Matrix Poles Research Articles

Non‐Hermitian Control of Topological Scattering Singularities Emerging from Bound States in the Continuum

AbstractLeveraging topological properties in the response of electromagnetic systems can greatly enhance their potential. Although the investigation of singularity‐based electromagnetics and non‐Hermitian electronics has considerably increased in recent years in the context of various scattering anomalies, their topological properties have not been fully assessed. In this work, it is theoretically and experimentally demonstrated that non‐Hermitian perturbations around bound states in the continuum can lead to singularities of the scattering matrix, which are topologically nontrivial and comply with charge conservation. The associated scattering matrix poles, zeros, and pole‐zero pairs delineate extreme scattering events, including lasing, coherent perfect absorption, and absorber‐lasers. The presented framework enables a recipe for generation, annihilation, and addition of these singularities in electric circuits, with potential for extreme scattering engineering across a broad range of the electromagnetic spectrum for sensing, wireless power and information transfer, polarization control, and thermal emission devices.

SS-HORSE extension of the no-core shell model: Application to resonances in He7

Theoretical ab initio studies of resonances in the unbound $^{7}\mathrm{He}$ nucleus are presented. We perform no-core shell-model calculations with $NN$ interactions Daejeon16 and JISP16 and utilize the SS-HORSE method to calculate the $S$ matrix for two-body channels $n+^{6}\mathrm{He}$ and $n+^{6}\mathrm{He}^{*}$ with $^{6}\mathrm{He}$ respectively in the ground and excited ${2}^{+}$ states as well as for the four-body democratic decay channel $^{4}\mathrm{He}+n+n+n$. The resonant energies and widths are obtained by numerical location of the $S$-matrix poles. We describe all experimentally known $^{7}\mathrm{He}$ resonances and suggest an interpretation of an observed wide resonance of unknown spin-parity.

The $$\varLambda (1405)$$ resonance as a genuine three-quark or molecular state

The mechanism for the formation of the $\Lambda(1405)$ resonance is studied in a chiral quark model that includes quark-meson as well as contact (four point) interactions. The negative-parity $S$-wave scattering amplitudes for strangeness $-1$ and $1$ are calculated within a unified coupled-channel framework that includes the $KN$, $\bar{K}N$, $\pi\Sigma$, $\eta\Lambda$, $K\Xi$, $\pi\Lambda$, and $\eta\Sigma$ channels and possible genuine three-quark bare singlet and octet states corresponding to $\frac12^-$ resonances. We show that in order to reproduce the scattering amplitudes in the $S_{01}$ partial wave it is important to include the pertinent three-quark octet states as well as the singlet state, while the inclusion of the contact term is not mandatory. The Laurent-Pietarinen expansion is used to determine the $S$-matrix poles. Following their evolution as a function of increasing interaction strength, the mass of the singlet state is strongly reduced due to the attractive self-energy in the $\pi\Sigma$ and $\bar{K}N$ channels; when it drops below the $KN$ threshold, the state acquires a dominant $\bar{K}N$ component which can be identified with a molecular state. The attraction between the kaon and the nucleon is generated through the $\bar{K}N\Lambda^*$ interaction rather than by meson-nucleon forces.

Use of transmission and reflection complex time delays to reveal scattering matrix poles and zeros: Example of the ring graph.

We identify the poles and zeros of the scattering matrix of a simple quantum graph by means of systematic measurement and analysis of Wigner, transmission, and reflection complex time delays. We examine the ring graph because it displays both shape and Feshbach resonances, the latter of which arises from an embedded eigenstate on the real frequency axis. Our analysis provides a unified understanding of the so-called shape, Feshbach, electromagnetically induced transparency, and Fano resonances on the basis of the distribution of poles and zeros of the scattering matrix in the complex frequency plane. It also provides a first-principles understanding of sharp resonant scattering features and associated large time delay in a variety of practical devices, including photonic microring resonators, microwave ring resonators, and mesoscopic ring-shaped conductor devices. Our analysis involves use of the reflection time difference, as well as a comprehensive use of complex time delay, to analyze experimental scattering data.

