Ground-state Eigenvalue Research Articles

Ground state and lowest eigenvalue of the Laplacian for non-compact hyperbolic surfaces

LetM be a complete Riemannian surface with constant curvature −1, infinite volume, and a finitely generated fundamental group. Denote by λ(M) the lowest eigenvalue of the Laplacian onM, and let Φ M be the associated eigenfunction. We estimate the size of λ(M) and the shape of Φ M by a finite procedure which has an electrical circuit analogue. Using the Margulis lemma, we decomposeM into its thick and thin parts. On the compact thick components, we show that Φ M varies from a constant value by no more thanO(\(\sqrt {\lambda (M)}\)). The estimate for λ(M) is calculable in terms of the topology ofM and the lengths of short geodesics ofM. An analogous theorem of the compact case was treated in [SWY].

Twentieth-order perturbation study of the nonadiabatic electric polarizabilities for H2 + via the perturbational-variational Rayleigh-Ritz formalism.

Large-order perturbation theory has been applied, for the first time, to the Stark effect for ${\mathrm{H}}_{2}^{+}$, yielding the Rayleigh-Schr\"odinger ground-state eigenvalue (polarizability) series through twentieth order; previous expansions were limited to fourth order. The calculations were performed nonadiabatically (i.e., without invoking the Born-Oppenheimer approximation) by means of the perturbational-variational Rayleigh-Ritz formalism. The leading terms of the Rayleigh-Schr\"odinger polarizability series so obtained provide the most accurate values thus far determined for ${\ensuremath{\alpha}}_{\mathrm{zz}}$ and ${\ensuremath{\gamma}}_{\mathrm{zzzz}}$.

Lower bounds to ground state eigenvalues of the Schr�dinger equation via optimized inner projection: Application to quartic and sextic anharmonic oscillators

Inner projection combined with bracketing techniques represents a very powerful method of calculating lower bounds to eigenvalues of the Schrodinger equation. The Hamiltonian is expressed as a sum of an “unperturbed” operator (whose eigenvalue equation is soluble) and a “perturbed” operator that should be positive definite. For small values of the coupling constant, the conventional splitting of the Hamiltonian is adequate. For strong coupling, however, it is advantageous to partition the Hamiltonian in an optimized fashion, accomplished by an appropriate scaling of the space coordinates and momenta. This optimized inner projection (OIP) method yields remarkably good results for all values of the coupling constant, even for manifolds of minimal dimensionality.

On the optimization of gaussian basis sets for hartree-fock-dirac calculations

Gaussian basis sets for atomic one-electron systems have been optimized by straight minimization of the electronic ground-state eigenvalue of the finite basis set representation of the Dirac operator, using the “kinetic energy balance” procedure in conjunction with appropriate additional variational freedom. The results, which are apparently upper bounds, are presented and compared with previous data.

A 30th-order perturbational study of the born-oppenheimer electronic polarizabilities for H2+ via the perturbational-variational Rayleigh-Ritz formalism

Large-order perturbation theory in the form of the perturbational-variational Rayleigh-Ritz formalism has been applied to the calculation of the Born-Oppenheimer electronic polarizabilities for H2+, yielding the Rayleigh-Schrödinger ground-state eigen-value (polarizability) series, at an internuclear separation of 2.0 au, through 30th order with unprecedented accuracy; previous Born-Oppenheimer polarizability expansions were limited to 4th order. Comparison of these purely electronic Born-Oppenheimer results with recent non-Born-Oppenheimer calculations, which include vibration, reveals the enormous effect of the nuclear motion on the high-order polarizabilities.

Constrained energy functionals obtained as solutions to a partial differential equation

The constrained energy functional, defined by E0(a) =minSa{a〈Ψ‖Ĥ0‖Ψ〉a}, ai=a〈Ψ‖Âi‖Ψ〉a, i=1, ..., M, ‖Ψ〉a∈Sa, with a={ai} a preassigned set of expectation values generated by any state vector ‖Ψ〉a in the class Sa, is shown to satisfy a first order, but nonlinear, partial differential equation. The equation results by replacing the Lagrange multipliers introduced in an earlier paper by the corresponding partial derivatives of the to-be-found energy functional with respect to the constraints. Two alternative expressions for E0(a), one focusing upon the ground state eigenvalue of a modified Hamiltonian operator and the other upon the corresponding eigenvector, lead via standard Rayleigh–Schrödinger perturbation theory to two approximate partial differential equations for the energy functional. Both equations together with the imposed subsidiary conditions lead in turn to the same approximate E0(a). An example based on the linear harmonic oscillator illustrates the concepts presented in the formalism.

