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A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical three-body system in one space dimension, comprised of one photon and two identical massive spin one-half Dirac particles, which can be thought of as two electrons (or alternatively, two positrons). Manifest covariance is achieved using Dirac’s formalism of multi-time wave functions, i.e., wave functions Ψ (x ph , x e 1 , x e 2) where x ph , x e 1 , x e 2 are generic spacetime events of the photon and two electrons, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifolds { x ph = x e 1 } and { x ph = x e 2 } compatible with conservation of probability current. The corresponding initial-boundary value problem is shown to be well-posed, and it is shown that the unique solution can be represented by a convergent infinite sum of Feynman-like diagrams, each one corresponding to the photon bouncing between the two electrons a fixed number of times. [ABSTRACT FROM AUTHOR]

We explore the construction of a generalized Dirac equation via the introduction of the notion of Clifford multivector-valued actions, which was inspired by the work of Kanatchikov [Kanatchikov IV. De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory. Rept Math Phys. 1999;43:157–170; Kanatchikov IV. De Donder-Weyl Hamiltonian formulation and precanonical quantization of Vielbein gravity. J Phys Conf Ser. 2013;442:012041] on the De Donder-Weyl formulation of field theory. Crucial in this construction is the evaluation of the exponentials of multivectors associated with Clifford (hypercomplex) analysis. Exact matrix solutions (instead of spinors) of the generalized Dirac equation in $ D = 2,3 $ D = 2 , 3 space–time dimensions are found. This formalism can be extended to curved space–time backgrounds. We conclude by proposing a wave-functional equation governing the quantum dynamics of branes living in C-spaces (Clifford spaces), and which is based on the De Donder–Weyl Hamiltonian formulation of field theory. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the existence of optimal solutions for control problems involving nonlinear stochastic Schrödinger equations perturbed by a multiplicative fractional Brownian motion. Finite-dimensional approximations and a linearization method are employed and lead to ε-optimal solutions for the considered problems. [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]

A closed form solution for the one-dimensional Schrödinger equation with a finite number of δ -interactions L q , I N y : = - y ″ + q (x) + ∑ k = 1 N α k δ (x - x k) y = λ y , 0 < x < b , λ ∈ C is presented in terms of the solution of the unperturbed equation L q y : = - y ″ + q (x) y = λ y , 0 < x < b , λ ∈ C and a corresponding transmutation (transformation) operator T I N f is obtained in the form of a Volterra integral operator. With the aid of the spectral parameter power series method, a practical construction of the image of the transmutation operator on a dense set is presented, and it is proved that the operator T I N f transmutes the second derivative into the Schrödinger operator L q , I N on a Sobolev space H 2 . A Fourier-Legendre series representation for the integral transmutation kernel is developed, from which a new representation for the solutions and their derivatives, in the form of a Neumann series of Bessel functions, is derived. [ABSTRACT FROM AUTHOR]

By the Klein-Gordon potential, we call a convolution-type integral with a kernel, which is the fundamental solution of the Klein-Gordon equation and also a solution of the Cauchy problem to the same equation. An interesting question having several important applications (in general) is what boundary condition can be imposed on the Klein-Gordon potential on the boundary of a given domain so that the Klein-Gordon equation with initial conditions complemented by this “transparent” boundary condition would have a unique solution within that domain still given by the Klein-Gordon potential. It amounts to finding the trace of the Klein-Gordon potential to the boundary of the given domain. In this article, we analyze this question and construct a novel initial boundary-value problem for the Klein-Gordon equation in characteristic coordinates.

The impurity segregation in diluted transition-metal bimetallic alloys is a subject of study in many fields of interest and still deserves more understanding. Using a tight-binding model for the case of a single substitutional metal impurity the segregation is investigated analysing the segregation energy. Different impurity positions are considered from the (001) top surface to a number of planes below it on a simple cubic lattice. With only one orbital per site, the impurity potentials are determined in order to satisfy charge neutrality through the Friedel sum rule. In particular, our attention will be mainly directed to the substitutional impurity located in the surface and in the bulk. Temperature effects are considered through the Sommerfeld Expansion. Interesting effects on the segregation energy are pointed out such as the appearance of a quantum interference on the electronic states due to the surface itself. The obtained trend of the segregation energy profile is in good agreement with the surface concentration experimental data. [ABSTRACT FROM AUTHOR]