Definite complete invariant parametrization of R -matrix theory

We establish a definite parametrization of $R$-matrix theory, which is complete and invariant. Compared to the traditional parametrization of Wigner and Eisenbud, our parametrization has the major advantage of having no arbitrary boundary condition ${B}_{c}$, and of being constituted of scattering matrix poles ${\mathcal{E}}_{\ensuremath{\lambda}}$, which are physical quantities and hence invariant with the choice of arbitrary channel radii ${a}_{c}$. Moreover, being the poles of the scattering matrix, the definite levels ${\mathcal{E}}_{\ensuremath{\lambda}}$ correspond exactly to the nuclear resonances. Our definite parametrization is also global and complete, meaning a finite number of definite parameters---the same number as the Wigner and Eisenbud ones, minus the boundary conditions---can fully describe the scattering matrix on the whole complex plane [it is thus not a local description restricted to an energy window as the previous Windowed Multipole Representation of Ducru et al., Phys. Rev. C 103, 064610 (2021)]. These benefits come at the cost of requiring all parameters to now be complex numbers without an explicit set of constraints, which significantly complicates their direct nuclear data evaluation. We show that our parametrization also gives rise to shadow poles, though we prove they can be ignored and still completely reconstruct the scattering matrix with all its poles, and thus describe nuclear cross sections exactly. This means our parametrization only requires as many scattering matrix poles ${\mathcal{E}}_{\ensuremath{\lambda}}$ as there are Wigner-Eisenbud resonance levels ${E}_{\ensuremath{\lambda}}$, thereby establishing a one-to-one correspondence between the traditional Wigner-Eisenbud and our definite parametrization of $R$-matrix theory. Remarkably, we show these same cross sections can also be obtained using the shadow poles instead of the principal poles. We observe evidence of these phenomena in the spin-parity group ${J}^{\ensuremath{\pi}}=1/{2}^{(\ensuremath{-})}$ of xenon isotope $^{134}\mathrm{Xe}$.

Consequences of increased hypertriton binding for s -shell Λ -hypernuclear systems

Consequences of increasing the binding energy of the hypertriton ground state ${_{\Lambda}^3}{\rm H}(J^P={\frac{1}{2}}^+)$ from the emulsion value $B^{\rm EMUL}_{\Lambda}({_{\Lambda}^3}{\rm H}_{\rm g.s.})$=0.13$\pm$0.05 MeV to the STAR value $B^{\rm STAR}_{\Lambda}(^{3}_{\Lambda}{\rm H}) = (0.41\pm 0.12 \pm 0.11)$ MeV are studied for $s$-shell hypernuclei within a pionless EFT approach at leading order, constrained by the binding energies of the $0^+$ and $1^+$ ${_{\Lambda}^4} {\rm H}$ states. The stochastic variational method is used in bound-state calculations, whereas the inverse analytic continuation in the coupling constant method is used to locate $S$-matrix poles of continuum states. It is found that the $\Lambda nn({\frac{1}{2}}^+)$ resonance becomes broader and less likely to be observed experimentally, whereas the ${_{\Lambda}^3}{\rm H}({\frac{3}{2}}^+)$ spin-flip virtual state moves closer to the $\Lambda d$ threshold to become a shallow bound state for specific $\Lambda N$ interaction strengths. The effect of such a near-threshold ${_{\Lambda}^3}{\rm H}({\frac{3}{2}}^+)$ state on femtoscopic studies of $\Lambda$-deuteron correlations, and its lifetime if bound, are discussed. Increasing $B_{\Lambda}({_{\Lambda}^3} {\rm H}_{\rm g.s.})$ moderately, up to $\sim$0.5 MeV, hardly affects calculated values of $B_{\Lambda}({_{\Lambda}^5}{\rm He})$.

