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We consider two-dimensional, non-relativistic quantum system with asymptotically straight soft waveguide. We show that the local deformation of the symmetric waveguide can lead to the emerging of the embedded eigenvalues living in the continuous spectrum. The main problem of this paper is devoted to the analysis of weak perturbation of the symmetric system. We show that the original embedded eigenvalues turn to the second sheet of the resolvent analytic continuation and constitute resonances. We describe the asymptotics of the real and imaginary components of the complex resonant pole depending on deformation. Finally, we generalize the problem to three dimensional system equipped with a soft layer. [ABSTRACT FROM AUTHOR]

The asymmetric quantum Rabi model (AQRM) is a fundamental model in quantum optics describing the interaction of light and matter. Besides its immediate physical interest, the AQRM possesses an intriguing mathematical structure which is far from being completely understood. In this paper, we focus on the distribution of the level spacing, the difference between consecutive eigenvalues of the AQRM in the limit of high energies, i.e. large quantum numbers. In the symmetric case, that is the quantum Rabi model (QRM), the spacing distribution for each parity (given by the Z 2 -symmetry) is fully clarified by an asymptotic expression derived by de Monvel and Zielinski, though some questions remain for the full spectrum spacing. However, in the general AQRM case, there is no parity decomposition for the eigenvalues. In connection with numerically exact studies and recent theoretical results, we describe the spacing distribution for the AQRM which is characterized by a new type of periodicity and symmetric behavior of the distribution with respect to the bias parameter, reflecting the hidden symmetry of the AQRM. In addition, we observe in the AQRM the excited state quantum phase transition for large values of the bias parameter, analogous to the QRM with large qubit energy, and an internal symmetry of the level spacing distribution for fixed bias. This novel symmetry is independent from the symmetry for half-integer bias and not explained by current theoretical knowledge. [ABSTRACT FROM AUTHOR]

This paper presents an investigation of the stability of a vortex with azimuthal velocity profile V ˉ = 1 âˆ' 1 âˆ' ε r 2 e âˆ' r 2 / r. When ε  = 0, the Lambâ€"Oseen vortex model is recovered. Although the Lambâ€"Oseen vortex supports propagating waves known as Kelvin waves, the flow is stable according to Rayleigh’s circulation criterion. In this paper, on the other hand, the modified vortex profile admits linearly unstable disturbances for ε  > 0 and we investigate their characteristics. These may be either axisymmetric or non-axisymmetric, but we find that the axisymmetric perturbations have the largest growth rates. When their growth rates are small, it becomes very difficult to solve the linear equation governing the axisymmetric perturbations because the eigenfunctions have a rapid exponential growth as one moves outward radially from the vortex center. To deal with such cases, a modified Riccati transformation was employed and found to be effective in solving the associated eigenvalue problem. [ABSTRACT FROM AUTHOR]

The authors replies to the comment made by Yang and Zhou (2024 J. Phys. A: Math. Theor.) on his 2015 paper entitled 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras' (Jacobsen 2015 J. Phys. A: Math. Theor. 48 454003). [ABSTRACT FROM AUTHOR]

In this paper, we use graph theory to solve wave scattering problems in the discrete dipole approximation. As a key result of this work, in the presence of active scatterers, we present a systematic method to find arbitrary large-order zero eigenvalue exceptional points (EPs). This is achieved by solving a set of non-linear equations that we interpret, in a graph theory picture, as vanishing sums of scattering events. We then show how the total field of the system responds to parameter perturbations at the EP. Finally, we investigate the sensitivity of the power output to imaginary perturbation in the design frequency. This perturbation can be employed to trade sensitivity for a different dissipation balance of the system. The purpose of the results of this paper is manifold. On the one hand, we aim to shed light on the link between graph theory and wave scattering. On the other hand, the results of this paper find application in all those settings where zero eigenvalue EPs play a unique role like in coherent perfect absorption structures. [ABSTRACT FROM AUTHOR]

The pretty good transmission (PGT) scenario implies that the probability of sending one excitation from one extreme of a spin chain to the other can reach values arbitrarily close to the unity just by waiting a time long enough. The conditions that ensure the appearance of this scenario are known for chains with different interactions and lengths. Sufficient conditions for the presence of PGT depend on the spectrum of the Hamiltonian of the spin chain. Some works suggest that the time t ɛ at which the PGT takes place scales as 1 / (| ε | ) f (N) , where ɛ is the difference between the probability that a single excitation propagates from one extreme of the chain to the other and the unity, while f (N) is an unknown function of the chain length. In this paper, we show that the exponent is not a simple function of the chain length but a power law of the number of linearly independent irrational eigenvalues of the one-excitation block of the Hamiltonian that enter into the expression of the probability of transmission of one excitation. We explicitly provide examples of a chain showing that the exponent changes when the couplings between the spins change while the length remains fixed. For centrosymmetric spin chains the exponent is at most N / 2. [ABSTRACT FROM AUTHOR]