We consider the Schrödinger operator with regular short range complex-valued potential in dimension d ≥ 1 . We show that, for d ≥ 2 , the unitarity of scattering operator for this Hamiltonian at high energies implies the reality of the potential (that is Hermiticity of Hamiltonian). In contrast, for d = 1 , we present complex-valued exponentially localized soliton potentials with unitary scattering operator for all positive energies and with unbroken PT symmetry. We also present examples of complex-valued regular short range potentials with real spectrum for d = 3 . Some directions for further research are formulated. [ABSTRACT FROM AUTHOR]

This paper concerns with the stabilizability for a quasilinear Klein–Gordon–Schrödinger system with variable coefficients in dimensionless form. The stabilizability of quaslinear Klein–Gordon-Wave system with the Kelvin–Voigt damping has been considered by Liu–Yan–Zhang (SIAM J Control Optim 61:1651–1678, 2023). Our main contribution is to find a suitable linear feedback control law such that the quasilinear Klein–Gordon–Schrödinger system is exponentially stable under certain smallness conditions. [ABSTRACT FROM AUTHOR]

In this paper, we report on the discovery of a previously-unknown type of long-range instability phenomenon for the one-dimensional linear Schrödinger (LS) equation on the vacuum spacetime quarter-plane. More specifically, the inhomogeneous LS on the half-line, with generic initial data, boundary conditions and forcing term, is addressed, as an illustrative paradigm of our techniques, in a classical, smooth context via the formula proposed by the linear Fokas' unified transform method. We, first, present a new and suitable decomposition of that formula in the complex plane in order to appropriately interpret various terms appearing in the formula, thus securing convergence in a strictly defined sense. We also write the solution in a form consistent with the fundamental principle of Ehrenpreis and Palamodov. This novel analysis then allows for the necessary rigorous a posteriori verification of the full initial-boundary-value problem, for the first time. This is followed by a thorough investigation of the behavior of the solution near the boundaries of the spatiotemporal domain. We prove that the integrals in this representation converge uniformly to 'prescribed' values and the solution admits a smooth extension up to the boundary only if certain compatibility conditions are satisfied by the data (with direct implications for efficient numerics, well-posedness and control). Importantly, moreover, based on our analysis, we perform an effective asymptotic study of far-field dynamics. This leads to new explicit asymptotic formulae which characterize the properties of the solution in terms of (in)compatibilities of the data at the 'corner' of the quadrant. In particular, we found out that the asymptotic behavior of the solution is sensitive to perturbations of the data at the corner. In all cases, even assuming the initial data to belong to the Schwartz class, the solution loses this property as soon as time becomes positive (implying an infinite-speed type of singularity propagation). Hereby, the recent discovery of a novel type of a long-range instability effect for the Stokes equation is further corroborated and elucidated by revisiting a celebrated lower-order linear evolution partial differential equation (PDE). It thence transpires that our rigorous analytical approach is straightforwardly extendable to other Schrödinger-like evolution equations as well as more general problems with dispersion formulated on domains with a quasi-infinite boundary. Finally, although occurrence of the new instability is most stunning in the case discussed herein, it is naturally conjectured that analogous phenomena shall too appear in variable-coefficient and nonlinear settings which remain to be accordingly investigated. [ABSTRACT FROM AUTHOR]

This paper introduces the Nucci reduction method, a novel and efficient approach for deriving exact solutions to the perturbed Gerdjikov–Ivanov equation, offering a significant advancement in the field. The suggested technique involves transforming the equation into real and imaginary components prior to application. We successfully obtained four distinct exact and explicit solutions, along with the corresponding first integrals. Explanations and presentations of solutions are given in a logical manner. We derive an analytical expression for the instability gain and examine its key features using linear stability analysis. Finally, we compare the correctness of the analytical and numerical solutions. We demonstrate the robustness and stability of solitary waves through numerical simulations. [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]

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]

Using the class of (μ,σ)\left(\mu ,\sigma )-nonexpansive mappings, we will effectively prove the uniqueness of the approximative fixed points set for equivalent nn-linear functional spaces F{\mathscr{F}}, where F{\mathscr{F}} is nonempty and identical in every bounded case in nn-Banach space.

