No abstract

It is shown that the F 4 rational and trigonometric integrable systems are exactlysolvable for arbitrary values of the coupling constants. Their spectra are found explicitly while eigenfunctions by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)triangular. These variables are invariant with respect to the Weyl group of F 4 root system and can be obtained by averaging over an orbit of the Weyl group. Alternative way of finding these variables exploiting a property of duality of the F 4 model is presented. It is demonstrated that in these variables the Hamiltonian of each model can be expressed as a quadratic polynomial in the generators of some infinite-dimensional Lie algebra of differential operators in a finite-dimensional representation. Both Hamiltonians preserve the same flag of spaces of polynomials and each subspace of the flag coincides with the finite-dimensional representation space of this algebra. Quasiexactly-solvable generalization of the rational F 4 model depending on two continuous and one discrete parameters is found.

Solvability of the rational quantum integrable systems related to exceptional root spaces G 2 , F 4 is re-examined and for E 6,7,8 is established in the framework of a unified approach. It is shown the Hamiltonians take algebraic form being written in a certain Weyl-invariant variables. It is demonstrated that for each Hamiltonian the finite-dimensional invariant subspaces are made from polynomials and they form an infinite flag. A notion of minimal flag is introduced and minimal flag for each Hamiltonian is found. Corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for arbitrary values of the coupling constants. The Hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite-dimensional but finitelygenerated Lie algebra of differential operators, taken in a finite-dimensional representation.

As a continuation of our previous work (Phys. Rev. A68, 012504 (2003)) an accurate study of the lowest 1σ g and the low-lying excited 1σ u , 2σ g , 1π u,g , 1δ g,u electronic states of the molecular ion H + 2 is made. Since the parallel configuration where the molecular axis coincides with the magnetic field direction is optimal, this is the only configuration which is considered. The variational method is applied and the same trial function is used for different magnetic fields. The magnetic field ranges from 10 9 G to 4.414 × 10 13 G where non-relativistic considerations are justified. Particular attention is paid to the 1σ u state which was studied for an arbitrary inclination. For this state a one-parameter vector potential is used which is then variationally optimized.

Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithiumlike ions with static nuclei for Z ≤ 20 the domain of applicability of Non-Relativistic QED (NRQED) is localized for ground state energy. It is demonstrated that for both helium-like and lithium-like ions with Z ≤ 20 the finite nuclear mass effects do not change 4-5 significant digits (s.d.) and the leading relativistic and QED effects leave unchanged 3-4 s.d. in the ground state energy. It is shown that the non-relativistic ground state energy can be interpolated with accuracy of not less than 6 decimal digits (d.d.) (or 7-8 s.d.) for Z ≤ 50 for helium-like and for Z ≤ 20 for lithium-like ions by a meromorphic function in, which is well inside the domain of applicability of NRQED. It is found that both the Majorana formula -a second degree polynomial in Z with two free parameters -and a fourth degree polynomial in λ (a generalization of the Majorana formula) reproduce the ground state energy of the helium-like and lithium-like ions for Z ≤ 20 in the domain of applicability of NRQED, thus, at least, 3 s.d. It is noted that 99.9% of the ground state energy is given by the variational energy for properly optimized trial function of the form of (anti)-symmetrized product of three (six) screened Coulomb orbitals for two-(three) electron system with 3 (7) free parameters for Z ≤ 20, respectively. It may imply that these trial functions are, in fact, exact wavefunctions in non-relativistic QED, thus, the NRQED effective potential can be derived. It is shown that the sum of relativistic and QED effects in leading approximation -3 s.d. -for both 2 and 3 electron systems is interpolated by 4th degree polynomial in Z