Formal Rayleigh-Schroedinger perturbation theory for both degenerate and non-degenerate energy states
Degenerate and nondegenerate energy states and wave functions formulated by Rayleigh- Schroedinger perturbation theory
Hirschfelder, J. O.↗
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Biased degenerate ground-state sampling of small Ising models with converged quantum approximate optimization algorithm
The quantum alternating operator ansatz, a generalization of the quantum approximate optimization algorithm (QAOA), is a quantum algorithm used for approximately solving combinatorial optimization problems. QAOA typically uses the transverse field mixer as the driving Hamiltonian. One of the interesting properties of the transverse field driving Hamiltonian is that it results in nonuniform sampling of degenerate ground states of optimization problems. In this study, we numerically examine the fair sampling properties of the transverse field mixer QAOA, and Grover mixer QAOA (GM-QAOA), which provides theoretical guarantees of fair sampling of degenerate optimal solutions, up to a large enough p such that the mean expectation value converges to an optimal approximation ratio of 1. This comparison is performed with high-quality heuristically computed, but not necessarily optimal, QAOA angles, which give strictly monotonically improving solution quality as p increases. These angles are computed using the Julia based numerical simulation software JuliQAOA. Fair sampling of degenerate ground states is quantified using the Shannon entropy of the ground-state amplitudes distribution. The fair sampling properties are reported on several quantum signature Hamiltonians from previous quantum annealing fair sampling studies. Small random fully connected spin glasses are shown, which exhibit exponential suppression of some degenerate ground states with transverse field mixer QAOA. The transverse field mixer QAOA simulations show that some problem instances clearly saturate the Shannon entropy of 0 with a maximally biased distribution that occurs when the learning converges to an approximation ratio of 1 while other problem instances never deviate from a maximum Shannon entropy (uniform distribution) at any p step. Published by the American Physical Society 2025
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Degenerate dwarf X-ray sources
X-ray emission by accreting degenerate dwarfs is examined qualitatively, considering bremsstrahlung and cyclotron fluxes and cooling. Also discussed are strongly magnetic degenerate dwarfs, which are characterized by a blackbody-limited UV cyclotron component and a hard X-ray bremsstrahlung component, both produced by the hot emission region, and a hard UV or soft X-ray blackbody component produced by cyclotron and bremsstrahlung photons. With regard to nonmagnetic and weakly magnetic degenerate dwarfs, it is concluded that degenerate dwarfs can produce larger luminosities than earlier estimates indicated, that even high mass stars can produce low temperature (less than 10 keV) X-ray spectra, and, for moderate and high accretion rates and high (greater than 1.2 solar masses) stellar masses, the spectra exhibit a high energy tail. Individual sources of each type of dwarf are considered, such as the AM Her, DQ Her and SS Cyg classes as well as Cyg X-2 and the Sco X-1 class, and the number of such sources in the Galaxy is estimated.
Lamb, D. Q.↗
X-ray and UV radiation from accreting nonmagnetic degenerate dwarfs. II
Numerical calculations of X-ray and UV emission from accreting nonmagnetic degenerate dwarfs are reported, which span the entire range of accretion rates and stellar masses. Calculations include the effects of bremsstrahlung, Compton cooling, radiation pressure, albedo of the stellar surface, Compton degradation and free-free abscription of the X-ray spectrum by the accreting matter. Maximum X-ray luminosity for degenerate dwarfs undergoing spherical accretion is found to be 2.2 x 10 to the 36th ergs/s, which is little changed if accretion occurs radially over only a fraction of the stellar surface, so that the emitted radiation escapes without significant scattering. The temperature characterizing the X-ray spectra produced by degenerate dwarfs strongly depends on the stellar mass and the accretion rate, and it is suggested that the correlation between spectral temperature and luminosity is an important signature of degenerate X-ray sources.
Kylafis, N. D.↗
Emerging monochromatic fluxes and colors of red degenerate stars
The emerging monochromatic fluxes and the B-V, V-I, J-H, and V-K color indices are presented for red degenerate stars with helium atmospheres which were calculated using an equation of state and opacities based on a hot Thomas-Fermi model of the helium gas. The effective temperature range is 4500-2500 K. It is found that although the emerging fluxes resemble blackbody curves, red degenerates emit more radiation than blackbodies in the short wavelength range and less in the long wavelength range. Thus, red degenerates appear bluer than blackbodies of the same temperature. The calculated colors of these models are compared to the colors of some of the coolest known non-DA degenerate stars. In particular it is found that the B-V and V-I colors of the cool white dwarf VB 11, whose temperature had been previously estimated to be higher than 4000 K, suggest a temperature of 3750 K. If this result is correct, then VB 11 is probably the coolest known white dwarf.