The envelope theory (ET) is a reliable and easy to implement method to solve time independent Schrödinger-like equations (eigenvalues and eigenvectors). It is particularly useful to solve many-body systems since the computational cost is independent from the number of particles. The purpose of this paper is twofold. First, we want to make known a method that is probably too little used. Second, we also want to show that this method can be used as a pedagogical tool, thanks to its simplicity and the reliable results that can be obtained. To reach these goals, the ET is applied to a simple problem in one dimension, the soft-Coulomb potential − k / x 2 + d 2 , characterised by a bias distance d. Such interaction is used for the study of excitons, electron–hole bound pairs where the two charges are kept separated in two different one-dimensional regions (quantum wires). In addition to its physical interest, this system has never been treated with the ET. [ABSTRACT FROM AUTHOR]

In this paper, we present a framework to compare the differences in the occupation probabilities of two random walk processes, which can be generated by modifications of the network or the transition probabilities between the nodes of the same network. We explore a dissimilarity measure defined in terms of the eigenvalues and eigenvectors of the normalized Laplacian of each process. This formalism is implemented to examine differences in the diffusive dynamics described by circulant matrices, the effect of new edges, and the rewiring in networks as well as to evaluate divergences in the transport in degree-biased random walks and random walks with stochastic reset. Our results provide a general tool to compare dynamical processes on networks considering the evolution of states and capturing the complexity of these structures. [ABSTRACT FROM AUTHOR]

This paper shows how the freely downloadable software package called SMath Studio can be used to solve the time-independent Schrödinger equation using a simple step-wise approximation. The solution is illustrated with reference to the hydrogen atom for which trial eigenvalues are supplied by the energy level formula of the Bohr atom. [ABSTRACT FROM AUTHOR]

In this paper we will solve an open problem raised by Manásevich and Mawhin twenty years ago on the structure of the periodic eigenvalues of the vectorial p -Laplacian. This is an Eulerâ€"Lagrangian equation on the plane or in higher dimensional Euclidean spaces. The main result obtained is that for any exponent p other than 2, the vectorial p -Laplacian on the plane will admit infinitely many different sequences of periodic eigenvalues with a given period. These sequences of eigenvalues are constructed using the notion of scaling momenta we will introduce. The whole proof is based on the complete integrability of the equivalent Hamiltonian system, the tricky reduction to two-dimensional dynamical systems, and a number-theoretical distinguishing between different sequences of eigenvalues. Some numerical simulations to the new sequences of eigenvalues and eigenfunctions will be given. Several further conjectures towards to the panorama of the spectral sets will be imposed. [ABSTRACT FROM AUTHOR]

Resonant sensors using coupled micro-cantilever arrays have found wide applications in the ultrasensitive mass detection of biomolecules and chemical analytes. Experimental observations indicate that a target mass deposited on one of the cantilevers can be detected by measuring the change in resonant frequencies or in eigenmodes. Analytical works have studied eigenvalue and eigenmode sensitivities, but for a single analyte only. Since a resonator array consists of several cantilevers, it offers an opportunity for the simultaneous detection of multiple analytes. However, multiple-analyte mass detection has not been investigated so far. In this paper, an analytical foundation for the detection of multiple analytes, through the measurement of eigenvalue shifts, is developed using matrix perturbation theory. The formulation presents a system of over-determined linear equations in terms of unknown analyte masses. A novel approach based on solving the equations in a least square sense is proposed and it is shown that it gives far better estimation accuracy than using a subset of equations for direct solution. The approach is demonstrated through numerical simulation for a typical three-cantilever array for the detection of two analyte masses. Estimation errors are studied for a range of analyte masses and presented in the form of an error surface. The effect of interconnection stiffness and array size on estimation error is also investigated. The robustness of the method is further tested against manufacturing variations and it is shown that an envelope guideline of maximum estimation error can be constructed for the user. [ABSTRACT FROM AUTHOR]

Low-lying collective quadrupole states in even–even nuclei are studied for the particular case of a γ-unstable potential within the Bohr Hamiltonian. In particular, the quasi-exactly solvable β-sextic potential is extended to cover the most relevant part of the low-lying spectra in nuclei. In previous papers (2004 Phys. Rev. C 69 014304, 2010 Phys Rev. C 81 044304), the same situation was solved for β-wavefunctions with up to one node (M = 0, 1), which are relevant for the first few low-lying states. Here, the model space is enlarged by including β-wavefunctions also with two nodes (M = 2), which generate many more states, in order to make it useful for actual fittings and more detailed checking of shape phase transitions between spherical and γ-unstable β-deformed shapes in nuclei. In addition to the energy eigenvalues and wavefunctions, closed analytical formulas are obtained for electric quadrupole and monopole transition probabilities too. The model is applied to the chains of even Ru and Pd isotopes to illustrate the transition between the spherical and deformed γ-unstable phases. These applications indicate that the optional extension of the model with a phenomenologic rotational term L ⋅ L is consistent with the experimental data. [ABSTRACT FROM AUTHOR]

