Related papers: Quantum singular operator limits of thin Dirichlet…
This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms, whose coefficients lie in critical…
The scaling limit and Schauder bounds are derived for a singular integral operator arising from a difference equation approach to monodromy problems.
In this paper, we study solutions $u$ of parabolic systems in divergence form with zero Dirichlet boundary conditions in the upper-half cylinder $Q_1^+\subset \mathbb{R}^{n+1}$, where the coefficients are weighted by $x_n^\alpha$,…
We study elliptic and parabolic problems governed by the singular elliptic operators $$ \mathcal L=y^{\alpha_1}\mbox{Tr }\left(QD^2_x\right)+2y^{\frac{\alpha_1+\alpha_2}{2}}q\cdot \nabla_xD_y+\gamma y^{\alpha_2}…
We consider an inverse spectral problem on a quantum graph associated with the square lattice. Assuming that the potentials on the edges are compactly supported and symmetric, we show that the Dirichlet-to-Neumann map for a boundary value…
This paper deals with the variational analysis of topological singularities in two dimensions. We consider two canonical zero-temperature models: the core radius approach and the Ginzburg-Landau energy. Denoting by $\varepsilon$ the length…
Let $T_\Omega$ be the singular integral operator with variable kernel $\Omega(x,z)$. In this paper, by using the atomic decomposition theory of weighted weak Hardy spaces, we will obtain the boundedness properties of $T_\Omega$ on these…
We consider the problems of extreming the first eigenvalue and the fundamental gap of a sub-elliptic operator with Dirichlet boundary condition, when the potential $V$ is subjected to a $p$-norm constraint. The existence results for weak…
The Schwinger model is used to study the artifacts of quenching in a controlled way. The model is solved on a finite-temperature cylinder of circumference $\beta=1/T$ with bag-inspired local boundary conditions at the two ends $x^1=0$ and…
We consider compact Hankel operators realized in $\ell^2(\mathbb Z_+)$ as infinite matrices $\Gamma$ with matrix elements $h(j+k)$. Roughly speaking, we show that, for all $\alpha>0$, the singular values $s_{n}$ of $\Gamma$ satisfy the…
We address the problem on the right definition of the Schroedinger operator with potential $\delta'$, where $\delta$ is the Dirac delta-function. Namely, we prove the uniform resolvent convergence of a family of Schroedinger operators with…
This chapter deals with the notion of the resolvent of a self-adjoint operator. We pay special attention to the convergence of unbounded self-adjoint operators in several resolvent senses, and how they are related to the convergence of…
A mixed Dirichlet-Neumann problem is regularized with a family of singularly perturbed Neumann-Robin boundary problems, parametrized by $\varepsilon > 0$. Using an asymptotic development by Gamma-convergence, the asymptotic behavior of the…
In this note, uniform bounds of the Birman-Schwinger operators in the discrete setting are studied. For uniformly decaying potentials, we obtain the same bound as in the continuous setting. However, for non-uniformly decaying potential, our…
We establish a $\Gamma$-convergence result for $h\to 0$ of a thin nonlinearly elastic 3D-plate of thickness $h>0$ which is assumed to be glued to a support region in the 2D-plane $x_3=0$ over the $h$-2D-neighborhood of a given closed set…
The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in $\Omega := \mathbb R^n \setminus \mathbb R^d$…
The determination of the spectrum of a Schr\"odinger operator is a fundamental problem in mathematical quantum mechanics. We discuss a series of results showing that Schr\"odinger operators can exhibit spectra that are remarkably thin in…
We consider inverse boundary value problems for elliptic equations of second order of determining coefficients by Dirichlet-to-Neumann map on subboundaries, that is, the mapping from Dirichlet data supported on $\partial\Omega\setminus…
We establish the $\Gamma$-convergence of some energy functionals describing nonlocal attractive interactions in bounded domains. The interaction potential solves an elliptic equation (local or nonlocal) in the bounded domain and the primary…
It is well known that a potential $q$ of the Sturm-Liouville operator $Ly= -y" +q(x)y$ on the finite interval $[0, \pi]$ can be uniquely recovered by the spectrum $\{\lambda_k\}_1^\infty$ and norming constants $\{\alpha_k\}_1^\infty$ of…