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Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time. [ABSTRACT FROM AUTHOR]

We present a multiscale method coupling the theory of open quantum systems with real-time ab initio treatment of electronic structure to study hot-carrier dynamics in photoexcited plasmonic systems. We combine the Markovian Stochastic Schrödinger equation with an ab initio GW coupled to the Bethe–Salpeter (BSE) equation description of the electronic degrees of freedom, interacting with a metallic nanoparticle modeled classically according to the polarizable continuum model. We apply this methodology to study the effect of relaxation (T1) and pure dephasing (T2) times on the hot-carrier dynamics in a system composed of a quantum portion described at GW/BSE level, i.e., a CHO fragment adsorbed on a vertex of a rhodium nanocube, and of the rest of the nanocube, treated classically, when irradiated with a 2.7 eV light pulse, inspired by the experimental results on plasmon-driven CO2 photoreduction. A net hole injection from rhodium to CHO is observed, with and without the classical portion of the nanocube. The nanocube effect is to enhance the generated charge population by two orders of magnitude. The nonradiative decay, via a relaxation time T1 based on the energy-gap law, produces a rapid decrease of the charge population. Results with T2 only show that a charge injection retarded with respect to the pulse, which is present in the coherent dynamics, disappears when coherence is erased. [ABSTRACT FROM AUTHOR]

A central aim of multiscale modeling is to use results from the Schrödinger equation to predict phenomenology on length scales that far exceed those of typical molecular correlations. In this work, we present a new approach rooted in classical density functional theory (cDFT) that allows us to accurately describe the solvation of apolar solutes across length scales. Our approach builds on the Lum–Chandler–Weeks (LCW) theory of hydrophobicity [K. Lum et al., J. Phys. Chem. B 103, 4570 (1999)] by constructing a free energy functional that uses a slowly varying component of the density field as a reference. From a practical viewpoint, the theory we present is numerically simpler and generalizes to solutes with soft-core repulsion more easily than LCW theory. Furthermore, by assessing the local compressibility and its critical scaling behavior, we demonstrate that our LCW-style cDFT approach contains the physics of critical drying, which has been emphasized as an essential aspect of hydrophobicity by recent theories. As our approach is parameterized on the two-body direct correlation function of the uniform fluid and the liquid–vapor surface tension, it straightforwardly captures the temperature dependence of solvation. Moreover, we use our theory to describe solvation at a first-principles level on length scales that vastly exceed what is accessible to molecular simulations. [ABSTRACT FROM AUTHOR]

Evaluation of the charge transport property of organic semiconductors requires exact quantum dynamics simulation of large systems. We present a numerically nearly exact approach to investigate carrier transport dynamics in organic semiconductors by extending the non-Markovian stochastic Schrödinger equation with complex frequency modes to a forward–backward scheme and by solving it using the matrix product state (MPS) approach. By utilizing the forward–backward formalism for noise generation, the bath correlation function can be effectively treated as a temperature-independent imaginary part, enabling a more accurate decomposition with fewer complex frequency modes. Using this approach, we study the carrier transport and mobility in the one-dimensional Peierls model, where the nonlocal electron–phonon interaction is taken into account. The reliability of this approach was validated by comparing carrier diffusion motion with those obtained from the hierarchical equations of motion method across various parameter regimes of the phonon bath. The efficiency was demonstrated by the modest virtual bond dimensions of MPS and the low scaling of the computational time with the system size. [ABSTRACT FROM AUTHOR]

Conical intersections (CIs) play a crucial role in photochemical reactions, offering an efficient channel for ultrafast non-adiabatic relaxation of excited states. This significantly influences the reaction pathways and the resulting products. In this work, we utilize the non-Markovian stochastic Schrödinger equation with complex modes method to explore the dynamics of electronic transitions through conical intersections (CIs) in pyrazine. The linear vibronic coupling model serves as the foundational framework, incorporating both intra-state and inter-state electron–vibrational interactions. The dynamics of the excited electronic transitions are analyzed across varying strengths of system–bath coupling and different bath relaxation times. The accuracy of this method is demonstrated by comparing its predictions with those from the hierarchical equations of motion method. [ABSTRACT FROM AUTHOR]

