Sharp constants in Lieb-Thirring inequalities

5 problems

• Optimal constant for counting bound states in three dimensions

  Open

  The number of bound states of a 3D potential well is at most a constant times the integral of $\mid V \mid^{3/2}$. The best possible constant is unknown.

• Exponent at which the semiclassical constant becomes optimal

  Open

  For large enough exponent $\gamma$ the best constant equals the simple semiclassical value; for small $\gamma$ it is larger. Find the crossover exponent in two and three dimensions.

• Semiclassical constant for the eigenvalue sum in three dimensions

  Open

  Prove that in three dimensions the total binding energy of all bound states is never larger than the simple semiclassical estimate. Semiclassical means counting states as if each occupies a fixed volume of position-momentum space.

• Sharp Lieb-Oxford constant for the indirect Coulomb energy

  Open

  The Lieb-Oxford inequality bounds how far the electrostatic energy of many charges can fall below the value for a smeared-out charge cloud, using only their density. The best constant lies between 1.44 and 1.58 and is unknown.

• One-bound-state optimality in one dimension for $1/2 < \gamma < 3/2$

  Open

  In one dimension the conjecture says the worst-case potential well, the one with the most binding for its size, has only a single bound state when the exponent $\gamma$ is between $1/2$ and $3/2$. Prove it.