ОЦІНЮВАННЯ ГЕОМЕТРИЧНИХ ПАРАМЕТРІВ ПОЛЮСНОЇ СИСТЕМИ МАТРИЦІ ЕЛЕКТРОМАГНІТНОГО СЕПАРАТОРА

Purpose. The choice of the geometric dimensions ratios of system of matrix poles of electromagnetic polygradient separator to increase productivity with maintaining the reliability of extracting of ferromagnetic impurities from bulk material.Methodology. To solve the dynamic problem of motion of a ferromagnetic body in the working gap of pole system of matrix of polygradient separator under the influence of an external magnetic field the known methods of solving linear inhomogeneous differential equations are used. To confirm the reliability of obtained results the method of experimental research is used.Findings. The formulation of dynamic problem of movement of ferromagnetic body in the working gap of plate pole system of matrix of polygradient separator is carried out. Parametric equation for the trajectory of ferromagnetic body removal and a calculated relation connecting the main geometric dimensions of the system of matrix poles are obtained. The calculation results are confirmed experimentally and by operating practice of known magnetic separating devices.Originality. The mathematical description of working process of a polygradient electromagnetic separator with a plate matrix was further developed, which made it possible to obtain an analytical expression that takes into account the main geometric dimensions of the working space of matrix of separator.Practical value. Accounting of obtained analytical dependences between the length of separation zone and air gap, which characterizes the thickness of the separated material layer through which the ferromagnetic body must pass during the separation process, will ensure the necessary purity and productivity of separation.

Statistics of Complex Wigner Time Delays as a Counter of S-Matrix Poles: Theory and Experiment.

We study the statistical properties of the complex generalization of Wigner time delay τ_{W} for subunitary wave-chaotic scattering systems. We first demonstrate theoretically that the mean value of the Re[τ_{W}] distribution function for a system with uniform absorption strength η is equal to the fraction of scattering matrix poles with imaginary parts exceeding η. The theory is tested experimentally with an ensemble of microwave graphs with either one or two scattering channels and showing broken time-reversal invariance and variable uniform attenuation. The experimental results are in excellent agreement with the developed theory. The tails of the distributions of both real and imaginary time delay are measured and are also found to agree with theory. The results are applicable to any practical realization of a wave-chaotic scattering system in the short-wavelength limit, including quantum wires and dots, acoustic and electromagnetic resonators, and quantum graphs.

Experimental observation of the avoided crossing of two S -matrix resonance poles in an ultracold atom collider

In quantum mechanics, collisions between two particles are captured by a scattering matrix which describes the transfer from an initial entrance state to an outgoing final state. Analyticity of the elements of this $S$-matrix enables their continuation onto the complex energy plane and opens up a powerful and widely used framework in scattering theory, where bound states and scattering resonances for a physical system are ascribed to $S$-matrix poles. In the Gedankenexperiment of gradually changing the potential parameters of the system, the complex energy poles will begin to move, and in their ensuing flow, two poles approaching will interact. An actual observation of this intriguing interaction between scattering poles in a collision experiment has, however, been elusive. Here, we expose the interplay between two scattering poles relating to a shape resonance and a magnetically tunable Feshbach resonance by studying ultracold atoms with a laser-based collider. We exploit the tunability of the Feshbach resonance to observe a compelling avoided crossing of the poles in their energies which is the hallmark of a strongly coupled system.