A new method for summation of divergent power series

A new method for summation of divergent power series is developed. It only requires the knowledge of the form of both the small and large λ-power expansion (λ being the perturbation parameter) and few coefficients of one of them to yield excellent results. Convergence is proved for a simple two-level model, and reasonable arguments are given for more complex and interesting models. The method is quite general and contains some resummation techniques reported previously as particular cases. The anharmonic, mean square, displacement function, the ground-state eigenvalue of the quantum-mechanical anharmonic oscillator, and the ground-state energy of the hydrogen atom in a magnetic field calculated in this way are shown to be of striking accuracy in the whole range of the perturbation parameter.

The four-nucleon problem with local realistic nn interactions

The ground state and the first excited 0 + state binding energies of the 4He are calculated within the framework of the hyperspherical method for the Gogny-Pirès-de Tourreil potential (GPDT), and for the two super soft core potentials (SSC.B and SSC.C) constructed by de Tourreil and Sprung. With neglect of the Coulomb effect, the ground state energies are found to be −30.10 MeV for GPDT, −25.88 MeV for SSC.B and −21.19 MeV for SSC.C. For the two first potentials we find an excited 0 + state. The location of this state above the ground state is found to be 25.66 MeV for GPDT and 25.11 MeV for SSC.B. We analyse the convergence of the ground state eigenvalues versus the grand orbital ( L = 2 K) in comparing the results of the coupled equation calculations (CE) with those of different approximations giving: a lower bound with the extreme adiabatic approximation (EAA), an upper bound with the uncoupled adiabatic approximation (UAA) and an accurate bound with the coupled adiabatic approximation (CAA).

Generalized Euler transformation for summing strongly divergent Rayleigh-Schrödinger perturbation series: The Zeeman effect

A generalized Euler transformation (GET) is introduced which provides a powerful alternative method of accurately summing strongly divergent Rayleigh-Schr\odinger (RS) perturbation series when other summability methods fail or are difficult to apply. The GET is simple to implement and, unlike a number of other summation procedures, requires no a priori knowledge of the analytic properties of the function underlying the RS series. Application of the GET to the difficult problem of the RS weak-field ground-state eigenvalue series of the hydrogen atom in a magnetic field (quadratic Zeeman effect) yields sums of good accuracy over a very wide range of field strengths up to the most intense fields of ${10}^{14}$ G. The GET results are compared with those obtained by other summing methods.

Coupled adiabatic approximation in the three-body problem

In the framework of the hyperspherical formalism, we present a study of the coupled adiabatic approximation for the case of three nucleons interacting via central spindependent two-body potentials. We analyze the convergence of the ground state eigenvalues versus the grand orbital quantum number ($2K$) and compare the results to that of the coupled equations. We also compare with two simpler but less accurate approximations: the uncoupled adiabatic approximation and the extreme adiabatic approximation. The former provides an upper and the latter provides a lower bound to the ground state energy.NUCLEAR STRUCTURE Few body bound states, reduction of hyperspherical equations applied to triton, quantum few-body problem.

Electronic structure of the pseudo-one-dimensional superconductor Tl 2Mo 6Se 6 according to SCF scattered-wave Xα computations

The electronic structure of Tl 2Mo 6Se 6 has been investigated using the Xα cluster approach. The ground-state eigenvalues show that the Fermi level lies in a single degenerated band which supports the experimental results. Contour maps show the anisotropic character of the wawe functions of this Fermi level. The metal-metal and metal-selenides interactions have been discussed.

The anharmonic oscillator: complex eigenvalues for the ground state with negative quartic or cubic energy distortion

The perturbed ground-state eigenvalues for the harmonic oscillator with negative quartic and with cubic energy distortion are calculated by numerical integration. These agree with the author's results using an asymptotic series.

Decay of correlations in surface models

A convergent low-temperature expansion for a variety of models of twodimensional surfaces is presented. It yields existence of the thermodynamic limit for the pressure and correlation functions as well as analyticity inz =e−β In addition, the estimates give exponential decay of truncated correlations, which proves the existence of a gap in the spectrum of the transfer matrix below the ground state eigenvalue. Two particular examples included in the general framework are the solid-on-solid and discrete Gaussian models.

DYNAMICS AND THERMODYNAMICS OF A ONE DIMENSIONAL NON-LINEAR LATTICE IN THE CONTINUUM LIMIT

The dynamics of a non-linear one dimensional lattice, with the on-site potential V(φ) = (ξ 2 /8) Cosn 4φ-ξ Cosh 2φ-(ξ 2 /8) is considered in the continuum limit. Exact classical kink solutions are obtained; for ξ<2, in which case the potential has the double well form. The free energy of the system is calculated using the ground state eigen value of the Schrodinger like equation for this potential.