Robust control of a quantum system is essential to utilize the current noisy quantum hardware to its full potential, such as quantum algorithms. To achieve such a goal, a systematic search for an optimal control for any given experiment is essential. The design of optimal control pulses requires accurate numerical models and, therefore, accurate characterization of the system parameters. We present an online Bayesian approach for quantum characterization of qutrit systems, which automatically and systematically identifies optimal experiments that provide maximum information on the system parameters, thereby greatly reducing the number of experiments that need to be performed on the quantum testbed. Unlike most characterization protocols that provide point-estimates of the parameters, the proposed approach is able to estimate their probability distribution. The applicability of the Bayesian experimental design technique was demonstrated on test problems, where each experiment was defined by a parameterized control pulse. In addition to this, we also present an approach for iterative pulse extension, which is robust under uncertainties in transition frequencies and coherence times, and shot noise, despite being initialized with wide uninformative priors. Furthermore, we provide a mathematical proof of the theoretical identifiability of the model parameters and present conditions on the quantum state under which the parameters are identifiable. The proof and conditions for identifiability are presented for both closed and open quantum systems using the Schrödinger equation and the Lindblad master equation, respectively. [ABSTRACT FROM AUTHOR]

Hydrogen clathrate hydrates are ice-like crystalline substances in which hydrogen molecules are trapped inside polyhedral cages formed by the water molecules. Small cages can host only a single H2 molecule, while each large cage can be occupied by up to four H2 molecules. Here, we present a neutron scattering study on the structure of the sII hydrogen clathrate hydrate and on the low-temperature dynamics of the hydrogen molecules trapped in its large cages, as a function of the gas content in the samples. We observe spectral features at low energy transfer (between 1 and 3 meV), and we show that they can be successfully assigned to the rattling motion of a single hydrogen molecule occupying a large water cage. These inelastic bands remarkably lose their intensity with increasing the hydrogen filling, consistently with the fact that the probability of single occupation (as opposed to multiple occupation) increases as the hydrogen content in the sample gets lower. The spectral intensity of the H2 rattling bands is studied as a function of the momentum transfer for partially emptied samples and compared with three distinct quantum models for a single H2 molecule in a large cage: (i) the exact solution of the Schrödinger equation for a well-assessed semiempirical force field, (ii) a particle trapped in a rigid sphere, and (iii) an isotropic three-dimensional harmonic oscillator. The first model provides good agreement between calculations and experimental data, while the last two only reproduce their qualitative trend. Finally, the radial wavefunctions of the three aforementioned models, as well as their potential surfaces, are presented and discussed. [ABSTRACT FROM AUTHOR]

The electron–phonon scattering plays a crucial role in determining the electronic, transport, optical, and thermal properties of materials. Here, we employ a non-Markovian stochastic Schrödinger equation (NMSSE) in momentum space, together with ab initio calculations for energy bands and electron–phonon interactions, to reveal the phonon-mediated ultrafast hole relaxation dynamics in the valence bands of monolayer black phosphorus. Our numerical simulations show that the hole can initially remain in the high-energy valence bands for more than 100 fs due to the weak interband scatterings, and its energy relaxation follows single-exponential decay toward the valence band maximum after scattering into low-energy valence bands. The total relaxation time of holes is much longer than that of electrons in the conduction band. This suggests that harnessing the excess energy of holes may be more effective than that of electrons. Compared to the semiclassical Boltzmann equation based on a hopping model, the NMSSE highlights the persistence of quantum coherence for a long time, which significantly impacts the relaxation dynamics. These findings complement the understanding of hot carrier relaxation dynamics in two-dimensional materials and may offer novel insights into harnessing hole energy in photocatalysis. [ABSTRACT FROM AUTHOR]

We examine the optical and electronic properties of a GaAs spherical quantum dot with a hydrogenic impurity in its center. We study two different confining potentials: (1) a modified Gaussian potential and (2) a power-exponential potential. Using the finite difference method, we solve the radial Schrodinger equation for the 1s and 1p energy levels and their probability densities and subsequently compute the optical absorption coefficient (OAC) for each confining potential using Fermis golden rule. We discuss the role of different physical quantities influencing the behavior of the OAC, such as the structural parameters of each potential, the dipole matrix elements, and their energy separation. Our results show that modification of the structural physical parameters of each potential can enable new optoelectronic devices that can leverage inter-sub-band optical transitions.

