Upper and lower bounds for the optimal constant in the extended Sobolev inequality. Derivation and numerical results

We prove and give numerical results for two lower bounds and eleven upper bounds to the optimal constant k 0 = k 0 (n,α) in the inequalitywhere for n = 1, 0 < α 1/2, and for n 2, 0 < α < 1.The lowest point in the point spectrum of the Schrödinger operator τ = -Δ + q on R n with the real-valued potential q can be expressed in λ n,α for all q -= max(0,-q) ∈ L p (R n ), for n = 1, 1 p < ∞, and n 2, n/2 < p < ∞, and the norm q -p .for 0 < α < 1/2, and k 0 (1, 1/2) = 1, Mathematics subject classification (2010): 26D15, 41A44.