Discrete versions of the Laplace and Dirac operators have been studied in the context of combinatorial models of statistical mechanics and quantum field theory. In this paper, we introduce several variations of the Laplace and Dirac operators on graphs, and we investigate graph-theoretic versions of the Schrödinger and Dirac equations. We provide a combinatorial interpretation for solutions of the equations and we prove gluing identities for the Dirac operator on lattice graphs, as well as for graph Clifford algebras. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new definition of the general conformable fractional (GCF) Laplace transform with respect to the function Φ generated by the fractional conformable function ϕ. By the new definition, the usual Laplace transform and the ρ - Laplace transform are special cases of the GCF Laplace transform. We prove several important properties of these GCF Laplace transforms. In addition, we also define the GCF cylindrical coordinates system and derive the GCF Laplace operator on these coordinates. Furthermore, we study the approximate solution of the GCF position-dependent mass (PDM) Schrödinger equation for a symmetrical modified Pöschl-Teller potential in the GCF cylindrical coordinate system by using the GCF Laplace transform method. The GCF radial wave function and GCF energy eigenvalue are obtained from the approximate solution of the GCF Schrödinger equation. The behavior of the bound state conformable fractional energy levels was demonstrated and analyzed by using the computational method for various values of the order of conformable fractionality α , quantum number, and mass of the molecule. Moreover, the maximum conformable fractional energy-eigenvalue is obtained with respect to the influence of the order of conformable fractionality α and mass for diatomic molecules H 2 , LiH, and HCl. [ABSTRACT FROM AUTHOR]

We discuss some simple Hückel-like matrix representations of non-Hermitian operators with antiunitary symmetries that include generalized P T (parity transformation followed by time-reversal) symmetry. One of them exhibits extremely broken antiunitary symmetry (complex eigenvalues for all nontrivial values of the model parameter) because of the degeneracy of the operator in the Hermitian limit. These examples illustrate the effect of point-group symmetry on the spectrum of the non-Hermitian operators. We construct the necessary unitary matrices by means of simple graphical representations of the non-Hermitian operators. [ABSTRACT FROM AUTHOR]

Solving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of O ε - 2 , where 0 < ε ≪ 1 is inversely proportional to the speed of light. Yongyong Cai and Yan Wang have shown, however, that such solutions can be approximated up to an error of O ε 2 by solving the semi-nonrelativistic limit system, which is a non-oscillatory problem. For this system, we construct a two-step method, called the explicit exponential midpoint rule, and prove second-order convergence of the semi-discretization in time. Furthermore, we construct a benchmark method based on standard techniques and compare the efficiency of both methods. Numerical experiments show that the new integrator reduces the computational costs per time step to 40% and within a given runtime improves the accuracy significantly. [ABSTRACT FROM AUTHOR]

In this paper, we study global stability dynamics for quasilinear damped Klein–Gordon equation with variable coefficients in R n for dimension n ≥ 1 , without assuming that the quasilinear term satisfies the null condition. Due to the influence of damping term, we are able to prove that if it admits a global smooth solution (including time quasi-periodic solution), then this quasilinear system must be asymptotic stable in Sobolev space under some assumptions on the variable coefficients of it. This result also implies that a smooth large solution can be constructed for such quasilinear system, which includes time quasi-periodic stable dynamics of Klein–Gordon equation. We note the global stability for a class of semilinear wave equation satisfying null condition has been shown by Zha (J Funct Anal 282:109219, 2022). [ABSTRACT FROM AUTHOR]

The familiar second derivative test for convexity, combined with resolvent calculus, is shown to yield a useful tool for the study of convex matrix-valued functions. We demonstrate the applicability of this approach on a number of theorems in this field. These include convexity principles which play an essential role in the Lieb–Ruskai proof of the strong subadditivity of quantum entropy. [ABSTRACT FROM AUTHOR]