Interpretation of charge transfer bands in Fe doped SrTiO 3 crystals

The electronic structures of SrTiO 3 crystals doped with Fe 3+, Fe 4+ and Fe 5+ ions have been investigated using the Xα cluster approach. The ground-state eigenvalues show the lower Fe acceptor level, of t 2 g ↓ symmetry, localized inside the SrTiO 3 band gap, respectively at 2.8 eV ( Fe 3+, S = 5 2 ), 1.6 eV ( Fe 4+, S = 1) or 1.1 eV ( Fe 4+, S = 0) and 1.1 eV ( Fe 5+, S = 3 2 ) above the valence band edge. Other acceptor levels, with eg↓ and eg↑ symmetries, appear inside the gap when the Fe nominal ionicity increases. The theoretical Xα excitation energies of O 2 p-Fe 3 d transitions confirms the experimental interpretations of acceptor charge transfer bands for the optical absorption spectra of SrTiO 3:Fe 4+ and SrTiO 3:Fe 5+ crystals. The large optical excitation energies compared with the thermal transitions are partly due to the O 2 p band width.

Disappearance of impurity levels in silicon and germanium due to screening

We have studied the disappearance of impurity levels in silicon and germanium due to free-carrier screening of the Coulomb field of the impurity ions. The ground-state eigenfunctions and eigenvalues have been calculated for electrons described by an ellipsoidal effective-mass Hamiltonian. A two-dimensional finite-element analysis was used to obtain the solutions. Only moderate carrier densities (1019 cm−3 for silicon and 1018 cm−3 for germanium) are needed to cause the impurity levels to disappear into the conduction band, the result at high doping densities being simply a degenerate semiconductor.

Solution of the four-nucleon Schrödinger equation

We present a solution of the Schrödinger equation for the case of four nucleons interacting via the Malfliet-Tjon potential. The Schrodinger equation is integrated by means of the Green function Monte Carlo method. The adaption of this method to treat finite systems is given. For the four-nucleon case, the ground-state eigenvalue is found to be −31.3 ± 0.2 MeV and the rms point mass radius 1.36 ± 0.01 fm. The point mass distribution is given and does not exhibit a central depression. The method is also applied to the problem of four helium atoms interacting via the Lennard-Jones potential.

Bounds on eigenvalues of related Hamiltonian operators

A theorem is presented which allows one to obtain a bound on the ground-state eigenvalue of an operator equation, given the eigenvalue of another related equation. The equations are related by the «potential» in one equation being the square of the «potential» in the other. The theorem is related to the Ritz variational principle, but allows one to obtain the bound without having to integrate a «trial function». Some examples are given of the application of this theorem to the Schrödinger equation.

Matrix moment methods in perturbation theory, boson quantum field models, and anharmonic oscillators

A matrix moment problem is considered in connection with anyx 2m(m=2, 3, 4, ...) anharmonic oscillator as well as the (:φ2m(x):g(x))2 (m=2, 3) field theory models, whose Rayleigh-Schrödinger perturbation expansions for the ground state eigenvalue are known to diverge. The approximants related to such a problem are proven to converge to the eigenvalue, when applied to an expansion of the Brillouin-Wigner type. These approximants, whose construction involves only matrix elements occurring in the Rayleigh-Schrödinger expansion, are the approximants of aJ-type matrix continued fraction, i.e. the [N−1,N] matrix Padé approximants. The explicit analytical expression of matrix continued fraction is found in the anharmonic oscillators case.

Eigenvalues of λx2m anharmonic oscillators

The ground state as well as excited energy levels of the generalized anharmonic oscillator defined by the Hamiltonian Hm = − d2/dx2+x2+ λx2m, m = 2,3, …, have been calculated nonperturbatively using the Hill determinants. For the λx4 oscillator, the ground state eigenvalues, for various values of λ, have been compared with the Borel-Padé sum of the asymptotic perturbation series for the problem. The agreement is excellent. In addition, we present results for some excited states for m = 2 as well as the ground and the first even excited states for m = 3 and 4. The behaviour of all the energy levels with respect to the coupling parameter shows a qualitative similarity to the ground state of the λx4 oscillator. Thus the results are model independent, as is to be expected from the WKB approximation. Our results also satisfy the scaling property that εn(m)(λ)/λ1/(m+1) tend to a finite limit for large λ, and always lie within the variational bounds, where available.