In industrial and engineering fields including lamination, melt-spinning, continuous casting, and fiber spinning, the flow caused by a continually moving surface is significant. Therefore, the problem of ternary hybrid nanofluid flow over a moving surface is studied. This study explores the stability and statistical analyses of the magnetohydrodynamics (MHD) forced flow of the ternary hybrid nanofluid with melting heat transfer phenomena. The impacts of viscous dissipation, Joule heating, and thermal radiation are also included in the flow. Different fluids including ternary hybrid nanofluid, hybrid nanofluids, and nanofluids with base fluid ethylene glycol (EG) are examined and compared, where magnetite (Fe3O4) and silica (SiO2) are taken as the magnetic nanomaterials while silver (Ag) is chosen as the nonmagnetic nanomaterial. The skin friction coefficient and the local Nusselt number are estimated through regression analysis. By employing similarity transformations, the governing partial differential equations are converted into non-linear ordinary differential equations. Then, the least square method is applied to solve the equations analytically. Dual solutions are established in a particular range of moving parameter λ. Due to this, a stability test is implemented to find the stable solution by using the bvp4c function in MATLAB software. It is found that the first solution is the stable one while the second is unstable. The use of ternary hybrid nanomaterials improves the heat transport rate. The increasing values of the Eckert number enlarge the heat passage. The fluid velocity and temperature profiles for nonmagnetic nanomaterials are higher than that of magnetic nanomaterials. The uniqueness and originality of this study stems from the fact that, to the best of the authors' knowledge, it is the first to use this combination technique. [ABSTRACT FROM AUTHOR]

We present an algorithm to compute the exact critical probability h (n) for an n × ∞ helical square lattice with random and independent site occupancy. The algorithm has time complexity O (n 2 c n) and space complexity O (c n) with c = 2.7459... and allows us to compute h (n) up to n = 24. Since the extrapolation result of h (n) is inconsistent with the current best estimation of pc, we also compute and extend the exact critical probability p c (n) for an n × ∞ cylindrical square lattice to n = 24. Our calculation shows that the current best result of p c = 0.592 746 050 792 10 (2) by Jacobsen (2015 J. Phys. A: Math. Theor. 48 454003) is incorrect and the corrected value should be 0.592 746 050 7896 (1) . [ABSTRACT FROM AUTHOR]

We show that the distribution of bulk spacings between pairs of adjacent eigenvalue real parts of a random matrix drawn from the complex elliptic Ginibre ensemble is asymptotically given by a generalization of the Gaudin-Mehta distribution, in the limit of weak non-Hermiticity. The same generalization is expressed in terms of an integro-differential Painlevé function and it is shown that the generalized Gaudin-Mehta distribution describes the crossover, with increasing degree of non-Hermiticity, from Gaudin-Mehta nearest-neighbor bulk statistics in the Gaussian Unitary Ensemble to Poisson gap statistics for eigenvalue real parts in the bulk of the Complex Ginibre Ensemble. [ABSTRACT FROM AUTHOR]

In this work, we obtain closed expressions for the transfer matrix and the transmittance of electromagnetic waves propagating in finite one-dimensional anisotropic periodic stratified media with an arbitrary number of cells. By invoking the Cayley–Hamilton theorem on the transfer matrix for the electromagnetic field in a periodic stratified media formed by N cells, we obtain a fourth-order recursive relation for the matrix coefficients that defines the so-called Tetranacci polynomials (TPs). In the symmetric case, corresponding to a unit-cell transfer matrix with a characteristic polynomial where the coefficients of the linear and cubic terms are equal, closed expressions for the solutions to the recursive relation, known as symmetric TPs, have recently been derived, allowing us to write the transfer matrix and transmittance in a closed form. We show as sufficient conditions that the 4 × 4 differential propagation matrix of each layer in the binary unit cell, Δ , a) has eigenvalues of the form ± p 1 , ± p 2 , with p 1 ≠ p 2 , and b) its off-diagonal 2 × 2 block matrices possess the same symmetric structure in both layers. Otherwise, the recursive relations are still solvable for any 4 × 4 matrix and provide an algorithm to compute the N th power of the transfer matrix without carrying out explicitly the matrix multiplication of N matrices. We obtain analytical expressions for the dispersion relation and transmittance, in closed form, for two finite periodic systems: the first one consists of two birefringent uniaxial media with their optical axis perpendicular to the z -axis, and the second consists of two isotropic media subject to an external magnetic field oriented along the z -axis and exhibiting the Faraday effect. Our formalism applies also to lossy media, magnetic anisotropy or optical activity. [ABSTRACT FROM AUTHOR]

