[en] A survey is given of several results obtained previously on the asymptotic properties in relation with its nodes. A new result on the behaviour of local solutions in the neighbourhood of its zero is announced. Let the domain Ω be element Rⁿ (n≥2), x₀ element Ω, V element C^∞(Ω); ψ notident 0; and ψ element C^∞(Ω) satisfy (-Δ+V)ψ=0; ψ(x₀)=0. The question arises if there exists a nodal domain D_r0 of ψ for which the corresponding S(r) (S^n-1 being the unit sphere in Rⁿ) shrinks, with r→0 into a subset of the nodal set of the Y_n, the spherical harmonics. Defining ψ₀=(∫_S(2)ψ²ds)^1/2, the answer is: ψ₀·r^-M and |S(2)| have finite non-zero limiting values for r→0. 14 refs. (qui)

[en] Results on nodal properties of L² solutions of two-dimensional Schroedinger equations obtained in a previous paper are refined. The generally unbounded nodal set of ψ is investigated for r → ∞ and shown that in this limit the nodal set consists of non-intersecting nodal lines which look asymptotically either like straight lines or like branches of parabolas. (G.Q.)

[en] It is shown that the local properties of the wave-function in the neighbourhood of a zero are determined to a certain extent by global properties of the nodal set of the corresponding surface harmonic. 8 refs

[en] A rigorous upper bound to the one-electron density at the nucleus rho(0) has been derived since the accuracy of the calculation of the electronic wavefunctions of atomic systems at the nucleus has been questioned in view of the discrepancy between experiment and theory of parity violation. The present results on small atomic systems indicate that the deviation of a factor five in the parity violation calculations is unlikely to be caused by numerical inaccuracy. Furthermore simple inequalities relating the kinetic energy to expectation values of multiplicative one-electron operators are given. (U.K.)

[en] Let Ω_R = Rⁿinverse slantB_R, where n ≥ 3 and B_R = {x element of Rⁿ:vertical strokexvertical stroke ≤ R}. We investigate the asymptotics of real valued solutions ψ element of L²(Ω_R) of the Schroedinger equation (-Δ+V-E)ψ = 0, where E < 0 and V(x) → 0 for vertical strokexvertical stroke → ∞: Let D denote an unbounded nodal domain of ψ (i.e. a component of Ω_Rinverse slant{x:ψ(x) = 0}), and let S(r) = {y element of S^n-1:ry element of D} with S^n-1 the unit sphere in Rⁿ. Under suitable assumptions on V it is shown that for some γ > 0, limsub(r → ∞) inf r^γ ∫sub(S(r)) ψ²dσ / ∫sub(S^n-1) ψ²dσ > 0 and limsub(r → ∞) inf ln (Volume(D intersection B_r))/ln r ≥ (n+1)/2. Results of this type are already non-trivial for radial problems with ψ satisfying non-radial boundary conditions on dΩ_R or for excited states of the Hydrogen atom if one considers linear combinations of different l-waves. (orig.)

[en] It is shown that the Hamiltonian H of the hydrogenic anion has no bound state at the threshold in the triplet S-sector. This extends a result of Hill (1977, J. Math. Phys. 18 2316) who showed that H has only an essential spectrum in the triplet sector. (author)

[en] We study L²-solutions of (-Δ+V-E)ψ = 0 in Rⁿ, n ≥ 2 and derive a sharp upper bound to l_x0(ψ) in terms of x₀, E and V under rather restrictive assumptions on V. We show for V smooth that an upper bound to l_x0(ψ) implies an upper bound to the dimension of the eigenspace associated to E. 16 refs. (Authors)

[en] The asymptotic behavior of ground states of two-electron atoms is investigated. Suppose psi(x₁,x₂) is the ground state of helium, rho(x₁) = ∫psi²(x₁,x₂) dx₂ the corresponding electron density, and Phi(x₂) the ground state of He⁺. We show that in the L²(dx₂)-sense, lim/sub vertical-barx/₁Vertical Bar→infinity psi(x₁,x₂)[rho(x₁)]/sup -1/2/ = Phi(x₂), and that psi[rho(x₁)]/sup -1/2/ solves for large Vertical Barx₁Vertical Bar the Schroedinger equation for He⁺ in the quadratic form sense. The rate of convergence of these limits is also discussed

[en] Let H = - Δ + V be defined on L²(Rsup(n)), n >= 3. Let V = V₁ + V₂, V₁ is element of Lsup(p)(Rsup(n)), for some p > 2n/3, V₂ is element of Lsup(infinity)(Rsup(n)) and |x|delta V/delta|x| relatively form bounded with respect to - Δ with relative bound < 2. It is proven that there exists an α₀ >= 0 such that for all α >= α₀, esup(α|x|)psi(x) is not element of L²(Rsup(n)), where psi denotes and L²-eigenfunction of H. Related results are also shown to hold for many body Schroedinger operators including atoms and molecules. (Author)

[en] It is shown that the Hamiltonian H of the hydrogenic anion has no bound state at threshold in the triplet S-sector. This extends a result of R.N. Hill (1977) who showed that H has only essential spectrum in the triplet sector. (Author)