Scattering matrix pole expansions for complex wave numbers inR-matrix theory

In this follow-up article to [Shadow poles in the alternative parametrization of R-matrix theory, Ducru (2020)], we establish new results on scattering matrix pole expansions for complex wavenumbers in R-matrix theory. In the past, two branches of theoretical formalisms emerged to describe the scattering matrix in nuclear physics: R-matrix theory, and pole expansions. The two have been quite isolated from one another. Recently, our study of Brune's alternative parametrization of R-matrix theory has shown the need to extend the scattering matrix (and the underlying R-matrix operators) to complex wavenumbers. Two competing ways of doing so have emerged from a historical ambiguity in the definitions of the shift $\boldsymbol{S}$ and penetration $\boldsymbol{P}$ functions: the legacy Lane \& Thomas "force closure" approach, versus analytic continuation (which is the standard in mathematical physics). The R-matrix community has not yet come to a consensus as to which to adopt for evaluations in standard nuclear data libraries, such as ENDF. In this article, we argue in favor of analytic continuation of R-matrix operators. We bridge R-matrix theory with the Humblet-Rosenfeld pole expansions, and unveil new properties of the Siegert-Humblet radioactive poles and widths, including their invariance properties to changes in channel radii $a_c$. We then show that analytic continuation of R-matrix operators preserves important physical and mathematical properties of the scattering matrix -- cancelling spurious poles and guaranteeing generalized unitarity -- while still being able to close channels below thresholds.

Shadow poles in the alternative parametrization of R -matrix theory

Nuclear data libraries (ENDF, JEFF, JENDL, CENDL, etc.) document our phenomenological knowledge of nuclear cross sections as interpreted by R-matrix theory. The R-matrix scattering model can parameterize the energy dependence of the scattering matrix in different ways. This article establishes new results on three such sets of parameters: the Wigner-Eisenbud, the Brune, and the Siegert-Humblet parameters. We show how the latter two arise from invariance to the arbitrary boundary condition, and how they can trade-off more complex parameters for a simpler energy dependence parameterization: the scattering matrix pole expansion of Humblet and Rosenfeld. We establish that the scattering matrix's invariance to channel radius sets a partial differential equation on the widths of the Kapur-Peierls operator, which enables us to derive an explicit transformation of the Siegert-Humblet radioactive residue widths under a change of channel radius. Considering the continuation of the scattering matrix to complex wavenumbers, several new results are established. We unveil there exist more Brune parameters than previously thought, depending on the way the scattering matrix is continued to complex energies. This points to a broader conundrum in the field: how to analytically continue the scattering matrix while closing sub-threshold channels. We argue that, contrary to the Lane and Thomas legacy of force-closing sub-threshold channels which introduces spurious poles, analytic continuation is the physically correct way of defining the scattering matrix for complex wave numbers. To back this claim we establish thee results: analytic continuation cancels spurious poles, enforces the generalized unitarity conditions of Eden and Taylor, and closes massive particle channels below threshold by both quantum tunneling and from the definition of the cross section as the ratio of probability currents.

On the Convergence of Oscillator Basis Calculations

The convergence of bound-state calculations performed via the oscillator basis expansions by means of locating $$S$$ -matrix poles for bound states within the HORSE and SS-HORSE approaches is examined. The convergence in question is studied both in the case of a sharp truncation of the potential matrix in the harmonic-oscillator space and in the case of smoothed matrix elements of the potential. As a result, a new method of extrapolation to the case of the infinite-dimensional model space is proposed. This method makes it possible to predict, on the basis of variational calculations, binding energies and asymptotic normalization coefficients for bound states to a high accuracy and to estimate the uncertainties of these predictions.

Jost matrices for some analytically solvable potential models

A family of analytically solvable potential models for the one- and two-channel problems is considered within the Jost matrix approach. The potentials are chosen to be constant in the interior region and to have different asymptotic behavior (tails) at large distances. The migration of the $S$-matrix poles on the Riemann surface of the energy, caused by variations of the potential strength, is studied. It is demonstrated that the long-range ($\ensuremath{\sim}1/{r}^{2}$) tails and Coulomb potential ($\ensuremath{\sim}1/r$) cause an unusual behavior of the $S$-matrix poles. It is found that in the two-channel problem with the long-range potentials the $S$-matrix poles may appear at complex energies on the physical Riemann sheet. The Coulomb tail not only changes the topology of the Riemann surface, but also breaks down the so-called mirror symmetry of the poles in both the single-channel and the two-channel problems.

Microarray profiling of circular RNAs in steroid-associated osteonecrosis of the femoral head: Observational study.