We evaluate, in the large- N limit, the complete probability distribution P (A , m) of the values A of the sum ∑ i = 1 N | λ i | m , where λ i ( i = 1 , 2 , ... , N ) are the eigenvalues of a Gaussian random matrix, and m is a positive real number. Combining the Coulomb gas method with numerical simulations using a matrix variant of the Wang–Landau algorithm, we found that, in the limit of N → ∞ , the rate function of P (A , m) exhibits phase transitions of different characters. The phase diagram of the system on the (A, m) plane is surprisingly rich, as it includes three regions: (i) a region with a single-interval support of the optimal spectrum of eigenvalues, (ii) a region emerging for m < 2 where the optimal spectrum splits into two separate intervals, and (iii) a region emerging for m > 2 where the maximum or minimum eigenvalue 'evaporates' from the rest of eigenvalues and dominates the statistics of A. The phase transition between regions (i) and (iii) is of second order. Analytical arguments and numerical simulations strongly suggest that the phase transition between regions (i) and (ii) is of (in general) fractional order p = 1 + 1 / | m − 1 | , where 0 < m < 2 . The transition becomes of infinite order in the special case of m = 1, where we provide a more complete analytical and numerical description. Remarkably, the transition between regions (i) and (ii) for m ⩽ 1 and the transition between regions (i) and (iii) for m > 2 occur at the ground state of the Coulomb gas which corresponds to the Wigner's semicircular distribution. [ABSTRACT FROM AUTHOR]

In this article, we compute and compare the statistics of the number of eigenvalues in a centred disc of radius R in all three Ginibre ensembles. We determine the mean and variance as functions of R in the vicinity of the origin, where the real and symplectic ensembles exhibit respectively an additional attraction to or repulsion from the real axis, leading to different results. In the large radius limit, all three ensembles coincide and display a universal bulk behaviour of O (R 2) for the mean, and O (R) for the variance. We present detailed conjectures for the bulk and edge scaling behaviours of the real Ginibre ensemble, having real and complex eigenvalues. For the symplectic ensemble we can go beyond the Gaussian case (corresponding to the Ginibre ensemble) and prove the universality of the full counting statistics both in the bulk and at the edge of the spectrum for rotationally invariant potentials, extending a recent work which considered the mean and the variance. This statistical behaviour coincides with the universality class of the complex Ginibre ensemble, which has been shown to be associated with the ground state of non-interacting fermions in a two-dimensional rotating harmonic trap. All our analytical results and conjectures are corroborated by numerical simulations. [ABSTRACT FROM AUTHOR]

We consider the spatially-dependent diffusion tensors that maximize the rate of convergence to equilibrium of solutions to the heat equation in inhomogeneous media. We formulate a variational problem for the optimal spectral gap, subject to an Lp constraint on the diffusion tensors, and solve a relaxed version of this variational problem. This provides an upper bound for the optimal spectral gap. The solution is characterized in terms of the extremals of Sobolev and Poincaré type inequalities. To illustrate this result, we give a presentation of the one-dimensional case. In this case, we can adapt the known closed solution of a related eigenvalue problem to write the solution to the relaxed variational problem, arriving at an expression for the upper bound on the optimal spectral gap. The sharpness of this upper bound is then investigated by comparison to numerical simulations. [ABSTRACT FROM AUTHOR]

We address the level of complexity that can be observed in the dynamics near a robust heteroclinic network. We show that infinite switching, which is a path towards chaos, does not exist near a heteroclinic network such that the eigenvalues of the Jacobian matrix at each node are all real. Furthermore, for a path starting at a node that belongs to more than one heteroclinic cycle, we find a bound for the number of such nodes that can exist in any such path. This constricted dynamics is in stark contrast with examples in the literature of heteroclinic networks such that the eigenvalues of the Jacobian matrix at one node are complex. [ABSTRACT FROM AUTHOR]

We employ a mathematical framework based on the Riemann-Hilbert approach developed by Bettelheim et al (2022 J. Phys. A: Math. Gen. 55 135001) to study logarithmic negativity of two intervals of free fermions in the case where the size of the intervals as well as the distance between them is macroscopic. We find that none of the eigenvalues of the density matrix become negative, but rather they develop a small imaginary value, leading to non-zero logarithmic negativity. As an example, we compute negativity at half-filling and for intervals of equal size we find a result of order (log (N)) − 1 , where N is the typical length scale in units of the lattice spacing. One may compute logarithmic negativity in further situations, but we find that the results are non-universal, depending non-smoothly on the Fermi level and the size of the intervals in units of the lattice spacing. [ABSTRACT FROM AUTHOR]

