Bound State Problem Research Articles

Quasinormal modes and the analytical continuation of non-self-adjoint operators

We briefly review the analytical continuation method for determining quasinormal modes (QNMs) and the associated frequencies in open systems. We explore two exactly solvable cases based on the Pöschl–Teller potential to show that the analytical continuation method cannot determine the full set of QNMs and frequencies of a given problem starting from the associated bound state problem in quantum mechanics. The root of the problem is that many QNMs are the analytically continued counterparts of solutions that do not belong to the domain where the associated Schrödinger operator is self-adjoint, challenging the application of the method for determining full sets of QNMs. We illustrate these problems through the physically relevant case of BTZ black holes, where the natural domain of the problem is the negative real line.

Determination of the Energy Eigenvalues of the Varshni-Hellmann Potential

In this paper, we solve the bound state problem for the Varshni-Hellmann potential via a useful technique. In our technique, we obtain the bound state solution of the Schrödinger equation for the Varshni-Hellmann potential via ansatz method. We obtain the energy eigenvalues and the corresponding eigenfunctions. Also, the behavior of the energy spectra for both the ground and the excited state of the two body systems is illustrated graphically. The similarity of our results to the accurate numerical values is indicative of the efficiency of our technique.

Continuum Energy Eigenstates via the Factorization Method

The factorization method was introduced by Schrödinger in 1940. Its use in bound-state problems is widely known, including in supersymmetric quantum mechanics; one can create a factorization chain, which simultaneously solves a sequence of auxiliary Hamiltonians that share common eigenvalues with their adjacent Hamiltonians in the chain, except for the lowest eigenvalue. In this work, we generalize the factorization method to continuum energy eigenstates. Here, one does not generically have a factorization chain—instead all energies are solved using a “single-shot factorization”, enabled by writing the superpotential in a form that includes the logarithmic derivative of a confluent hypergeometric function. The single-shot factorization approach is an alternative to the conventional method of “deriving a differential equation and looking up its solution”, but it does require some working knowledge of confluent hypergeometric functions. This can also be viewed as a method for solving the Ricatti equation needed to construct the superpotential.

Bethe–Salpeter kernel and properties of strange-quark mesons

Focusing on the continuum meson bound-state problem, a novel method is used to calculate closed-form Bethe–Salpeter kernels that are symmetry consistent with any reasonable gluon-quark vertex, $$\varGamma _\nu $$ , and therewith deliver a Poincaré-invariant treatment of the spectrum and decay constants of the ground- and first-excited states of u, d, s mesons. The predictions include masses of as-yet unseen states and many unmeasured decay constants. The analysis reveals that a realistic, unified description of meson properties (including level orderings and mass splittings) requires a sound expression of emergent hadron mass in bound-state kernels; alternatively, that such properties may reveal much about the emergence of mass in the standard model.

Solving hadron structures using the basis light-front quantization approach on quantum computers

Quantum computing has demonstrated the potential to revolutionize our understanding of nuclear, atomic, and molecular structure by obtaining forefront solutions in nonrelativistic quantum many-body theory. In this work, we show that quantum computing can be used to solve for the structure of hadrons, governed by strongly interacting relativistic quantum field theory. Following our previous work on light unflavored mesons as a relativistic bound-state problem within the nonperturbative Hamiltonian formalism, we present the numerical calculations on simulated quantum devices using the basis light-front quantization approach. We implement and compare the variational quantum eigensolver and the subspace-search variational quantum eigensolver to find the low-lying mass spectrum of the light meson system and its corresponding light-front wave functions as quantum states from ideal simulators, noisy simulators, and IBM quantum computers. Based on obtained quantum states, we evaluate the meson decay constants and parton distribution functions directly on the quantum circuits. Our calculations on the quantum computers and simulators are in reasonable agreement with accurate numerical solutions solved on classical computers when noises are moderately small, and our overall results are comparable with the available experimental data.

