This paper introduces a rigorously defined fractional Heun operator constructed through a symmetric composition of left and right Riemann–Liouville fractional derivatives. By deriving a compatible fractional Pearson-type equation, a new weight function and Hilbert space setting are established, ensuring the operator’s self-adjointness under natural fractional boundary conditions. Within this framework, we prove the existence of a real, discrete spectrum and demonstrate that the corresponding eigenfunctions form a complete orthogonal system in Lωα2(a,b). The central theoretical result shows that the fractional eigenpairs (λn(α),un(α)) converge continuously to their classical Heun counterparts (λn(1),un(1)) as α→1−. This provides a rigorous analytic bridge between fractional and classical spectral theories. A numerical study based on the fractional Legendre case confirms the predicted self-adjointness and spectral convergence, illustrating the smooth deformation of the classical eigenfunctions into their fractional counterparts. The results establish the fractional Heun operator as a mathematically consistent generalization capable of generating new families of orthogonal fractional functions.

Abstract Advanced tokamak regimes, featuring extended regions of low magnetic shear, are promising candidates for future fusion reactors but are also more prone to specific kinds of MHD instabilities. The proximity to a rational surface in a very low shear region weakens field line bending stabilisation and amplifies the effects of toroidal coupling between modes, leading to the emergence of long-wavelength resistive infernal modes. These modes can grow collectively as a discrete spectrum, leading to a cascade of different perturbations for single mode numbers ( m , n ), with subdominant modes showing increasingly oscillatory radial structures. These spectra of fast-growing modes are significant for developing stable scenarios in future reactors, and for the understanding of global reconnection events like sawteeth, motivating a deeper investigation into their fundamental physics. Deriving new analytic solutions, including a generalisation of the ideal interchange dispersion relation to non monotonic q profiles, and extending a modular linear resistive MHD solver, we investigate how resistivity, compressibility, toroidal effects, and shaping influence stability, especially in reversed shear q profiles. It is also shown that common assumptions in numerical calculations prevent the observation of the full variety of modes present in these advanced scenarios.

The nonlinear Schrödinger equation is a classical nonlinear evolution equation with wide applications. This paper explores the asymptotic behavior of solutions to the nonlinear Schrödinger equation with non-zero boundary conditions in the presence of a pair of second-order discrete spectra. We analyze the Riemann–Hilbert problem in the inverse scattering transform by the Deift–Zhou nonlinear steepest descent method. Then we propose a proper deformation to deal with the growing time term and give the conditions for the series in the process of deformation by the Laurent expansion. Finally, we provide the characterization of the interactions between the solitary waves corresponding to second-order discrete spectra and the coherent oscillations produced by the perturbation. Numerical verifications are also performed.

Abstract We introduce a renormalization procedure necessary for the complete description of the energy spectra of a one-dimensional stationary Schr"odinger equation with a potential that exhibits inverse-square singularities. We apply and extend the methods introduced in our recent paper on the hyperbolic P"oschl-Teller potential (with a single singularity) to its trigonometric version. This potential, defined between two singularities, is analyzed across the entire bidimensional coupling space. The fact that the trigonometric P"oschl-Teller potential is supersymmetric and shape-invariant simplifies the analysis and eliminates the need for self-adjoint extensions in certain coupling regions. However, if at least one coupling is strongly attractive, the renormalization is essential to construct a discrete energy spectrum family of one or two parameters. We also investigate the features of a singular symmetric double well obtained by extending the range of the trigonometric P"oschl-Teller potential. It has a non-degenerate energy spectrum and eigenstates with well-defined parity.

Given a quantum state in the finite-dimensional Hilbert space Cn, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such observable is identified with (i) an 'ortho-measurable' function defined on the Boolean 'ortho-algebra' generated by the eigenspaces that form an orthogonal decomposition of Cn, and (ii) a 'numerically identified' orthogonal decomposition of Cn. The latter means that each subspace of the orthogonal decomposition can be uniquely identified by its own attached real number, just as each eigenspace of a Hermitian matrix can be uniquely identified by the corresponding eigenvalue. Furthermore, any density matrix on Cn is identified with a Bayesian prior 'ortho-probability' measure defined on the linear subspaces that make up the Boolean ortho-algebra induced by its eigenspaces. Then any pure quantum state is identified with a degenerate density matrix, and any mixed state with a probability measure on a set of orthogonal pure states. Finally, given any quantum observable, the relevant Bayesian posterior probabilities of measured outcomes can be found by the usual trace formula that extends Born's rule. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