We describe a weak magnetic field sensor operating at room temperature based on the magnonic crystal (MC). MC consisting of periodic stripes of cobalt (Co) and permalloy (Py) in one dimension is studied. The magnonic bandgaps are calculated by solving the eigenvalue problem of the Landau–Lifshitz equation using the finite element method. Magnonic bandgap frequency shifts depend upon the external magnetic field and this phenomenon is utilized for magnetic field sensing. The sensitivity characteristics of MCs with dispersion spectra in the gigahertz (GHz) frequency range are studied. It is found that the sensor's performance gets enhanced for smaller thickness and larger periodicity. The sensitivity reaches a magnitude as large as 66.0 GHz T−1 for 10 nm thickness and 1 μ m periodicity. Our analysis indicates that a limit of detection (LOD) of the order of 10−11 T can be achieved for all the geometric configurations considered in the 0–1 T range. The results are explained in terms of corresponding fundamental concepts and phenomena. Further, our simulation results show that the typical gap (e.g. 1 nm) between Co and Py stripes does not significantly affect the sensitivity of the sensor. The results also indicate that any small variation (e.g. 1 nm) in MC thickness may lead to reasonable variation in sensitivity magnitudes. Moreover, the proposed sensor's performance is significantly superior (in terms of sensitivity, LOD, miniaturization, and material, etc) to the currently available state-of-the-art magnetometers. [ABSTRACT FROM AUTHOR]

We study the Schrödinger operators H λ μ (K) , with K ∈ T 2 the fixed quasimomentum of the particles pair, associated with a system of two identical fermions on the two-dimensional lattice Z 2 with first and second nearest-neighboring-site interactions of magnitudes λ ∈ R and μ ∈ R , respectively. We establish a partition of the (λ , μ) − plane so that in each its connected component, the Schrödinger operator H λ μ (0) has a definite (fixed) number of eigenvalues, which are situated below the bottom of the essential spectrum and above its top. Moreover, we establish a sharp lower bound for the number of isolated eigenvalues of H λ μ (K) in each connected component. [ABSTRACT FROM AUTHOR]

Symmetric matrices with zero row sums occur in many theoretical settings and in real-life applications. When the offdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions converge to a peculiar universal curve p z r s (λ) that looks like a cross between the Wigner semicircle and a Gaussian distribution. An analytic theory for this curve, originally due to Fyodorov, can be developed using supersymmetry-based techniques. We extend these derivations to the case of sparse matrices, including the important case of graph Laplacians for large random graphs with N vertices of mean degree c. In the regime 1 ≪ c ≪ N , the eigenvalue distribution of the ordinary graph Laplacian (diffusion with a fixed transition rate per edge) tends to a shifted and scaled version of p z r s (λ) , centered at c with width ∼ c . At smaller c, this curve receives corrections in powers of 1 / c accurately captured by our theory. For the normalized graph Laplacian (diffusion with a fixed transition rate per vertex), the large c limit is a shifted and scaled Wigner semicircle, again with corrections captured by our analysis. [ABSTRACT FROM AUTHOR]

We present a general mathematical procedure to handle interactions described by a Morse potential in the presence of a strong harmonic excitation. We account for permanent and field-induced terms and their gradients in the dipole moment function, and we derive analytic formulae for the bond-length change and for the shifted energy eigenvalues of the vibrations, by using the Kramers–Henneberger frame. We apply these results to the important cases of H2 and LiH, driven by a near- or mid-infrared laser in the 1013 W cm−2 intensity range. [ABSTRACT FROM AUTHOR]

We study the inverse eigenvalue problem for finding doubly stochastic matrices with specified eigenvalues. By making use of a combination of Dykstra's algorithm and an alternating projection process onto a non-convex set, we derive hybrid algorithms for finding doubly stochastic matrices and symmetric doubly stochastic matrices with prescribed eigenvalues. Furthermore, we prove that the proposed algorithms converge, and linear convergence is also proved. Numerical examples are presented to demonstrate the efficiency of our method. [ABSTRACT FROM AUTHOR]

We establish the theory for pretty good state transfer in discrete-time quantum walks. For a class of walks, we show that pretty good state transfer is characterized by the spectrum of certain Hermitian adjacency matrix of the graph; more specifically, the vertices involved in pretty good state transfer must be strongly cospectral relative to this matrix, and the arccosines of its eigenvalues must satisfy some number theoretic conditions. Using normalized adjacency matrices, cyclic covers, and the theory on linear relations between geodetic angles, we construct several infinite families of walks that exhibits this phenomenon. [ABSTRACT FROM AUTHOR]

