17 problems

Let $c(\lambda,a)$ be the coefficient appearing in the leading terms of the distant perturbation asymptotics for window-coupled waveguides, where $a$ is the window parameter and…

Consider two waveguides of widths $d_1$ and $d_2$, let … and define the waveguide-asymmetry parameter … Let $E(\ell)$ denote the eigenvalue associated with a narrow window of width…

The nonlinear Schrödinger equation is considered on a waveguide manifold in the setting of Theorem, with nonlinearity exponent $\alpha$ and mass-critical endpoint…

Absence-of-trapped-modes conjecture. The absence of non-trivial square-integrable solutions, equivalently trapped modes, still holds for such junctions; the restriction to angles g…

Let $\Omega$ be the period-$1/2$ waveguide structure, and let $-\Delta$ denote its operator. Suppose the spectral properties listed in Assumption 0 hold, including analyticity and…

Let $Q_\ell\subset Q_\ell'$ be the cross-sections of a tube and its local enlargement, and let $\mathcal{A}_s^{Q'}$ be the corresponding restricted fractional Laplacian. The discre…

Let $\mathcal{A}^\Omega_s$ be the restricted fractional Laplacian of order $s$ in a tube $\Omega$, and let the threshold denote the bottom of its essential spectrum. A threshold is…

Duclos–Exner conjecture. The discrete spectrum of this operator is empty when the magnetic field is sufficiently strong and uniform.

For each $n\geq 2$, let $\lambda_\bullet(\Theta^n_2)$ be the eigenvalue below the continuous spectrum of the cross domain $\Theta^n_2$. The sequence…

Let $\Theta^n_1$ and $\Theta^n_2$ be the corner and cross domains in $\mathbb{R}^n$, respectively, and let $\lambda_\bullet(\Theta^n_j)$ denote an eigenvalue below the continuous s…

An open waveguide in a doubly periodic medium is considered, with embedded eigenvalues meaning eigenvalues lying in the continuous spectrum. Embedded-eigenvalue conjecture. Such an…

Let $\Omega$ be a globally twisted waveguide with twist perturbation $\varepsilon$ vanishing at infinity, and let $-\Delta_D^{\Omega}$ and $\lambda_1$ be as above. A Hardy inequali…

Let $\Omega$ be a globally twisted waveguide with twist derivative $\dot\theta=\beta+\varepsilon$, where $\beta$ is the constant twisting rate and $\varepsilon$ is a nonzero pertur…

Let $\mathcal{H}_\varepsilon$ be the waveguide operator with non-periodically alternating Dirichlet and Robin boundary conditions, and let its essential spectrum be decomposed into…

Threshold-eigenfunction projection conjecture. In this case the projection to the associated eigenfunction will be the leading term in the asymptotic expansion for the resolvent.

Let $S_\pi(t)$ be the heat semigroup for the twisted Dirichlet–Neumann waveguide, and let $\gamma_\pi$ denote the decay exponent in the estimate of Theorem. Decay-exponent conjectu…

Let $d>0$, let $\beta$ be a suitable boundary deformation, and let $unavailable?]$?