A simple eigenvalue formula for the quartic anharmonic oscillator

1985 ◽  
Vol 63 (3) ◽  
pp. 311-313 ◽  
Author(s):  
Richard L. Hall

The eigenvalues Enl(λ) of the Hamiltonian H = −Δ + r2 + λr4 are analysed with the help of "potential envelopes" and "kinetic potentials." The result is the following simple approximate eigenvalue formiula:[Formula: see text]where E ≥ P = (4n + 2l − 1) and Q = 3(An + Bl − C)4/322/3. E is a lower bound to Enl if (A, B, C) = (1, 1/2, 1/4) and a good approximation if (A, B, C) = (1.125, 0.509, 0.218).

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.


10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.


10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.


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