We investigate the survival probability of a particle diffusing between two parallel reflecting planes toward a periodic array of absorbing pillars. We approximate the periodic cell of this system by a cylindrical tube containing a single pillar. Using a mode matching method, we obtain an exact solution of the modified Helmholtz equation in this domain that determines the Laplace transform of the survival probability and the associated distribution of first-passage times (FPTs). This solution reveals the respective roles of several geometric parameters: the height and radius of the pillar, the inter-pillar distance, and the distance between confining planes. This model allows us to explore different asymptotic regimes in the probability density of the FPT. In the practically relevant case of a large distance between confining planes, we argue that the mean FPT is much larger than the typical time and thus uninformative. We also illustrate the failure of the capacitance approximation for the principal eigenvalue of the Laplace operator. Some practical implications and future perspectives are discussed. [ABSTRACT FROM AUTHOR]

We consider the eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble, which are known to form a Pfaffian point process in the plane. It was recently discovered that the limiting correlation kernel of the symplectic Ginibre ensemble in the vicinity of the real line can be expressed in a unified form of a Wronskian. We derive scaling limits for variations of the symplectic Ginibre ensemble and obtain such Wronskian structures for the associated universality classes. These include almost-Hermitian bulk/edge scaling limits of the elliptic symplectic Ginibre ensemble and edge scaling limits of the symplectic Ginibre ensemble with boundary confinement. Our proofs follow from the generalised Christoffel–Darboux formula for the former and from the Laplace method for the latter. Based on such a unified integrable structure of Wronskian form, we also provide an intimate relation between the function in the argument of the Wronskian in the symplectic symmetry class and the kernel in the complex symmetry class which form determinantal point processes in the plane. [ABSTRACT FROM AUTHOR]

We investigate the peculiar feature of non-Hermitian operators, namely, the existence of spectral branch points (also known as exceptional or level crossing points), at which two (or many) eigenmodes collapse onto a single eigenmode and thus loose their completeness. Such branch points are generic and produce non-analyticities in the spectrum of the operator, which, in turn, result in a finite convergence radius of perturbative expansions based on eigenvalues and eigenmodes that can be relevant even for Hermitian operators. We start with a pedagogic introduction to this phenomenon by considering the case of 2 Ă— 2 matrices and explaining how the analysis of more general differential operators can be reduced to this setting. We propose an efficient numerical algorithm to find spectral branch points in the complex plane. This algorithm is then employed to show the emergence of spectral branch points in the spectrum of the Blochâ€"Torrey operator âˆ' ∇ 2 âˆ' i g x , which governs the time evolution of the nuclear magnetization under diffusion and precession. We discuss their mathematical properties and physical implications for diffusion nuclear magnetic resonance experiments in general bounded domains. [ABSTRACT FROM AUTHOR]

We present two types of systems of differential equations that can be derived from a set of discrete integrable systems which we call the closed geometric crystal chains. One is an extended Lotkaâ€"Volterra type system, and the other seems to be generally new but reduces to a previously known system in a special case. Both equations have Lax representations associated with what are known as the loop elementary symmetric functions, which were originally introduced to describe products of affine type A geometric crystals for symmetric tensor representations. Examples of the derivations of the continuous time Lax equations from a discrete time one are described in detail, where a novel method of taking a continuum limit by assuming asymptotic behaviors of the eigenvalues of the Lax matrix in Puiseux series expansions is used. [ABSTRACT FROM AUTHOR]

Recent work has shown a deep connection between semilocal approximations in density functional theory and the asymptotics of the sum of the Wentzelâ€"Kramersâ€"Brillouin (WKB) semiclassical expansion for the eigenvalues. However, all examples studied to date have potentials with only real classical turning points. But systems with complex turning points generate subdominant (SD) terms beyond those in the WKB series. The simplest case is a pure quartic oscillator. We show how to generalize the asymptotics of eigenvalue sums to include SD contributions to the sums, if they are known for the eigenvalues. These corrections to WKB greatly improve accuracy for eigenvalue sums, especially for many levels. We obtain further improvements to the sums through hyperasymptotics. For the lowest level, our summation method has error below 2 Ă— 10âˆ'4. For the sum of the lowest ten levels, our error is less than 10âˆ'22. We report all results to many digits and include copious details of the asymptotic expansions and their derivation. [ABSTRACT FROM AUTHOR]