We present a numerically exact approach for evaluating vibrationally resolved electronic spectra at finite temperatures using the coherence thermofield dynamics. In this method, which avoids implementing an algorithm for solving the von Neumann equation for coherence, the thermal vibrational ensemble is first mapped to a pure-state wavepacket in an augmented space, and this wavepacket is then propagated by solving the standard, zero-temperature Schrödinger equation with the split-operator Fourier method. We show that the finite-temperature spectra obtained with the coherence thermofield dynamics in a Morse potential agree exactly with those computed by Boltzmann-averaging the spectra of individual vibrational levels. Because the split-operator thermofield dynamics on a full tensor-product grid is restricted to low-dimensional systems, we briefly discuss how the accessible dimensionality can be increased by various techniques developed for the zero-temperature split-operator Fourier method. [ABSTRACT FROM AUTHOR]

The N–H photodissociation dynamics of the pyridinyl radical upon continuous excitation to the optically bright, first excited ππ* electronic state by an ultra-violet (UV) laser pulse has been investigated within the mathematical framework of optimal control theory. The genetic algorithm (GA) is employed as the optimization protocol. We considered a three-state and three-mode model Hamiltonian, which includes the reaction coordinate, R (a1 symmetry); the coupling coordinates (namely, out-of-plane bending coordinate of the hydrogen atom of azine group), Θ (b1 symmetry); and the wagging mode, Q9 (a2 symmetry). The three electronic states are the ground, ππ*, and πσ* states. The πσ* state crosses both the ground state and the ππ* state, and it is a repulsive state on which N–H dissociation occurs upon photoexcitation. Different vibrational wave functions along the coupling coordinates, Θ and Q9, of the ground electronic state are used as the initial condition for solving the time-dependent Schrödinger equation. The optimal UV laser pulse is designed by applying the GA, which maximizes the dissociation yield. We obtained over 95% dissociation yield through the πσ* asymptote using the optimal pulse of a time duration of ∼30 000 a.u. (∼725.66 fs). [ABSTRACT FROM AUTHOR]

Arising from the harmonic approximation in solving the vibrational Schrödinger equation, normal modes dissect molecular vibrations into distinct degrees of freedom. Normal modes are widely used as they give rise to descriptive vibrational notations and are convenient for expanding anharmonic potential energy surfaces as an alternative to higher-order Taylor series representations. Usually, normal modes are expressed in Cartesian coordinates, which bears drawbacks that can be overcome by switching to internal coordinates. Considering vibrational notations, normal modes with delocalized characters are difficult to denote, but internal coordinates offer a route to clearer notations. Based on the Hessian, normal mode decomposition schemes for a given set of internal coordinates can describe a normal mode by its contributions from internal coordinates. However, choosing a set of internal coordinates is not straightforward. While the Hessian provides unique sets of normal modes, various internal coordinate sets are possible for a given system. In the present work, we employ a normal mode decomposition scheme to choose an optimal set. Therefore, we screen reasonable sets based on topology and symmetry considerations and rely on a metric that minimizes coupling between internal coordinates. Ultimately, the Nomodeco toolkit presented here generates internal coordinate sets to find an optimal set for representing molecular vibrations. The resulting contribution tables can be used to clarify vibrational notations. We test our scheme on small to mid-sized molecules, showing how the space of definable internal coordinate sets can significantly be reduced. [ABSTRACT FROM AUTHOR]

Chemical and photochemical reactivity, as well as supramolecular organization and several other molecular properties, can be modified by strong interactions between light and matter. Theoretical studies of these phenomena require the separation of the Schrödinger equation into different degrees of freedom as in the Born–Oppenheimer approximation. In this paper, we analyze the electron–photon Hamiltonian within the cavity Born–Oppenheimer approximation (CBOA), where the electronic problem is solved for fixed nuclear positions and photonic parameters. In particular, we focus on intermolecular interactions in representative dimer complexes. The CBOA potential energy surfaces are compared with those obtained using a polaritonic approach, where the photonic and electronic degrees of freedom are treated at the same level. This allows us to assess the role of electron–photon correlation and the accuracy of CBOA. [ABSTRACT FROM AUTHOR]

A high-order implicit–explicit (IMEX) finite element method with energy conservation is constructed to solve the regularized logarithmic Schrödinger equation (RLogSE) with a periodic boundary condition. The discrete scheme consists of the relaxation-extrapolated Runge–Kutta (RERK) method in the temporal direction and the finite element method in the spatial direction. Choosing a proper relaxation parameter for the RERK method is the key technique for energy conservation. The optimal error estimates in the L 2 -norm and H 1 -norm are provided without any restrictions between time step size τ and mesh size h by temporal–spatial splitting technology. Numerical examples are given to demonstrate the theoretical results. • Establishing an energy-preserving discretization for Schrödinger equations. • Adopting FEMs and IMEX RERK in time-domain direction, respectively. • Obtaining the optimal convergent rate without any restrictions by the temporal-spatial splitting technology. [ABSTRACT FROM AUTHOR]

