9 problems

Feenberg-scaling invariance conjecture. Since the true characteristic polynomials, which depend only on the total Hamiltonian, are invariant under Feenberg scaling, estimates obtai…

Let $|\Psi_\mathrm{2e}\rangle$ be a pure state of $N=2$ fermions in $2K$ modes, and let $\mathcal{B}$ be any orthonormal basis of the one-particle Hilbert space. In the natural-orb…

Consider quantum-chemistry Hamiltonians containing Coulomb potentials, and input states for which the Trotter error exhibits slower-than-standard scaling. Generic Coulomb-scaling c…

Let $n_s$ and $n_d$ denote the numbers of single and double excitations, respectively. For each doubly excited determinant $Phi_{ij}^{ab}$, let $k=k(i,j,a,b)$ be its canonically or…

For two rigid molecules with orientations and separation encoded by $\tau=(L,U,V)$, let $\mathcal{E}(\tau)$ be the Born–Oppenheimer ground-state energy. In the limit $L\to\infty$,…

In the neutral Born–Oppenheimer model, suppose a mountain-pass construction produces a transition state $(\tilde Y,\tilde\Psi)$ at the mountain-pass level $c$, with $\tilde Y$ a nu…

Let $N=|Z|$ be the neutral case, let $\bar Y_0$ and $\bar Y_1$ be two local minima of $Y\mapsto E_N(Y,Z)$, and define … Assume $c>\max\{E_N(\bar Y_0,Z),E_N(\bar Y_1,Z)\}$. Compactn…

Let $Z=(Z_1,Z_2)$ describe two rigid molecular subsystems and let $N$ electrons be distributed as $N_1+N_2=N$. In the dissociation limit, the energy is the minimum over all such di…

A natural orbital is an eigenstate of the one-body reduced density matrix, whose eigenvalues are the natural occupation numbers. Löwdin's conjecture. The natural orbitals would be…