An acceleration-dependent Hamiltonian operator [Formula: see text] and its corresponding Hamiltonian operator [Formula: see text] are introduced based on the new notion of quantum acceleratum operator. The new Hamiltonian operator consists of two independent parts: the Berry–Keating (BK) operator [Formula: see text] and the new operator part [Formula: see text] which depends on the acceleratum operator [Formula: see text]. The Hamiltonian [Formula: see text] offers new features for the zeros of the Riemann zeta function. The operator couples [Formula: see text] with a higher-order interaction involving the position and acceleratum operators, introducing nonlinear effects beyond conventional canonical mechanics. The structure of this coupling determines whether the eigenvalue spectrum remains discrete or transitions to a continuous regime, reflecting the balance between confinement and scaling symmetry. Establishing conditions for spectral discreteness provides key insight into the existence of bound states, the boundedness of energy levels, and the physical realizability of acceleration-dependent quantum dynamics. Such analysis extends the mathematical framework of self-adjoint operators and spectral theory to higher-order Hamiltonians, offering new perspectives on quantization in extended phase-space formulations and on the emergence of discrete energy spectra in nonstandard dynamical systems.

Diagnosis of pine wilt disease using unattended UAV hyperspectral imager: A comparison of discrete bands and continuous spectrum -based methods

Recent discoveries of ultra-faint dwarf galaxies (UFDs) infalling onto the Milky Way, namely Leo K & M at r ≃ 450 kpc, considerably strengthens the case that UFDs constitute a distinct galaxy class that is inherently smaller and fainter, and metal-poorer than the classical dwarf spheroidals (dSph). This distinction is at odds with the inherent continuity of galaxy halo masses formed under scale-free gravity for any standard dark-matter (DM) model. Here, we show that distinct galaxy classes do evolve in cosmological simulations of multiple light bosons representing the “Axiverse” proposal of string theory, where a discrete mass spectrum of axions is generically predicted to span many decades in mass. In this context, the observed UFD class we show corresponds to a relatively heavy boson of 3 × 10 −21 eV, including Leo K & M, whereas a lighter axion of 10 −22 eV comprises the bulk of DM in all larger galaxies including the dSphs. Although Leo M is larger in size than Leo K, we predict its velocity dispersion to be smaller ( ≃ 1.7 km/s) than that of Leo K ( ≃ 4.5 km/s) because of the inverse de Broglie scale dependence on momentum. This scenario can be definitively tested using millisecond pulsars close to the Galactic center, where the Compton frequencies of the heavy and light bosons imprint monotone timing residuals that may be detected by the Square Kilometre Array (SKA) on timescales of approximately one week and four months, respectively.

A bstract Quark-gluon string fragmentation is usually seen as a universal process with its physics remaining unchanged for different colliding systems and center-of-mass energy. However, there is a way to go beyond this assumption by considering the angular momentum of the fragmenting system. An accurate calculation of the spin of the string requires a specific definition of the initial state of the string. In this paper, I present a new method for defining the initial data for the Nambu-Goto relativistic string and derive the basics of its fragmentation process using the so-called Virasoro conditions. It is shown that very non-trivial consequences arise for non-zero mass strings, including the discrete spectrum of string spin, the discreteness of the possible string breaking points, the close-to-Regge behavior of the spin-mass relation for the light string fragments, and the natural stopping mechanism for fragmentation.

This article analyzes Legendre, Laguerre, and Hermite polynomials as eigenfunctions of self-adjoint differential operators defined on Hilbert spaces of the type , with the goal of strengthening their conceptual understanding in advanced educational settings. By reformulating their differential equations as Sturm–Liouville problems, we identify the structural components that allow for a spectral interpretation: weight functions, domains, eigenvalues, and orthogonality. This unified framework presents these polynomials not only as formal solutions but also as pedagogical tools to explain key concepts in functional analysis, such as orthonormality, discrete spectra, and complete bases. The approach combines mathematical rigor with visual and comparative resources to support their integration into the teaching of upper-level mathematics and physics courses. It is concluded that the spectral interpretation of these polynomial systems can significantly enhance students' conceptual comprehension by connecting topics from linear algebra, differential equations, and functional spaces. The study proposes their inclusion as didactic resources in university-level mathematics education, promoting meaningful learning around abstract structures.

In this work, we show that the completeness relation for the eigenvectors, which is an essential assumption of quantum mechanics, remains true if the Hamiltonian, having a discrete spectrum, is modified by a delta potential (to be made precise by a renormalization scheme) supported at a point in two- and three-dimensional compact manifolds or Euclidean spaces. The formulation can be easily extended to an N center case and the case where delta interaction is supported on curves in the plane or space. We finally give an interesting application for the sudden perturbation of the support of the delta potential.

