9 problems

Let $\widetilde{p}(t,x,y)$ be the heat kernel associated with the fractional-type Schrödinger operator with singular potential, and let $p(t,x,y)$ be the corresponding free heat ke…

Solenoid-limit conjecture. In some well-defined limit, largely independent of the details of the approximating potentials, the limiting situation should be described by a singular…

Failure conjecture. The family of approximating Hamiltonians will not yield the sought approximation when $a(\epsilon)\sim\epsilon^{\nu}$ with $\nu>1$. In this regime the three pot…

Marino's conjecture. Singularities and symmetries pair down the test function space, expanding the space of distributions to accommodate the variety of quantum fields observed in n…

Existence and uniqueness conjecture. There exists a unique solution to equation with the contact-line conditions and.

Let $u$ be the solution and let $E_2u$ denote the energy appearing in Proposition B.asymp. Let $\kappa$ be the parameter in the weighted boundary quantity, and let an appropriate f…

Let $B'$ be the tangential component of the reflected Brownian motion in the half-space $\mathbb{R}^N_+$, let $L^N_s$ denote its boundary local time, and let … For…

Singularity-threshold conjecture. The threshold value separating the existence and nonexistence of the runaway effect is $\alpha=2$.

Let $\Gamma\subset\mathbb{R}^3$ be a single smooth curve, and suppose a family of scaled potential wells in the normal plane has a zero-energy resonance, as in approximations of a…