sturm-Liouville operators. Main tools are the Aronszajn-Donoghue theory for rank-one perturbations, a reduction process of the resolvent based on the Feshbach-Livsic formula, the Fermi golden rule, and a careful analysis of the Fourier transform of quasi-Lorentzian functions. We relate these results to sojourn time estimates and spectral concentration phenomena. [ABSTRACT FROM AUTHOR]

We present a detailed analysis of the orbital stability of the Pais-Uhlenbeck oscillator, using Lie-Deprit series and Hamiltonian normal form theories. In particular, we explicitly describe the reduced phase space for this Hamiltonian system and give a proof for the existence of stable orbits for a certain class of self-interaction, found numerically in previous studies, by using singular symplectic reduction. [ABSTRACT FROM AUTHOR]

In the present work, we consider a variety of two-component, one-dimensional states in nonlinear Schrödinger equations in the presence of a parabolic trap, inspired by the atomic physics context of Bose-Einstein condensates. The use of Lyapunov-Schmidt reduction methods allows us to identify persistence criteria for the different families of solutions which we classify as (m, n), in accordance with the number of zeros in each component. Upon developing the existence theory, we turn to a stability analysis of the different configurations, using the Krein signature and the Hamiltonian-Krein index as topological tools identifying the number of potentially unstable eigendirections for each branch. A perturbation expansion for the eigenvalue problems associated with nonlinear states found near the linear limit permits us to obtain explicit asymptotic expressions for the eigenvalues. Finally, when the states are found to be unstable, typically by virtue of Hamiltonian Hopf bifurcations, their dynamics is studied in order to identify the nature of the respective instability. The dynamics is generally found to lead to a vibrational evolution over long time scales. [ABSTRACT FROM AUTHOR]

Perturbation problems for operators with embedded eigenvalues are generally challenging since the embedded eigenvalues cannot be separated from the rest of the spectrum. In this paper, we study a perturbation problem for embedded eigenvalues for a magnetic Schrödinger operator, when the underlying domain is a cylinder. The magnetic potential is C² with an algebraic decay rate as the unbounded variable of the cylinder tends to ±∞. In particular, no analyticity of the magnetic potential is assumed. We also assume that the embedded eigenvalue of the unperturbed problem is not the square of an integer, thus avoiding the thresholds of the continuous spectrum of the unperturbed operator. We show that the set of nearby potentials, for which a simple embedded eigenvalue persists, forms a smooth manifold of finite codimension. [ABSTRACT FROM AUTHOR]

In this work, we analyze a system consisting in two-dimensional position-dependent massive particles in the presence of a Morse-like potential in two spatial dimensions. We obtain the exact wavefunctions and energies for a complete set of eigenstates for a given dependence of the mass with the spatial variables. Furthermore, we argue that this scenario can play an important role to construct more realistic ones by using their solution in perturbative approaches. [ABSTRACT FROM AUTHOR]

We consider the global classical solution near a global Maxwellian to the one-species Vlasov-Poisson-Landau system in the whole space Rx³. It is shown that our global solvability result is obtained under the weaker smallness condition on the initial perturbation than that of Duan et al., [preprint arXiv:1112.3261 (2011)] and Lei et al., [Kinet. Relat. Models 7(3), 551-590 (2014)]. [ABSTRACT FROM AUTHOR]

The paper focuses on a transmission eigenvalue problem for Maxwell's equations with cubic nonlinearity that describes the propagation of transverse magnetic waves along the boundaries of a dielectric layer filled with nonlinear (Kerr) medium. Using an original approach, it is proved that even for small values of the nonlinearity coefficient, the nonlinear problem has infinitely many nonperturbative solutions (eigenvalues and eigenwaves), whereas the corresponding linear problem always has a finite number of solutions. This fact implies the theoretical existence of a novel type of eigenwaves that do not reduce to the linear ones in the limit in which the nonlinear coefficient reduces to zero. Asymptotic distribution of the eigenvalues is found, periodicity of the eigenfunctions is proved, the exact formula for the period is found, and the zeros of the eigenfunctions are determined. [ABSTRACT FROM AUTHOR]

We study a class of Anderson Hamiltonians with heavy-tailed independent and identically distributed random potential on graphs with sub-exponential growth of the balls and of the number of self-avoiding paths connecting pairs of points. We show that for a class of marginal distributions, Anderson localization occurs non-perturbatively, i.e., for any nonzero amplitude of the potential, like in one-dimensional systems. The proof is based on the moment analysis of the Green functions via large deviations estimates. [ABSTRACT FROM AUTHOR]