The modified Korteweg–de Vries (mKdV) equation is an important internal solitary wave model in ocean engineering. In this paper, a new time-dependent coefficient complex mKdV-type (tdccmKdV-type) equation with nonzero boundaries is proposed and analytically solved by using the Riemann–Hilbert (RH) method. First, the related spectral problems of the tdccmKdV-type equation are given. Then, a Riemann surface is introduced to handle the multivalued problem of spectral parameter, and a singularly valorized variable is used to restore the infeasible operations on the Riemann surface to the complex plane. The analyticity, symmetry and asymptotic property of the Jost solutions and spectral matrix are further analyzed. Subsequently, a RH problem associated with nonzero asymptotic boundaries is presented. Finally, by analyzing the discrete spectra and residual conditions of the poles in the presented RH problem, and calculating the trace formula corresponding to the reflection coefficients and discrete spectra, N-soliton solution of the tdccmKdV-type equation is derived. By simulating specific single- and two-soliton solutions, it is indicated that the time-dependent coefficients have a regulatory effect on the velocity and spatiotemporal structure of solitons. The novelty of this work lies in proposing the tdccmKdV-type equation with nonzero boundaries, applying the RH method for the first time to such a third-order time-dependent coefficient model with nonzero boundaries, and reporting novel nonlinear dynamical characteristics of nonzero boundary soliton solutions. This not only enriches the content of nonzero boundary solitons and nonlinear integrable systems with potential practical applications, but also expands the applicability of the RH method.

The possibility of controlling the spectrum of high harmonics in the range of 50–200 eV under the excitation of a gas jet by a pair of intense femtosecond near- (1.24 μm) and mid-infrared (4.5 μm) laser pulses has been experimentally demonstrated. It has been established that an increase in the intensity of long-wavelength radiation from ~10 9 to ~10 12 W/cm 2 under the peak intensity 10 14 W/cm 2 of the short-wavelength pulse leads to a transition from a discrete spectrum containing sum and difference frequencies to a quasi-continuous broadband spectrum. This transition has been explained by a change in the regime of influence of the mid-infrared field on the dynamics of the generating electron. The obtained results demonstrate the prospects for generating radiation with the quasi-continuous broadband spectrum up to the soft X-ray region when generating high harmonics by a synthesized laser field.

Discrete spectrum of infinite and finite cylindrical shells

Controlling quantum systems with time-dependent fields opens avenues for engineering novel states of matter and exploring nonequilibrium phenomena. Landau levels in two-dimensional electron gases (2DEGs), with their discrete energy spectrum and characteristic cyclotron dynamics, provide an important platform for realizing and studying such driven quantum systems. While exact solutions for driven Landau levels exist, they have been limited to specific gauges or representations. In this work, we present an algebraic, gauge- and representation-independent exact solution for driven Landau levels in 2DEGs subject to arbitrary time-dependent electromagnetic fields. Our approach, based on a time-dependent unitary transformation via the displacement operator, provides clear physical insights into the driven quantum dynamics. We apply this method to derive the exact Floquet states and quasienergies for periodically driven Landau levels, and we extend our analysis to the resonant driving regime, where the Floquet picture breaks down and the electron wave function exhibits unbounded spatial spreading. Furthermore, we calculate the instantaneous energy absorption rate, revealing distinct absorption behaviors between coherent states and Fock or thermal states, stemming from quantum interference effects.

A new method for video protection using phase digital watermarks is proposed in this work. Unlike the previously described method, it employs a secondary container (a discrete Fourier spectrum) that provides additional security for video content. The protective information (a binary QR code) is embedded into the discrete spectrum of the watermark, which is then converted into a spatial domain representation as a halftone image visually resembling noise. This image is subsequently embedded into the phases of time-varying sinusoids. The paper presents experiments analyzing the dependence of watermark extraction accuracy on the amount of embedded data and its quality. It also examines the possibility of placing the QR code in different regions of the discrete spectrum. Experimental results demonstrate the effectiveness of the proposed method and its robustness against compression using the H.264 codec. In conclusion, we compare the efficiency of the proposed method with that of the original phase watermarking approach.