The aim of this study was to elucidate the molecular mechanisms and to identify the differential expression of circular RNAs (circRNAs) for steroid-associated osteonecrosis of the femoral head (SONFH) using bioinformatics analysis.circRNA microarray was performed with 3 SONFH tissues and the adjacent normal tissues, and differentially expressed circRNA were identified by limma package in R. Gene ontology (GO) and Kyoto Encyclopedia of Genes and Genomes (KEGG) pathway enrichment analyses were performed using the Database for Annotation, Visualization and Integrated Discovery database. In addition, a differentially expressed genes (DEG)-associated circRNA/microRNA (miRNA) interaction was predicted by combination of TargetScan and miRanda, and the circRNA/miRNA interaction network generated by the cytoscape software.A total of 647 differentially expressed circRNAs, including 433 upregulated and 214 downregulated circRNA were identified. The most enriched GO terms for upregulated and downregulated circRNA were extracellular matrix organization and leukocyte activation in biological process; extracellular matrix and spindle pole in cellular component; integrin binding and ATP binding in molecular function, and KEGG pathway enrichment analyses showed that the upregulated and downregulated circRNA were strongly associated with Protein digestion and absorption and Cell cycle. Moreover, a total of 212 differentially expressed messenger RNAs (mRNAs), including 113 upregulated and 99 downregulated genes were identified. In addition, from the analysis of miRNA, long noncoding RNAs, mRNA, and circRNA networks, we found that hsa_circ_0008136 and hsa_circ_0074758 were respectively the upregulated and downregulated circRNA with highest degrees.The identified circRNA and mRNA could be implicated in the progression of human SONFH. The findings could lead to a better understanding of SONFH pathogenesis.

Bottomonium resonances with I=0 from lattice QCD correlation functions with static and light quarks

We discuss, how to study $I = 0$ quarkonium resonances decaying into pairs of heavy-light mesons using static potentials from lattice QCD. These static potentials can be obtained from a set of correlation functions containing both static and light quarks. As a proof of concept we focus on bottomonium with relative orbital angular momentum $L = 0$ of the $\bar{b} b$ pair corresponding to $J^{P C} = 0^{- +}$ and $J^{P C} = 1^{- -}$. We use static potentials from an existing lattice QCD string breaking study and compute phase shifts and $\mbox{T}$ matrix poles for the lightest heavy-light meson-meson decay channel. We discuss our results in the context of corresponding experimental results, in particular for $\Upsilon (10860)$ and $\Upsilon (11020)$.

Resonant states 0+ of the Boron isotope 8B from the Jost-matrix analysis of the partial cross-sections

The available [Formula: see text]-matrix parametrization of experimental data on the excitation functions for the elastic and inelastic [Formula: see text] scattering at the collision energies up to 3.4[Formula: see text]MeV is used to generate the corresponding partial-wave cross-sections in the states with [Formula: see text]. Thus, obtained data are fitted using the semi-analytic two-channel Jost matrix with a proper analytic structure and some adjustable parameters. Then the spectral points are sought as zeros of the Jost matrix determinant (which correspond to the [Formula: see text]-matrix poles) at complex energies. The correct analytic structure makes it possible to calculate the fitted Jost matrix on any sheet of the Riemann surface whose topology involves not only the square-root but also the logarithmic branching caused by the Coulomb interaction. In this way, two overlapping [Formula: see text] resonances at the excitation energies [Formula: see text] and [Formula: see text][Formula: see text]MeV have been found.