We discuss spectral properties of a periodic quantum graph consisting of an array of rings coupled either tightly or loosely through connecting links, assuming that the vertex coupling is manifestly non-invariant with respect to the time reversal and a homogeneous magnetic field perpendicular to the graph plane is present. It is shown that the vertex parity determines the spectral behavior at high energies and the Bandâ€"Berkolaiko universality holds whenever the edges are incommensurate. The magnetic field influences the probability that an energy belongs to the spectrum in the tight-chain case, and also it can turn some spectral bands into infinitely degenerate eigenvalues. [ABSTRACT FROM AUTHOR]

We analyze a recent discussion of one-dimensional quantum-mechanical models with a perturbation produced by a delta-function potential. We show that the authors failed to derive a correct transcendental equation for the eigenvalues in the case of the Morse oscillator because they omitted the contribution of the continuum spectrum in the basis set. [ABSTRACT FROM AUTHOR]

In this study, we investigate the numerical stability of the covariant BSSN (cBSSN) formulation proposed by Brown. We calculate the constraint amplification factor (CAF), which is an eigenvalue of the coefficient matrix of the evolution equations of the constraints on the cBSSN formulation and on some adjusted formulations with constraints added to the evolution equations. The adjusted formulations have a higher numerical stability than the cBSSN formulation from the viewpoint of the CAF. [ABSTRACT FROM AUTHOR]

The spectral properties of two-dimensional Schrödinger operators with δ ′-potentials supported on star graphs are discussed. We describe the essential spectrum and give a complete description of situations in which the discrete spectrum is non-trivial but finite. A more detailed study is presented for the case of a star graph with two branches, in particular, the small angle asymptotics for the eigenvalues is obtained. [ABSTRACT FROM AUTHOR]

About two dozens of exactly solvable Markov chains on one-dimensional finite and semi-infinite integer lattices are constructed in terms of convolutions of orthogonality measures of the Krawtchouk, Hahn, Meixner, Charlier, q -Hahn, q -Meixner and little q -Jacobi polynomials. By construction, the stationary probability distributions, the complete sets of eigenvalues and eigenvectors are provided by the polynomials and the orthogonality measures. An interesting property possessed by these stationary probability distributions, called ‘convolutional self-similarity,’ is demonstrated. [ABSTRACT FROM AUTHOR]

For Schrödinger operators with suitable 1D potentials, focussing particularly on those that go to infinity at infinity, a characteristic function is constructed, via shooting functions. It is proved to be entire and its zeroes to be the eigenvalues. [ABSTRACT FROM AUTHOR]

We develop explicit formulae for the eigenvalues of various invariants for highest weight irreducible representations of the quantum supergroup U q [ gl (m | n)]. The techniques employed make use of modified characteristic identity methods and allow for the evaluation of generator matrix elements and reduced Wigner coefficients. [ABSTRACT FROM AUTHOR]

Mathematical analysis on the existence of eigenvalues is essential because it is deeply related to localization, which is an exceptionally crucial property of quantum walks (QWs). We construct the method for the eigenvalue problem via the transfer matrix for space-inhomogeneous three-state QWs in one dimension with a self-loop, which is an extension of the technique in a previous study (Kiumi and Saito 2021 Quantum Inf. Process. 20 171). This method reveals the necessary and sufficient condition for the eigenvalue problem of a two-phase three-state QW with one defect whose time evolution varies in the negative part, positive part, and at the origin. [ABSTRACT FROM AUTHOR]

The local spectral statistics of random matrices forms distinct universality classes, strongly depending on the position in the spectrum. Surprisingly, the spacing between consecutive eigenvalues at the spectral edges has received little attention, where the density diverges or vanishes, respectively. This different behaviour is called hard or soft edge. We show that the spacings at the edges are almost indistinguishable from the spacing in the bulk of the spectrum. We present analytical results for consecutive spacings between the k th and (k + 1)st smallest eigenvalues in the chiral Gaussian unitary ensemble, both for finite- and large- n. The result depends on the number of the generic zero modes ν and the number of flavours N f, which are given in terms of characteristic polynomials, as motivated by quantum chromodynamics (QCD). We find that the convergence in n is very rapid. The same can be said separately about the limit k â†' ∞ (limit to the bulk) and ν â†' ∞ (limit to the soft edge). Interestingly, the Wigner surmise is a very good approximation for all these cases and, apart from k = 1, shows a deviation below one percent. These findings are corroborated with Monte-Carlo simulations. We finally compare for k = 1 with data from QCD on the lattice, being in this symmetry class. [ABSTRACT FROM AUTHOR]