The application of topological defects to the study of different molecular potential models has been a focus of interest in recent times due to its significant role in shaping the behavior and interactions of molecular systems. In lieu of this, the asymptotic iteration method is employed to obtain the eigensolutions of the Schrödinger equation of the modified shifted Morse potential model with topological defect. Vibrational energies for carbon monoxide have been presented numerically for various topological defects and the results are compared with experimental data. With the help of the energy expression of modified shifted Morse potential, the q-deformed partition function and other superstatistical function expressions of modified shifted Morse potential for carbon monoxide within the modified Dirac delta distribution are obtained, using the Euler–MacLaurin’s formula. For various temperature values considered, the superstatistical plots show a high level of dependence on the topological defect parameter and deformation parameter. The superstatistical functions for a deformation parameter of zero correspond to the conventional thermodynamic functions. Our results also agree with the available results in the literature. [ABSTRACT FROM AUTHOR]

Over the last four decades, Sinc methods have occupied an important place in numerical analysis due to their simplicity and great performance. An incorporation of the Sinc collocation method with double exponential transformation is used to solve the two-dimensional time dependent Schrödinger equation. Numerical comparison between the double exponential and single exponential approaches is made to illustrate the superiority of the double exponential Sinc method. [ABSTRACT FROM AUTHOR]

The Schrödinger–Virasoro algebra was originally introduced during the process of investigating the free Schrödinger equations. In order to investigate vertex representations of the Schrödinger–Virasoro algebra, Unterberger introduced a new infinite-dimensional Lie algebra s v ̃ called the extended Schrödinger–Virasoro algebra in the context of two-dimensional conformal field theory and statistical physics. In this paper, we study simple smooth modules over s v ̃ , which can be viewed as an extension of the Schrödinger–Virasoro algebra by a conformal current with conformal weight 1. More specifically, these modules are determined by simple modules over finite-dimensional solvable Lie algebras. Moreover, we present several equivalent descriptions for simple smooth modules over s v ̃. [ABSTRACT FROM AUTHOR]

We consider the following higher-order Schrödinger equation involving supercritical growth and competing potentials: * (- Δ) m u + V (y) u = Q (y) u m ∗ - 1 + ε , u > 0 , in R N , u ∈ D m , 2 (R N) , where m ∗ = 2 N N - 2 m , N ≥ 4 m + 1 , m ≥ 2 is an integer, (y ′ , y ′ ′) ∈ R 2 × R N - 2 , V (y) = V (| y ′ | , y ′ ′) and Q (y) = Q (| y ′ | , y ′ ′) ≢ 0 are two bounded nonnegative functions. By using the finite-dimensional reduction argument and local Pohozaev-type identities, under some suitable assumptions on the potentials V and Q, we prove that for any small ε > 0 , the problem (∗) has a large number of bubble solutions whose functional energy is in the order ε - N - 4 m (N - 2 m) 2 . [ABSTRACT FROM AUTHOR]

In this work, a precise regularization technique based on the mollification method is proposed to solve the ill-posed Schr $ \ddot {o} $ o ¨ dinger Cauchy problem with potential-free field. In the sense of Hadamard, this problem is severely ill-posed. Using the mollification regularization method based on Poisson kernel, some new error estimates are obtained under the priori and posteriori parameter selection rules. The regular approximate solutions are created, and the convergence evidences are provided. The good performance and high accuracy of this technique are demonstrated through various examples. The solution works well and is resistant to data disruption noise. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKEs) for water waves models. This problem has received intense attention in recent years in the context of semilinear models, such as Schrödinger equations or multidimensional KdV‐type equations. However, our situation here is different since the water waves equations are quasilinear and solutions cannot be constructed by iteration of the Duhamel formula due to unavoidable derivative loss. This is the first of two papers in which we design a new strategy to address this issue. We investigate solutions of the gravity water waves system in two dimensions. In the irrotational case, this system can be reduced to an evolution equation on the one‐dimensional interface, which is a large torus TR${\mathbb {T}}_R$ of size R≥1$R\ge 1$. Our first main result is a deterministic energy inequality, which provides control of (possibly large) Sobolev norms of solutions for long times, under the condition that a certain L∞$L^\infty$‐type norm is small. This energy inequality is of "quintic" type: if the L∞$L^\infty$ norm is O(ε)$O(\varepsilon)$, then the increment of the high‐order energies is controlled for times of the order ε−3$\varepsilon ^{-3}$, consistent with the approximate quartic integrability of the system. In the second paper in this sequence, we will show how to use this energy estimate and a propagation of randomness argument to prove a probabilistic regularity result up to times of the order ε−3R3/2−${\varepsilon }^{-3} R^{3/2-}$, in a suitable scaling regime relating ε→0${\varepsilon }\rightarrow 0$ and R→∞$R \rightarrow \infty$. For our second main result, we combine the quintic energy inequality with a bootstrap argument using a suitable Z$Z$‐norm of Strichartz‐type to prove that deterministic solutions with Sobolev data of size ε≪1${\varepsilon }\ll 1$ are regular for times of the order ε−3min(ε−3,R3/4)${\varepsilon }^{-3} \min \big ({\varepsilon }^{-3}, R^{3/4} \big)$. In particular, on the real line, solutions exist for times of order O(ε−6)$O(\varepsilon ^{-6})$. This improves substantially on all the earlier extended lifespan results for 2D gravity water waves with small Sobolev data. [ABSTRACT FROM AUTHOR]