We consider the perturbed Stark operator H q φ = - φ ′ ′ + x φ + q (x) φ , φ (0) = 0 , in L 2 (R +) , where q is a real function that belongs to A r = q ∈ A r ∩ AC [ 0 , ∞) : q ′ ∈ A r , where A r = L R 2 (R + , (1 + x) r d x) and r > 1 is arbitrary but fixed. Let λ n (q) n = 1 ∞ and κ n (q) n = 1 ∞ be the spectrum and associated set of norming constants of H q . Let { a n } n = 1 ∞ be the zeros of the Airy function of the first kind, and let ω r : N → R be defined by the rule ω r (n) = n - 1 / 3 log 1 / 2 n if r ∈ (1 , 2) and ω r (n) = n - 1 / 3 if r ∈ [ 2 , ∞) . We prove that λ n (q) = - a n + π (- a n) - 1 / 2 ∫ 0 ∞ Ai 2 (x + a n) q (x) d x + O (n - 1 / 3 ω r 2 (n)) and κ n (q) = - 2 π (- a n) - 1 / 2 ∫ 0 ∞ Ai (x + a n) Ai ′ (x + a n) q (x) d x + O (ω r 3 (n)) , uniformly on bounded subsets of A r . In order to obtain these asymptotic formulas, we first show that λ n : A r → R and κ n : A r → R are real analytic maps. [ABSTRACT FROM AUTHOR]

One of the crucial properties of a quantum system is the existence of bound states. While the existence of eigenvalues below zero, that is, below the essential spectrum, is well understood, the situation of zero energy bound states at the edge of the essential spectrum is far less understood. We present complementary sharp criteria for the existence and nonexistence of zero energy ground states. Our criteria give a straightforward explanation for the folklore that there is a spectral phase transition with critical dimension four, concerning the existence versus nonexistence of zero energy ground states.

Recently, it has been proved that a set of spherically symmetric densities uniquely determines the Coulomb external potential. Utilizing this theorem, a previous method is extended to construct an orbital-free density functional theory. Auxilary spherical non-interacting systems and Kohn–Sham-like equations are constructed. A set of spherical generating functions are defined. These functions have two extra variables and are reduced to the set of spherically symmetric densities if the extra variables are equal to zero. Euler equations are rewritten as Schrödinger-like equations for the square root of the spherically symmetric generating functions. Generalized Pauli potentials are defined, and it is shown that these potentials can be calculated in the knowledge of the generating functions. The Euler equations can be solved with these Pauli potentials. [ABSTRACT FROM AUTHOR]

In this paper, we study the critical fractional Choquard equation with a local perturbation (- Δ) s u = λ u + μ | u | q - 2 u + (I α ∗ | u | 2 α , s ∗ ) | u | 2 α , s ∗ - 2 u , x ∈ R N , having prescribed mass ∫ R N u 2 d x = a 2 , where I α (x) is the Riesz potential, s ∈ (0 , 1) , N > 2 s , 0 < α < min { N , 4 s } , 2 < q < 2 s ∗ = 2 N N - 2 s is the fractional critical Sobolev exponent, and 2 α , s ∗ = 2 N - α N - 2 s is the fractional Hardy–Littlewood–Sobolev critical exponent, a > 0 , μ ∈ R. . Under some L 2 -subcritical, L 2 -critical and L 2 -supercritical perturbation μ | u | q - 2 u , respectively, we prove several existence and non-existence results. The qualitative behavior of the ground states as μ → 0 + is also studied. The mathematical analysis carried out in this paper can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions for fractional Choquard equation. In this framework, several related results are extended and improved. [ABSTRACT FROM AUTHOR]

We present several numerical methods and establish their error estimates forthe discretization of the nonlinear Dirac equation in the nonrelativistic limitregime, involving a small dimensionless parameter $0

We consider the $\mathbb {T}^{4}$ cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross–Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [24, 35] and does not require the existence of a solution in Strichartz-type spaces. We prove U-V multilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel–Born expansion. The new combinatorics and the U-V estimates then seamlessly conclude the $H^{1}$ unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove $H^{1}$ uniqueness for the $ \mathbb {R}^{3}/\mathbb {R}^{4}/\mathbb {T}^{3}/\mathbb {T}^{4}$ energy-critical Gross–Pitaevskii hierarchies and thus the corresponding NLS.