Bound states of a one-dimensional Dirac equation with multiple delta-potentials

Two approaches are developed for the study of the bound states of a one-dimensional Dirac equation with the potential consisting of N δ-function centers. One of these uses Green’s function method. This method is applicable to a finite number N of δ-point centers, reducing the bound state problem to finding the energy eigenvalues from the determinant of a 2 N × 2 N matrix. The second approach starts with the matrix for a single delta-center that connects the two-sided boundary conditions for this center. This connection matrix is obtained from the squeezing limit of a piecewise constant approximation of the delta-function. Having then the connection matrices for each center, the transmission matrix for the whole system is obtained by multiplying the one-center connection matrices and the free transfer matrices between neighbor centers. An equation for bound state energies is derived in terms of the elements of the total transfer matrix. Within both approaches, the transcendental equations for bound state energies are derived, the solutions to which depend on the strength of delta-centers and the distance between them, and this dependence is illustrated by numerical calculations. The bound state energies for the potentials composed of one, two, and three delta-centers (N = 1, 2, 3) are computed explicitly. The principle of strength additivity is analyzed in the limits as the delta-centers merge at a single point or diverge to infinity.

Λnn bound state and resonance

We perform the ab initio no-core shell model (NCSM) calculation to investigate the bound state problem of the three-body $\Lambda nn$ system in chiral next-to-next-to-leading-order NN and chiral leading-order YN interactions. The calculations show that no $\Lambda nn$ bound state exists, but predict a low-lying $\Lambda nn$ resonant state near the threshold with the energy of $E_r= 0.124$ MeV and the width of about $\Gamma=1.161$ MeV. In searching for $\Lambda nn$ resonances, we extend the NCSM calculation to the continuum state by employing the J-matrix formalism in the scattering theory with the hyperspherical oscillator basis.

Nucleon axial-vector and pseudoscalar form factors and PCAC relations

We use a continuum quark+diquark approach to the nucleon bound-state problem in relativistic quantum field theory to deliver parameter-free predictions for the nucleon axial and induced pseudoscalar form factors, $G_A$ and $G_P$, and unify them with the pseudoscalar form factor $G_5$ or, equivalently, the pion-nucleon form factor $G_{\pi NN}$. We explain how partial conservation of the axial-vector current and the associated Goldberger-Treiman relation are satisfied once all necessary couplings of the external current to the building blocks of the nucleon are constructed consistently; in particular, we fully resolve the seagull couplings to the diquark-quark vertices associated with the axial-vector and pseudoscalar currents. Among the results we describe, the following are worth highlighting. A dipole form factor defined by an axial charge $g_A=G_A(0)=1.25(3)$ and a mass-scale $M_A = 1.23(3) m_N$, where $m_N$ is the nucleon mass, can accurately describe the pointwise behaviour of $G_A$. Concerning $G_P$, we obtain the pseudoscalar charge $g_p^\ast = 8.80(23)$, and find that the pion pole dominance approach delivers a reliable estimate of the directly computed result. Our computed value of the pion-nucleon coupling constant, $g_{\pi NN}/m_N =14.02(33)/{\rm GeV}$ is consistent with recent precision determinations.

Gauge dependence of spontaneous radiation spectrum in a time-dependent relativistic non-perturbative Coulomb field

We extend the "gauge choice" problem Lamb noticed to include a time-dependent relativistic non-perturbative Coulomb field, which can be produced by a cluster of relativistic charged particles. If adiabatic conditions are carefully maintained, such a field must be included along side the nuclear Coulomb potential when defining the atomic state. We reveal that when taking the external field approximation, the gauge choice for this time-dependent relativistic non-perturbative Coulomb field cannot be overcome by previous method, and leads to considerable gauge-dependence of the transient spontaneous radiation spectrum. We calculate explicitly with a simple one-dimensional charged harmonic oscillator that such a gauge-dependence can be of a measurable magnitude of 10 MHz or larger for the commonly used Coulomb, Lorentz, and multipolar gauges. Contrary to the popular view, we explain that this gauge dependence is not really a disaster, but actually an advantage here: The relativistic bound-state problem is so complicated that a fully quantum-field method is still lacking, thus the external field approximation cannot be derived and hence not guaranteed. However, by fitting to the experimental data, one may always define an effective external field, which may likely be parameterized with the gauge potential in a particular gauge. This effective external field would not only be of phenomenological use, but also shed light on the physical significance of the gauge field.