We consider the analysis of singular waveguides separating insulating phases in two‐space dimensions. The insulating domains are modeled by a massive Schrödinger equation and the singular waveguide by appropriate jump conditions along the one‐dimensional interface separating the insulators. We present an integral formulation of the problem and analyze its mathematical properties. We also implement a fast multipole and sweeping‐accelerated iterative algorithm for solving the integral equations, and demonstrate numerically the fast convergence of this method. Several numerical examples of solutions and scattering effects illustrate our theory. [ABSTRACT FROM AUTHOR]

The current study employs the Nikiforov-Uvarov method to solve the Schrödinger equation for quarkonium systems, utilizing the radial scalar power potential. The eigenvalues of energy and their corresponding wave functions are determined by including the spin–spin, spin–orbit, and tensor interactions in the radial scalar power potential. The mass spectra of charmonia, bottomonia, and bottom-charm in their S, P, D, and F states were determined. Our theoretical states for quarkonium systems align with experimental data across a range of spin levels, as evidenced by our comparison. The total percentage error of our work was computed, yielding a high level of accuracy. The cumulative percentage error for the meson masses of charmonia and bottomonia was determined to be 0.324% and 0.333%, respectively. The masses of the bottom-charm mesons had a total percentage error of 0.012%. Consequently, the present potential yields favorable outcomes for the quarkonium masses, surpassing previous theoretical studies and aligning well with experimental data. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fourth-order Schrödinger equations with focusing, $ L^2 $-supercritical nonlinearity in one dimension. We prove the global existence and scattering of even solutions below the ground state threshold, under the assumption that the threshold for even solutions is less than a certain constant times the threshold for general solutions. [ABSTRACT FROM AUTHOR]

We are concerned with the existence and mass collapse of standing waves with prescribed mass for the Schrödinger equation with general potential and nonlinearities. For local nonlinearities, this equation arises in the theory of Bose–Einstein condensates. For nonlocal nonlinearities, this equation is the Choquard euqation, which appears in the quantum theory of a polaron at rest. We used a unified approach (local minizing method) to study the existence of standing waves for the local, nonlocal and dipolar type cases. And then we established the mass collapse result. [ABSTRACT FROM AUTHOR]

We consider the existence of normalized solutions of the following nonlinear Schrödinger equation - Δ u + λ u = g (u) in Ω , ∫ Ω u 2 d x = ρ 2 , u ∈ H 0 1 (Ω) , where Ω ⊂ R N is a bounded domain with smooth boundary, N ≥ 3 , the nonlinearity g is Sobolev subcritical near infinity and at least mass critical growth near zero. We prove the existence of a solution, which is a local minimizer and obtain a second one of mountain pass type if Ω is a star-shaped domain in R N . Moreover, we uncover a relation between normalized solutions in R N and the corresponding normalized solutions on bounded domain B R (0) by analyzing the behavior of these solutions as R → ∞ . Our results are more general and we propose a different variational approach to deal with nonlinear Schrödinger equations on bounded domains with prescribed L 2 -norm. [ABSTRACT FROM AUTHOR]