Violation of the no-signaling principle may occur in PT -symmetric evolutions, that is, evolutions that stem from Hamiltonians that are symmetric with respect to parity and time-reversal, of bipartite entangled states. The violation can be avoided by using a non-conventional inner product. We show that even within the formalism that utilizes the conventional inner product between physical state vectors in a separable Hilbert space, it is possible to obtain instances of local preservation of the no-signaling principle for evolutions corresponding to PT -symmetric non-Hermitian Hamiltonians with real eigenvalues, of multiparty entangled states, whose bipartite versions still violate the principle. The phenomenon can be witnessed already by using the Greenbergerâ€"Horneâ€"Zeilinger state. Interestingly, the generalized W states do not support such a local preservation of no-signaling. [ABSTRACT FROM AUTHOR]

We consider corner scattering for the operator ∇ â‹... Îł (x)∇ + k 2 ρ (x) in R 2 , with Îł a positive definite symmetric matrix and ρ a positive scalar function. A corner is referred to one that is on the boundary of the (compact) support of Îł (x) âˆ' I or ρ (x) âˆ' 1, where I stands for the identity matrix. We assume that Îł is a scalar function in a small neighborhood of the corner. We show that any admissible incident field will be scattered by such corners, which are allowed to be concave. Moreover, we provide a brief discussion on the existence of non-scattering waves when Îł âˆ' I has a jump across the corner. In order to prove the results, we construct a new type of complex geometric optics solutions. [ABSTRACT FROM AUTHOR]

We investigate an inverse scattering problem for a thin inhomogeneous scatterer in R m , m = 2, 3, which we model as an m âˆ' 1 dimensional open surface. The scatterer is referred to as a screen. The goal is to design target signatures that are computable from scattering data in order to detect changes in the material properties of the screen. This target signature is characterized by a mixed Steklov eigenvalue problem for a domain whose boundary contains the screen. We show that the corresponding eigenvalues can be determined from appropriately modified scattering data by using the generalized linear sampling method. A weaker justification is provided for the classical linear sampling method. Numerical experiments are presented to support our theoretical results. [ABSTRACT FROM AUTHOR]

We study, analytically and numerically, a simple P T -symmetric tight-binding ring with an onsite energy a at the gain and loss sites. We show that if a ≠0, the system generically exhibits an unbroken P T -symmetric phase. We study the nature of the spectrum in terms of the singularities in the complex parameter space as well as the behavior of the eigenstates at large values of the gain and loss strength. We find that in addition to the usual exceptional points (EPs), there are ‘diabolical points’, and inverse EPs at which complex eigenvalues reconvert into real eigenvalues. We also study the transport through the system. We calculate the total flux from the source to the drain, and how it splits along the branches of the ring. We find that while usually the density flows from the source to the drain, for certain eigenstates a stationary ‘backflow’ of density from the drain to the source along one of the branches can occur. We also identify two types of singular eigenstates, i.e. states that do not depend on the strength of the gain and loss, and classify them in terms of their transport properties. [ABSTRACT FROM AUTHOR]

The fast optimal attitude matrix (FOAM) for minimizing Wahba's loss function solves the optimal attitude matrix directly and efficiently. This method is more intuitive compared with the method of calculating attitude quaternion. A method named super-FOAM (SFOAM) is proposed with a more precise initial iteration value and a faster iterative algorithm. A condition number estimation method of third-order matrices is presented for cases without exact singular value. When the matrix condition number is not particularly large, the root-seeking formula for the quartic equation is directly used to obtain the maximum characteristic root. Otherwise, under large condition numbers, a high-accuracy estimation algorithm for the maximum eigenvalue is given to achieve a high-accuracy initial value set, which is beneficial to iteration reduction and convergence speed improvement. Finally, some comparative simulations show that the SFOAM method has the same accuracy as the singular value decomposition (Matlab) method even in the case of a very large matrix condition number, while the SFOAM's computation is only 50%–60% of the traditional FOAM algorithm. This algorithm has good generalization and application values. [ABSTRACT FROM AUTHOR]

Most studies for postselected weak measurement in optomechanical system focus on using a single photon as a measured system. However, we find that using weak coherent light instead of a single photon can also amplify the mirror's position displacement of one photon. In the weak value amplification (WVA) regime, the weak value of one photon can lie outside the eigenvalue spectrum, proportional to the difference of the mirror's position displacement between the successful and failed postselection and its successful postselection probability is dependent of the mean photon number of the coherent light and improved by adjusting it accordingly. Outside the WVA regime, the amplification limit can reach the level of the vacuum fluctuations, and when the mean photon number and the optomechanical coupling parameter are selected appropriately, its successful postselection probability becomes higher, which is beneficial to observe the maximum amplification value under the current experimental conditions. This result breaks the constraint that it is difficult to detect outside the WVA regime. This opens up a new regime for the study of a single photon nonlinearity in optomechanical system. [ABSTRACT FROM AUTHOR]