Estimating Scattering Potentials in Inverse Problems with a Non-Causal Volterra Model

In this paper, a finite memory, non-causal Volterra model is proposed to estimate the potential functions in various inverse quantum mechanical problems, where the bound or scattered wave functions are used as inputs of the Volterra system, while the potential is the desired output. Two simple examples are given to show the model capabilities, where in both cases, a really good match is achieved for a very wide range of potential functions. The first example is a simple one-dimensional bound state problem, where the wave function of the first bound state is used as input to determine the model potential. The second example is a one-dimensional scattering problem, where the scattered wave is used as the system input. In both cases, a higher order, non-causal description is needed to be able to give a good estimation to the solution of the inverse problem. The model sensitivity to input perturbations is also examined, showing that the Volterra representation is capable of giving a robust estimate to the underlying dynamical system. The model could be useful in real-life situations, where the scattering potential should be found from measured data, where the precise equations that govern the dynamics of the system are not known.

Infinite bound states and hydrogen atom-like energy spectrum induced by a flat band

In this work, we investigate the bound-state problem in a one-dimensional spin-1 Dirac Hamiltonian with a flat band. It is found that the flat band has significant effects on the bound states. For example, for Dirac delta potential gδ(x), there exists one bound state for both the positive and negative potential strength g. Furthermore, when the potential is weak, the bound-state energy is proportional to the potential strength g. For square well potential, the flat band results in the existence of infinite bound states for arbitrarily weak potential. In addition, when the bound-state energy is very near the flat band, the energy displays a hydrogen atom-like spectrum, i.e. the bound-state energies are inversely proportional to the square of the natural number n (e.g., E n ∝ 1/n 2, n = 1, 2, 3, …). Most of the above nontrivial behaviors can be attributed to the infinitely large density of states of the flat band and its ensuing 1/z singularity of the Green function. The combination of a short-ranged potential and flat band provides a new possibility to get an infinite number of bound states and a hydrogen atom-like energy spectrum. In addition, our findings provide some useful insights and further our understanding of the many-body physics of the flat band.

The status of $$\Xi _\mathrm{{cc}}^{++}$$ baryon: investigating quark–diquark model

Inspired by the discovery of the doubly charmed and doubly charged $$\Xi _{\text {cc}}^{++}$$ baryon, in this present work, we investigate spin-1/2 and spin-3/2 mass spectra and magnetic moments of ccu baryon in quark–diquark model with a nonrelativistic potential. In the quark–diquark picture, the three-body problem of the ccu baryon is reduced to the two-body bound state problem. We obtained ground and radially excited mass spectra and magnetic moments of the ground state. Ground state mass spectra agree well with the results in the literature, especially with lattice QCD computations. Predictions for the magnetic moment are compatible with the literature. According to our results, cu diquark clustering in the $$\Xi _{\text {cc}}^{++}$$ baryon is more favorable than the cc diquark clustering.

An improved quantum condition for the bound state problem in non-relativistic quantum mechanics using the NUFA method

An improved quantum condition for the bound state problem in non-relativistic quantum mechanics using the NUFA method

Semileptonic transitions: B(s) → π(K); Ds → K; D → π,K; and K → π

Continuum Schwinger function methods for the strong-interaction bound-state problem are used to arrive at a unified set of parameter-free predictions for the semileptonic K→π, D→π,K and Ds→K, B(s)→π(K) transition form factors and the associated branching fractions. The form factors are a leading source of uncertainty in all such calculations: our results agree quantitatively with available data and provide benchmarks for the hitherto unmeasured Ds→K0, B¯s→K+ form factors. The analysis delivers a value of |Vcs|=0.974(10) and also predictions for all branching fraction ratios in the pseudoscalar meson sector that can be used to test lepton flavour universality. Quantitative comparisons are provided between extant theory and the recent measurement of BBs0→K−μ+νμ. Here, further, refined measurements would be useful in moving toward a more accurate value of |Vub|.

Wave function collapses and 1/n energy spectrum induced by a Coulomb potential in a one-dimensional flat band system

We investigate the bound state problem in a one-dimensional flat band system with a Coulomb potential. It is found that, in the presence of a Coulomb potential of type I (with three equal diagonal elements), similarly to that in the two-dimensional case, the flat band could not survive. At the same time, the flat band states are transformed into localized states with a logarithmic singularity near the center position. In addition, the wave function near the origin would collapse for an arbitrarily weak Coulomb potential. Due to the wave function collapses, the eigen-energies for a shifted Coulomb potential depend sensitively on the cut-off parameter. For a Coulomb potential of type II, there exist infinite bound states that are generated from the flat band. Furthermore, when the bound state energy is very near the flat band, the energy is inversely proportional to the natural number, e.g., En ∝ 1/n, n = 1,2,3,… It is expected that the 1/n energy spectrum could be observed experimentally in the near future.