We show that the dynamics of a quantum system can be represented by the dynamics of an underlying classical systems obeying the Hamilton equations of motion. This is achieved by transforming the phase space of dimension 2n into a Hilbert space of dimension n which is obtained by a peculiar canonical transformation that changes a pair of real canonical variables into a pair of complex canonical variables which are complex conjugate of each other. The probabilistic character of quantum mechanics is devised by treating the wave function as a stochastic variable. The dynamics of the underlying system is chosen so as to preserve the norm of the state vector. [ABSTRACT FROM AUTHOR]

We consider a partial data Calderón problem for elliptic boundary value problems with unbounded coefficients. In particular, we will show that partial measurement of the Neumann-Dirichlet data for the operator $ \Delta + q $ can uniquely determine $ q\in L^{n/2}(\Omega) $. This requires that we construct an explicit Green's function for the conjugated Laplacian with specified Neumann boundary conditions. The explicit construction of the Green's function allows us to deduce $ L^p $ type estimates which would otherwise be out of reach using the previous methods for constructing such Green's functions for bounded potentials. [ABSTRACT FROM AUTHOR]

In this article, we prove a bilinear estimate for Schrödinger equations on 2d waveguide, R × T. We hope it may be of use in the further study of concentration compactness for cubic NLS on R × T. [ABSTRACT FROM AUTHOR]

This paper explores the Riemann–Hilbert method for deriving exact N‐soliton solutions of the sixth‐order nonlinear Schrödinger (6th‐NLS) equation with nonzero boundary condition. The analytical process comprises three fundamental steps. First, transformations are used to simplify the nonzero boundaries. Next, the inverse scattering method establishes a crucial link between the solutions of the 6th‐NLS equation and the corresponding Riemann–Hilbert problem. Finally, this Riemann–Hilbert problem is systematically solved. Additionally, selected parameter values in the solutions generate graphical representations, vividly illustrating the solutions to the 6th‐NLS equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a heavy quarkonium state in a hot QCD medium, namely the ground state of charmonium and bottomonium. The quark–antiquark potential is proposed in the form of a temperature-dependent Cornell potential, and is extended to include a temperature-dependent parameter. Then, for the proposed potential with different parametrizations of the Debye mass, the N-dimensional Schrödinger equation is analytically solved using the biconfluent Heun equation. The obtained energy eigenvalues and wave functions are employed to study the flavor dependence, the effect of dimensionality and the influence of the additional term on the mass and binding energy of 1S state of charmonium and bottomonium at finite temperature. The findings of this research are compared to those from past studies. We concluded that this work provides a simple and consistent description of quarkonium states at finite temperature in a hot QCD medium, highlighting the effect of noncompact extra dimensions and the flavor dependence of the binding energies of heavy quarkonia. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stochastic (3 + 1)-dimensional nonlinear Schrödinger equation (NLSE), a critical model for understanding wave dynamics under the influence of random perturbations. We apply He’s semi-inverse method to establish a variational formula for the stochastic (3 + 1)-dimensional NLSE and we use this formula to construct conservation laws corresponding to the symmetries of the model. Based on this formulation, a solitary solution can be easily obtained using the Ritz method. We explore approximate solutions of the stochastic (3 + 1)-dimensional NLSE equation by choice of a trial function in the region of the rectangular box in three cases. Subsequently, we approximate the trial function using a piecewise linear ansatz in one case of the two-box potential, followed by an approximation using quadratic polynomials with two free parameters instead of the piecewise linear ansatz. We demonstrate the effectiveness of the analytical approaches, namely, the improved modified extended tanh-function method and He’s semi-inverse variational principle method for seeking more exact solutions via the stochastic (3 + 1)-dimensional NLSE equation. As a result, bright solitons, singular solitons, periodic wave solution and dark solitons solutions are obtained. Furthermore, the dynamical behaviors of the model are discussed by investigating bifurcations at equilibrium points. The paper not only advances the theoretical understanding of the stochastic (3 + 1)D NLSE but also provides practical methodologies for solving this complex equation, offering new insights into wave phenomena influenced by stochastic effects. In addition, we employ the fractional reduced differential transform (F-RDT) method, interpreted through Caputo fractional derivatives, to construct a numerical solution utilizing hyperbolic functions. We further analyze the influence of the fractional order parameter, denoted as α, on the behavior of the numerical solution. This investigation includes a detailed examination of the discrepancies between the exact solitary wave solutions and the corresponding numerical approximations, allowing us to identify the absolute errors. By comparing the two, we provide insights into the accuracy and stability of the numerical approach for solving the studied equation, emphasizing how variations in α impact the precision of the model. [ABSTRACT FROM AUTHOR]