163 problems

Laptev–Safronov conjecture. The estimate should remain valid throughout the larger range $q\le d$, rather than only for $\frac d2<q\le\frac12(d+1)$.

Let … be the Schrödinger operator with potential $u(x)=ix^3$. Bessis–Zinn-Justin conjecture. The spectrum of $H$ is real. This conjecture concerns the spectral reality of a non-sel…

Let $(M,g)$ be a complete Riemannian manifold, let $\Delta$ be its negative definite Laplace–Beltrami operator, and let $L^2$ positivity preservation for $-\Delta+1$ mean that … fo…

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…

Let $\Gamma$ be the orbifold fundamental group of a compact good orbifold, acting on its universal orbifold cover $\widetilde{M}$, and let $H_V=\Delta+V$ be a $\Gamma$-invariant Sc…

Turbiner's second conjecture. If the symbol of $H$ engenders a Euclidean geometry, then the spectral equation $H\psi=E\psi$ can be solved by separation of variables. The paper stat…

Fundamental gap conjecture. The fundamental gap is at least $3\pi^2/D^2$.

Last's conjecture. In general, $\dim_B^+\Sigma_\lambda$ does not bound $\tilde\alpha^\pm_u(\lambda)$ from above.

Let $\gamma$ be a closed curve in $\mathbb{R}^3$ of length $L$, with curvature $\kappa$. Let $\mu_1$ denote the first eigenvalue of the periodic Sturm–Liouville operator … on the i…

Conditional-basis conjecture. For nonreal coupling constant, the eigenfunctions of the harmonic oscillator do not form a conditional basis.

Let $Q(x)<0$ on some interval $(0,x_0)$. Suppose that $Q(x)$ changes sign one or more times as $x$ increases, eventually decays to zero as $x\rightarrow\infty$, and satisfies…

For a one-dimensional Schrödinger operator, consider the claim that the absolutely continuous spectrum is preserved in general under the relevant perturbations, with the perturbed…

Simon's conjecture. The absolutely continuous spectrum of $H_V$ is the positive real axis:

A quasi-exactly-solvable or exactly-solvable problem in $\mathbb{R}^2$ is one containing the Laplace–Beltrami operator with a flat-space metric tensor; its variables are non-separa…

Barry Simon's conjecture. The absolutely continuous spectrum of $H_V$ should fill the entire positive half-axis, that is, it should contain all of $(0,\infty)$.

Let $\gamma$ be a smooth closed curve of length $2\pi$ in $\mathbb{R}^3$, parametrized by arclength $s$, and let $\kappa(s)$ be its curvature. Define the one-dimensional Schrödinge…

Perturbed Fourier expansion conjecture. An $L^2$ function admits a similar expansion in terms of “eigenfunctions” of $H$.

Let $H=-\Delta+V$ be the Schrödinger operator with the barrier potential $V$ defined in the paper, and let $H_0=-\Delta$ be the free Schrödinger operator. For parameters $p$ and…

Let $V\in L^2(\mathbb R^3)$ be a potential. Ionescu–Jerison's conjecture. Their main result, asserting that if $u\in W^{1,2}_{\rm loc}(\mathbb R^3)$ satisfies…

Generalized Lamé operators conjecture. The generalized Lamé operators are all algebraically integrable.

Let $M$ be a Riemannian manifold, let $E$ be a Hermitian vector bundle over $M$, and let $V$ be a potential satisfying Assumption A: $V=V_{+}+V_{-}$, with $V_{+}(x)8e0$ and…

Coulomb-rate absence conjecture. Singular continuous spectrum cannot occur for such $V$.

Let $H_0=-\Delta+q(x)$ be a Schrödinger operator with periodic potential $q$, let $F_{\,\lambda}(q)$ be its Fermi variety, and let the Condition mean that, for every $\lambda$ in t…

Let $q(z)=z^4+a^4$. Consider the eigenfunctions of the associated one-dimensional Schrödinger operator. Titchmarsh's conjecture. All non-real zeros of these eigenfunctions lie on t…

Consider a planar embedding of a periodic graph with equal angles between the tangent lines of the edges meeting at each vertex, and let $\Omega_\varepsilon$ be the associated fatt…