Model of electromagnetic interactions with spin-1/2 neutral particles

We address the bound-state dynamics of relativistic spin-1/2 neutral particles (in this paper, Dirac neutrinos) with anomalous magnetic dipole moment in the presence of an electromagnetic (EM) field described by a generalized Dirac–Pauli equation. This equation of motion is derived including appropriate couplings between Lorentz scalar and pseudoscalar fields with the EM field in the Lagrangian of the system. Specifically, we exactly solve the bound-state problem of neutrinos in the presence of a homogeneous magnetic field in cylindrical coordinates. We comment on the relevance of this approach to study Dirac neutrino self-interactions.

A new quantization rule to the bound state problem in non-relativistic quantum mechanics

In this paper, a new quantization rule to calculate energy spectrum is presented for the bound state problem in non-relativistic quantum mechanics based on the parametric Nikiforov-Uvarov (NU) method. Moreover, it is observed that its final form is conveniently expressed in terms of the original parameters that appear in the general form of the Schrödinger equation. As a result, an analytical formula to the bound state problem is written out explicitly in a straightforward manner without difficulty. The fool-proof nature of the method will appeal to the theorist who requires an analytical solution on the fly. A few special cases arising from this formula are also provided to demonstrate its accuracy.

Ξ hypernuclear states predicted by next-to-leading-order chiral baryon–baryon interactions

Abstract The $\Xi$ single-particle potential obtained in nuclear matter with next-to-leading-order baryon–baryon interactions in chiral effective field theory is applied to finite nuclei by an improved local-density approximation method. As a premise, phase shifts of $\Xi N$ elastic scattering and the results of Faddeev calculations for the $\Xi NN$ bound state problem are presented to show the properties of the $\Xi N$ interactions in the present parametrization. First, the $\Xi$ states in $^{14}$N are revisited because of recent experimental progress, including discussion on the $\Xi N$ spin–orbit interaction that is relevant to the location of the $p$-state. Then the $\Xi$ levels in $^{56}$Fe are calculated. In particular, the level shift which is expected to be measured experimentally in the near future is predicted. The smallness of the imaginary part of the $\Xi$ single-particle potential is explicitly demonstrated.

Regarding the Distribution of Glue in the Pion

Understanding why the scale of emergent hadron mass is obvious in the proton but hidden in the pion may rest on mapping the distribution functions (DFs) of all partons within the pion and comparing them with those in the proton; and since glue provides binding in quantum chromodynamics, the glue DF could play a special role. Producing reliable predictions for the proton’s DFs is difficult because the proton is a three-valence-body bound-state problem. As sketched herein, the situation for the pion, a two-valence-body problem, is much better, with continuum and lattice predictions for the valence-quark and glue DFs in agreement. This beginning of theory alignment is timely because experimental facilities now either in operation or planning promise to realize the longstanding goal of providing pion targets, thereby enabling precision experimental tests of rigorous theory predictions concerning Nature’s most fundamental Nambu–Goldstone bosons.

A Realistic Approach to the $$\varXi NN$$ Bound-State Problem Based on Faddeev Equation

The Faddeev equations for the \(\varXi NN\) bound-state problem are solved where the three \(S=-2\) baryon–baryon interactions of Jülich-Bonn-München chiral EFT, HAL QCD and Nijmegen ESC08c are used. The T-matrix \(T_{\varXi N, \varXi N}\) obtained within the original \(\varLambda \varLambda -\varXi N-\varSigma \varSigma /\varXi N-\varLambda \varSigma -\varSigma \varSigma \) coupled-channel framework is employed as an input to the equations. We found no bound state for Jülich-Bonn-München chiral EFT and HAL QCD but ESC08c generates a bound state with the total isospin and spin-parity \((T,J^{\pi })=(1/2, 3/2^+)\) where the decays into \(\varLambda \varLambda N\) are suppressed.