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Exact solutions to the d-dimensional Schrödinger equation, d >= 2, for Coulomb plus harmonic oscillator potentials V(r) = -a/r + br2, b > 0, and a ≠ 0 are obtained. The potential V(r) is considered both in all space, and under the condition of spherical confinement inside an impenetrable spherical box of radius R. With the aid of the asymptotic iteration method, the exact analytic solutions under certain constraints, and general approximate solutions, are obtained. These exhibit the parametric dependence of the eigenenergies on a, b, and R. The wave functions have the simple form of a product of a power function, an exponential function, and a polynomial. Polynomial solutions are found for differential equations of the form py″ + qy′ - ry = 0, where p, q, and r are given polynomials with degrees 4, 3, and 2, respectively. [ABSTRACT FROM AUTHOR]

We study the lowest energy E of a relativistic system of N identical bosons bound by pair potentials of the form V(rij) = g(rij²) in thee spatial dimensions. In natural units &hstroke; = c = 1 the system has the semirelativistic ‘spinless-Salpeter’ Hamiltonian H = ∑I=1N √m² + pi² + ∑j>i=jNg(¦ri-rj¦²), where g is monotone increasing and has convexity gn≥0. We use ‘envelope theory’ to derive formulas for general lower energy bounds and we use a variational method to find complementary upper bounds valid for all N≥2. In particular, we determine the energy of the N-body oscillator g(r²) = cr² with error less than 0.15% for all m≥0, N≥2, and c>0. [ABSTRACT FROM AUTHOR]

We study the lowest energy E of a relativistic system of N identical bosons bound by pair potentials of the form V(rij) = g(rij²) in thee spatial dimensions. In natural units &hstroke; = c = 1 the system has the semirelativistic ‘spinless-Salpeter’ Hamiltonian H = ∑I=1N √m² + pi² + ∑j>i=jNg(¦ri-rj¦²), where g is monotone increasing and has convexity gn≥0. We use ‘envelope theory’ to derive formulas for general lower energy bounds and we use a variational method to find complementary upper bounds valid for all N≥2. In particular, we determine the energy of the N-body oscillator g(r²) = cr² with error less than 0.15% for all m≥0, N≥2, and c>0. [ABSTRACT FROM AUTHOR]

A system of N identical bosons is studied, each having mass m, which interact in R³ via attractive central pair potentials and obey nonrelativistic quantum mechanics. A lower energy bound is found by the equivalent two-body method. An upper energy bound established previously on the basis of field theory is now derived by variational methods within conventional quantum mechanics. In the case of the linear potential V[sub ij] = γr[sub i] - r[sub j] the bounds imply that the ground-state energy is given by E = C(N) (N - 1) (h²/m)[sup 1/3](γN/ 2)[sup 2/3], where 2.3381 < C(N) < 2.343 52. The energy is therefore determined in this case with error < 0.116% for all N≥2. Similar results are given for other power-law potentials. [ABSTRACT FROM AUTHOR]

It is supposed that a single fermion with Hamiltonian H=α·p+βμ(r)+[lowercase_phi_synonym](r), where μ(r) and [lowercase_phi_synonym](r) are central potentials, obeys the Dirac equation. If ψ1(r) and ψ2(r) are the radial factors in the Dirac spinor, then the graph {ψ1(r), ψ2(r)} for r∈(0,∞) is called a spinor orbit. In cases where discrete eigenvalues exist, the corresponding spinor orbit eventually returns to the origin. However, if there is a constant a≥0 such that, for r>a, the three functions [lowercase_phi_synonym](r), [lowercase_phi_synonym](r)/μ(r), and rμ(r) increase monotonically without bound, then it is proved that the spinor orbit must eventually be confined to an annular region excluding the origin. Consequently, the spinor orbit approaches a ‘‘spinor circle,’’ the spinor is not L2, and there are no eigenvalues. This happens, for example, if μ is constant and [lowercase_phi_synonym](r) is any monotone increasing and unbounded potential. In such cases the radius of the spinor circle is sensitive to the energy, and instead of eigenvalues one finds a sequence of resonant energies for which the radii of the spinor circles are local minima. [ABSTRACT FROM AUTHOR]

If E is an eigenvalue of the quantum-mechanical Hamiltonian H= 1/2 Δ+V(r) in N spatial dimensions, then large-N theory, the potential-envelope method, and scale-optimized variational energies all lead to quasiclassical approximations having the same form given by E(α) = minr>0[ 1/2 αr-2+V(r)], where α depends on the quantum numbers and on N. Energy bounds provided by the envelope method allow us to prove that in many cases the large-N results are lower energy estimates. For pure power-law potentials all these energies approach the exact eigenvalue in either of the limits l → ∞ or N → ∞. [ABSTRACT FROM AUTHOR]

We consider a hydrogenlike system in which the Coulomb potential is replaced by the more general central potential V(r) = vf(r/b) = vg(-b/r), where g is monotone increasing and convex. The method of potential envelopes is applied to this problem and approximations are obtained for the energy trajectories based on the expression ∈[sub nj] = min [D[sub nj](u) - uD'[sub nj](u) u ∈ (0,1) + vf {-1/bD'[sub nj](u)}], where D[sub nj] (u) is the known exact trajectory function for the hydrogenic atom. General formulas are given for linear combinations of power-law potentials and the log potential. Some graphical results are presented in the case of the Coulomb-plus-linear potential f(r) = -α/r + βr. [ABSTRACT FROM AUTHOR]