Kapranidis, S.↗
Spectroscopic observations of cool degenerate star candidates
Spectroscopic observations are reported for 23 Luyten Half-Second degenerate star candidates and for 13 Luyten-Palomar common proper-motion pairs containing possible degenerate star components. Twenty-five degenerate stars are identified, 20 of which lack previous spectroscopy. Most of these stars are cool - Luyten color class g or later. One star, LP 77-57, shows broad continuum depressions similar to those in LHS 1126, which Liebert and Dahn attributed to pressure-shifted C2. A second degenerate star, LHS 290, exhibits apparent strong Swan bands which are blueshifted about 75 A. Further observations, including polarimetry and photometry, are required to appraise the spectroscopic peculiarities of these stars. Finally, five cool, sharp-lined DA white dwarfs have been observed to detect lines of metals and to determine line strengths. None of these DAs show signs of Mg b or the G band, and four show no evidence of Ca II K. The attempt to detect Ca MI in the fifth star, G199-71, was inconclusive.
Hintzen, P.↗
Three very cool degenerate stars in Luyten common proper motion binaries - Implications for the age of the galactic disk
During the course of a spectroscopic study of Luyten common proper motion (CPM) stars, spectrophotometric observations have been obtained of three binaries containing degenerate stars with estimated absolute magnitudes M(V) of about 16. Each of the three pairs consists of a yellow degenerate star primary and a DC 13 + secondary 1.4-2.3 mag fainter. One of the primary stars is spectral class DC 7, another is a sharp-lined DA 8, and the third shows peculiar broad absorption features which we interpret as pressure-shifted C2 Swan bands. The LP 701 - 69/70 system has survived for over 8 billion years without disruption by passing stars, despite its 1500 a.u. orbital major axis. The three cool degenerate companions nearly double the available sample of stars at the low-luminosity terminus of the white dwarf cooling sequence. These findings appear consistent with the conclusion that degenerate stars in the old disk population have not had time to evolve to a luminosity fainter than M(V) about 16.2.
Hintzen, Paul↗
Multi-Disciplinary, Multi-Fidelity Discrete Data Transfer Using Degenerate Geometry Forms
In a typical multi-fidelity design process, different levels of geometric abstraction are used for different analysis methods, and transitioning from one phase of design to the next often requires a complete re-creation of the geometry. To maintain consistency between lower-order and higher-order analysis results, Vehicle Sketch Pad (OpenVSP) recently introduced the ability to generate and export several degenerate forms of the geometry, representing the type of abstraction required to perform low- to medium-order analysis for a range of aeronautical disciplines. In this research, the functionality of these degenerate models was extended, so that in addition to serving as repositories for the geometric information that is required as input to an analysis, the degenerate models can also store the results of that analysis mapped back onto the geometric nodes. At the same time, the results are also mapped indirectly onto the nodes of lower-order degenerate models using a process called aggregation, and onto higher-order models using a process called disaggregation. The mapped analysis results are available for use by any subsequent analysis in an integrated design and analysis process. A simple multi-fidelity analysis process for a single-aisle subsonic transport aircraft is used as an example case to demonstrate the value of the approach.
Olson, Erik D.↗
Degenerate coupled-cluster theory
A size-extensive, converging, black-box, ab initio coupled-cluster (ΔCC) ansatz is introduced that computes the energies and wave functions of states from any degenerate or nondegenerate Slater-determinant references with any numbers of α- and β-spin electrons, any patterns of orbital occupancy, any spin multiplicities, and any spatial symmetries. For a nondegenerate reference, it reduces to the single-reference coupled-cluster ansatz. For a degenerate multireference, it is a natural coupled-cluster extension of degenerate Møller–Plesset perturbation (ΔMP) theory. For ionized and electron-attached references, it is a coupled-cluster Green’s function, although the present theory is convergent toward the full-configuration-interaction limits, while the Feynman–Dyson many-body Green’s function (MBGF) theory generally is not. Its single-excitation instance is a projection Hartree–Fock theory as per the Thouless theorem, which may be useful for core ionizations, high-spin states, and possibly electron affinities. Additionally, a new multireference coupled-cluster theory for a general model space is developed. This quasidegenerate coupled-cluster (QCC) theory is exactly converging, but not black-box, and intended for strong correlation. Determinant-based, general-order algorithms of ΔCC and QCC theories are implemented and compared with configuration-interaction (CI) and equation-of-motion coupled-cluster (EOM-CC) theories through octuple excitations and with ΔMP and MBGF theories up to the nineteenth order. An algebraic, optimal-scaling algorithm of the ΔCC theory is computer-synthesized at the levels of single excitations (ΔCCS) and of single and double excitations (ΔCCSD). As a result, the order of performance is QCC ≈ ΔCC > EOM-CC > CI at the same order or QCC ≈ ΔCC > ΔMP > MBGF at the same cost scaling.
