β Scribed by Elliott H Lieb; Walter E Thirring
No coin nor oath required. For personal study only.
We consider a single particle in a negative potential V: H = --d + V(x). A lower bound is found for the quantity +I -EW, where cm is the ground-state energy of H in all space and where EA is the ground-state energy of H in a bounded domain A with Dirichlet ($ = 0) boundary conditions. Our estimate for EA -cm involves only e4 and the volume, 1 A I, but does not depend upon V or upon the shape of A.
π SIMILAR VOLUMES
Recent work by Bernasconi, Damm, and Shparlinski showed that the set of square-free numbers is not in AC 0 and raised as an open question whether similar (or stronger) lower bounds could be proved for the set of prime numbers. We show that the Boolean majority function is AC 0 -Turing reducible to t
## Abstract For any graph __G__, let __i__(__G__) and ΞΌ;(__G__) denote the smallest number of vertices in a maximal independent set and maximal clique, respectively. For positive integers __m__ and __n__, the lower Ramsey number __s__(__m, n__) is the largest integer __p__ so that every graph of or
We show lower bounds on the worst-case complexity of Shellsort. In particular, Ε½ Ε½ 2 . Ε½ . 2 . we give a fairly simple proof of an β n lg n r lg lg n lower bound for the size of Shellsort sorting networks for arbitrary increment sequences. We also show an identical lower bound for the running time o
Let G be an n-vertex graph with list-chromatic number Ο . Suppose that each vertex of G is assigned a list of t colors. Albertson, Grossman, and Haas [1] conjecture that at least t n /Ο vertices can be colored from these lists. We prove a lower bound for the number of colorable vertices. As a coroll
A non-isolated vertex of a graph G is called a groupie if the average degree of the vertices connected to it is larger than or equal to the average degree of the vertices in G. An isolated vertex is a groupie only if all vertices of G are isolated. While it is well known that every graph must contai