Analytical version of the resonance coupled-channel model for D + T → 5He** → α + n reaction and its application for the description of low-energy D-T and D- 3He scattering

The purpose of the present paper is the formulation of the analytical version of the resonance coupled-channel model (RCCM) originally developed for D + T → 5He** → α + n nuclear fusion reaction. The integral in the denominator of the Breit-Wigner type is examined in the expression for S-matrix elements of binary processes in this model. Imaginary part of this integral determines the energy-dependent decay width for the near-threshold channel. It is demonstrated that this integral can be calculated explicitly with the Binet representation for the ψ-function (the logarithmic derivation of the gamma function). As the result the explicit expression for the S-matrix elements in the form of analytical functions of the channel momenta are obtained and the equivalence of the RCCM and the effective range approximation (Landau – Smorodinsky – Bethe approximation) is established on this basis. This allows expressing the parameters of the RCCM through the model independent system characteristics: the complex scattering length and the complex effective range. Several sets of model parameters of both approaches that provide a good description of the measured data on D + T → α + n reaction and D-T elastic scattering are derived. By this means we find the location of the S – matrix poles on different Riemann sheets which corresponds to Jπ = (3/2)+ state of 5He and 5Li nuclei. In particular, the location of the resonance (R) and shadow (S) poles is determined: 5He**: ZR = 46.9 – i37.2 (keV) ZS = 81.7 – i3.5 (keV) 5Li**: ZR = 205.7 – i146.8 (keV) ZS = 264.4 + i112.0 (keV). Our results agree well with previous findings. The possible generalizations of the results obtained are discussed.

Dynamical generation of resonances in the P33 partial wave

We investigate the formation of resonances in the P33 partial wave with the emphasis on possible emergence of dynamically generated quasi-bound states as a consequence of a strong $p$-wave pion attractive interaction in this partial wave, as well as their possible interaction with the genuine quark excited states. By using the Laurent-Pietarinen expansion we follow the evolution of the $S$-matrix poles in the complex energy plane as a function of the interaction strength. Already without introducing a genuine quark resonant state, two physically interesting resonances emerge with pole masses around 1200 MeV and 1400 MeV, with the dominant $\pi N$ and $\pi\Delta$ component, respectively. The added genuine resonant state in the $(1s)^3$ quark configuration mixes with the lower dynamically generated resonance forming the physical $\Delta(1232)$ resonance, and pushes the second dynamical resonance to around 1500 MeV, which allows it to be identified with the $\Delta(1600)$ resonance. Adding a second resonant state with one quark promoted to the $2s$ orbit generates another pole whose evolution remains well separated from the lower two poles. We calculate the helicity amplitudes at the pole and suggest that their $Q^2$ dependence could be a decisive test to discriminate between different models of the $\Delta(1600)$ resonance.

Jost-matrix analysis of the resonance 5He∗(3 2+) near the dt-threshold

Partial cross-sections of the [Formula: see text] and [Formula: see text] collisions in the quantum state [Formula: see text] near the [Formula: see text]-threshold, taken from an available [Formula: see text]-matrix analysis, are fitted using the semi-analytic multi-channel Jost matrix with proper analytic structure and some adjustable parameters. Then the spectral points are sought as zeros of the Jost matrix determinant (which correspond to the [Formula: see text]-matrix poles) at complex energies. The correct analytic structure makes it possible to calculate (analytically continue) the fitted Jost matrix on any sheet of the Riemann surface whose topology involves not only the square-root but also the logarithmic branching caused by the Coulomb interaction. As a result of such an analytic continuation, it was found that the previously established [Formula: see text]-resonance (at [Formula: see text]) and its shadow pole (at [Formula: see text]) both are split in overlapping pairs. Apart from studying the properties of a specific nuclear state, it is also proved for a general multi-channel problem that the Coulomb forces change the topology of the Riemann surface as well as destroy the so-called mirror symmetry of the [Formula: see text]-matrix.

Doubly heavy tetraquark resonances in lattice QCD

We study tetraquark resonances with lattice QCD potentials computed for two static quarks and two dynamical quarks, the Born-Oppenheimer approximation and the emergent wave method of scattering theory. We compute the phase shifts and search for S and T matrix poles in the second Riemann sheet. We predict a new tetraquark resonance for l = 1, decaying into two B mesons, with quantum numbers I(JP) = 0(1−), mass and decay width .