We study the two-dimensional steady-state creeping flow in a converging–diverging channel gap formed by two immobile rollers of identical radius. For this purpose, we analyse the Stokes equation in the streamfunction formulation, i.e. the biharmonic equation, which has homogeneous and particular solutions in the roll-adapted bipolar coordinate system. The analysis of existing works, investigating the particular solutions allowing arbitrary velocities at the two rollers, is extended by an investigation of homogeneous solutions. These can be reduced to an algebraic eigenvalue problem, whereby the associated discrete but infinite eigenvalue spectrum generates symmetric and asymmetric eigenfunctions with respect to the centre line between the rollers. These represent nested viscous vortex structures, which form a counter-rotating chain of vortices for the smallest unsymmetrical eigenvalue. With increasing eigenvalue, increasingly complex finger-like structures with more and more layered vortices are formed, which continuously form more free stagnation points. In the symmetrical case, all structures are mirror-symmetrical to the centre line and with increasing eigenvalues, finger-like nested vortex structures are also formed. As the gap height in the pressure gap decreases, the vortex density increases, i.e. the number of vortices per unit length increases, or the length scales of the vortices decrease. At the same time the rate of decay between subsequent vortices increases and reaches and asymptotic limit as the gap vanishes.

Nonlinear frequency division multiplexing (NFDM) systems utilize the nonlinear Fourier transform to address Kerr nonlinearity in optical fiber communications. However, they are highly sensitive to signal impairments such as amplified spontaneous emission (ASE) noise and phase noise, which pose significant challenges for practical implementation in real-world systems. To address the key issue of NFDM systems, this paper first analyzes the modeling characteristics of ASE noise in the discrete spectrum NFDM system. Based on this analysis, it designs an olive-shaped 32-ary quadrature amplitude modulation (32QAM) geometric constellation to improve noise robustness. The simulation results show that the proposed Olive-32QAM constellation increases the transmission distance by 133 km compared to traditional 32-ary amplitude phase shift keying (32APSK) and by 157 km compared to traditional 32QAM. Meanwhile, its phase noise tolerance is increased to over 1.9 kHz, demonstrating a stronger ability to withstand phase noise. Furthermore, leveraging the geometric characteristics of the Olive-32QAM constellation, this paper optimizes the carrier phase recovery (CPR) algorithm using principal component analysis and maximum likelihood, reducing its computational complexity. Compared with the blind phase search (BPS) CPR algorithm, the proposed CPR algorithm achieves an optical signal-to-noise ratio gain of up to 3.2 dB while maintaining a computational complexity of only about 1.2% of that of the BPS algorithm.

Abstract We analyze linear fluctuations of five-dimensional Einstein-Dilaton theories dual to holographic quantum field theories defined on four-dimensional de Sitter and Anti-de Sitter space-times. We identify the physical propagating scalar and tensor degrees of freedom. For these, we write the linearized bulk field equations as eigenvalue equations. In the dual QFT, the eigenstates correspond to towers of spin-0 and spin-2 particles propagating on (A)dS 4 associated to gauge-invariant composite states. Using particular care in treating special “zero-modes,” we show in general that, for negative curvature, the particle spectra are always discrete, whereas for positive curvature they always have a continuous component starting at m 2 = (9/4)α −2, where α is the (A)dS 4 radius. We numerically compute the spectra in a concrete model characterized by a polynomial dilaton bulk potential admitting holographic RG-flow solutions with a UV and IR fixed points. In this case, we find no discrete spectrum and no perturbative instabilities.

Abstract The continuum of holographic dual gravitational charges is recovered out of the discrete spectrum of U(N) $$ \mathcal{N} $$ N = 4 SYM on ℝ × S 3. In such a limit, the free energy of the free gauge theory is computed up to logarithmic contributions and exponentially suppressed contributions. Assuming the supergravity dual prediction to correctly capture strong-coupling results in field theory, the answer is bound to encode a complete low-temperature expansion of the Gibbons-Hawking gravitational on-shell action, valid well beyond the vicinity of supersymmetric black hole solutions. The formula recovers the long awaited Schwarzian contribution at low enough temperatures for certain choices of flows to the continuum. One such choice identifies the chemical potentials and thermodynamic charges in field theory with the chemical potentials and thermodynamic charges of the dual black holes. For such a flow the computed mass-gap kinematically matches the conjectured strong-coupling result obtained by Boruch, Heydeman, Iliesiu, and Turiaci in supergravity, including small 1/λ-corrections. The emergent reparameterizations, broken by the selection of the Schwarzian, correspond to redefinitions of the relevant cutoff scale. Observations are made regarding the existence of $$ \frac{1}{8} $$ 1 8 -BPS black holes and how this is in tension with BPS inequalities. The RG-flow procedure leading to these results opens a way to understanding the emergence of chaos in gauge theories and its relation to non-extremal and non-supersymmetric black hole physics.