Hirata, So [University of Illinois at Urbana-Champ↗
Degenerate domain walls in supersymmetric theories
In supersymmetric Yang–Mills theories tension-degenerate domain walls are typical. Adding matter fields in fundamental representation, we arrive at supersymmetric quantum chromodynamics (SQCD) supporting similar walls. We demonstrate that the degenerate domain walls can belong to one of two classes: i) locally distinguishable, i.e. those which differ from each other locally (which could be detected in local measurements); and ii) those which have identical local structure and are differentiated only topologically, through a judicially chosen compactification of R 4 . Depending on the number of flavors F and the pattern of Higgsing, both classes can coexist among SQCD k walls interpolating between the vacua n and n + k . We prove that the overall multiplicity of the domain walls obtained after accounting for both classes is ν N , k walls = N ! / [ ( N − k ) ! k ! ] , as was discovered previously in limiting cases. (Here, N is the number of colors.) Thus, ν N , k walls is a peculiar index. For the locally distinguishable degenerate domain walls, we observe two-wall junctions, a phenomenon specific for supersymmetry with central extensions. This phenomenon does not exist for topological replicas.
Chen, Shi↗
Triaxial degenerate dwarfs and pulsing X-ray sources
It is shown that triaxial configurations for rotating degenerate dwarfs consisting entirely of fluid are not limited to forms analogous to the Jacobi ellipsoids. Arguments are presented which rule out only the specialized class of uniformly rotating analogs of the Jacobi ellipsoid for some degenerate dwarfs. Several ways of forming triaxial degenerate dwarfs are examined along with the characteristic lifetimes of such objects. Noting that some of the alternative forms may exist in mass-transfer binary systems, consideration is given to the secondaries in Her X-1 and Cen X-3. It is suggested that the secondary in Her X-1 must be close to an analog of a Dedekind ellipsoid, and plausible triaxial white-dwarf configurations are proposed for the secondary in Cen X-3.
Durisen, R. H.↗
X-ray spectra of accreting degenerate stars
X-ray emission from accreting nonmagnetic degenerate dwarfs is discussed. Emphasis is placed on the numerical calculations by Kylafis and Lamb (1979) with regard to the entire range of accretion rates and stellar masses. It is concluded that at moderate and high accretion rates the spectral temperature varies dramatically and the star exhibits a pronounced correlation between X-ray spectral temperature and luminosity. Also considered are the results of Lamb and Masters (1979) of calculations of high-harmonic cyclotron emission from a hot plasma, from which they developed a model of X-ray and UV emission from accreting magnetic degenerate dwarfs. It is found that the position and relative strength of the spectral components change with variations in the accretion rate and differ for different magnetic fields; in addition, magnetic degenerate dwarfs are predicted to be strong UV sources with only a few percent of the total accretion luminosity ordinarily appearing as soft and hard X-rays.
Kylafis, N. D.↗
Pulsational analyses of post planetary nebula central stars and degenerate dwarfs
Recent observational and theoretical studies of the ZZ Ceti variabls (DA degenerate dwarfs), the DBV variables (DB degenerate dwarfs), and the GW Vir variables (D0 degenerate dwarfs) have shown them to be pulsating in nonradial g-modes. The pulsation mechanism has been identified for each class of variable star. For the ZZ Ceti and DBV variables it is both the kappa and gamma effects in the partial-ionization regions of either H or He and also a a recently identified pulsation-driving mechanism called 'convection blocking'. For the GW Vir variables, it is the kappa and gamma effects in the partial-ionization region of C and O. The ZZ Ceti variables must have pure H surface layers, the DBV stars must have pure He surface layers, and the GW Vir stars must have C- and O-rich surface layers with only a small amount of He present. The accuracy of the prediction for the GW Vir stars is limited by the lack of observational determinations of their luminosity, effective temperature, and composition.
Starrfield, Sumner↗
Extending Orbital-Optimized Density Functional Theory to L-Edge XPS and Beyond: Spin–Orbit Coupling via Nonorthogonal Quasi-Degenerate Perturbation Theory
Quantum mechanical calculations of core electron binding energies (CEBEs) are relevant to interpreting X-ray photoelectron spectroscopy (XPS). Orbital-optimized density functional theory (OO-DFT) accurately predicts K-edge CEBEs but is challenged by the presence of significant spin–orbit coupling (SOC) at L- and higher edges involving inner-shell orbitals with nonzero angular momentum. Here, to extend OO-DFT to L-edges and higher, our method utilizes scalar-relativistic, spin-restricted open-shell OO-DFT to construct a minimal, quasi-degenerate basis of core-hole states corresponding to a chosen inner-shell (e.g., ionizing all six possible 2p spin orbitals). Nonorthogonal configuration interaction (NOCI) is then used to obtain the matrix elements of the full Hamiltonian including SOC in this quasi-degenerate model space of determinants. Using a screened 1-electron SOC operator parametrized with the Dirac-Coulomb-Breit (DCB) Hamiltonian results in doublet splitting (DS) values for third row elements that are nearly in quantitative agreement with experiment. The resulting NOCI eigenvalues are shifted by the average of the (scalar) OO-DFT CEBEs to yield CEBEs (split by SOC) corrected for dynamic correlation. Comparing calculations on gas phase molecules with experimental results establishes that NO-QDPT with the SCAN functional (NO-QDPT/SCAN), using the DCB screened 1-electron SOC operator is accurate to about 0.2 eV for L-edge CEBEs of molecules containing third row atoms. However, this NO-QDPT approach becomes less accurate for fourth-row elements starting in the middle of the 3d transition metal series, with errors increasing as atomic number increases.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Kinematic control of robot with degenerate wrist
Kinematic resolved rate equations allow an operator with visual feedback to dynamically control a robot hand. When the robot wrist is degenerate, the computed joint angle rates exceed operational limits, and unwanted hand movements can result. The generalized matrix inverse solution can also produce unwanted responses. A method is introduced to control the robot hand in the region of the degenerate robot wrist. The method uses a coordinated movement of the first and third joints of the robot wrist to locate the second wrist joint axis for movement of the robot hand in the commanded direction. The method does not entail infinite joint angle rates.
Barker, L. K.↗
IUE spectroscopy of the degenerate components in the Hyades close binaries V471 Tauri and HZ9
Effective temperatures and gravities have been determined for the degenerate components in the Hyades close binaries V471 Tauri and HZ9 from model atmosphere analysis of IUE ultraviolet spectra. For V471 Tauri, T(e) = 35,000 K and log g is about 8. For HZ9, T(e) = 20,000 K and log g = 8. Both systems contain DA2 white dwarfs. No photospheric metal lines were detected at low IUE resolution in either DA2 star. The 1400 A and 1600 A features are absent in the HZ9 degenerate, despsite being within the temperature range of some other DA stars with the 1400 A absorption feature.
Guinan, E. F.↗
Cygnus X-2 - Neutron star or degenerate dwarf?
Some conflicting models have been proposed for Cyg X-2: a degenerate dwarf model which predicts a distance of 250 + or 50 pc; and a neutron star model which implies a distance of about 8000 pc. Based on a reddening study, it is found that the distance to Cyg X-2 is greater than 1100 pc, which rules strongly against the degenerate dwarf model. This conclusion is based on observations of the 2200 A feature in the spectrum of Cyg X-2 made with the International Ultraviolet Explorer (IUE), and UBV and spectroscopic observations of 38 field stars. For the reddening of Cyg X-2 values of E(B-V) = 0.40 + or - 0.07 (1 sigma) are found and are consistent with the reddening to infinity in that direction inferred from radio data. Consequently, Cyg X-2 may be located in the halo at about 8 kpc as proposed in 1979 by Cowley, Crampton, and Hutchings.
Mcclintock, J. E.↗
The influence of velocity-changing collisions on resonant degenerate four-wave mixing
The phase-conjugate signal observed in resonant degenerate four-wave mixing on the 6 3P2 to 7 3S1 transition of atomic Hg in an Hg-Ar discharge is investigated. At a fixed Ar pressure the variation of the signal with pump powers is explained by a model that includes the effects of velocity-changing collisions (VCCs). As the Ar pressure was varied from 0 to 1 torr, an increase in the phase-conjugate signal was observed and is ascribed to a change in the discharge dynamics with Ar pressure and to the influence of VCCs. To further clarify the role of collisions and optical pumping, degenerate four-wave mixing spectra are examined as a function of pump power. Line shapes are briefly discussed.
Richardson, W. H.↗