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A popular approach to modeling bimolecular reactions between diffusing molecules is through the use of reactive boundary conditions. One common model is the Smoluchowski partial absorption condition, which uses a Robin boundary condition in…
We announce a new result which shows that under either Dirichlet, Neumann, or Robin boundary conditions, the corners in a planar domain are a spectral invariant of the Laplacian. For the case of polygonal domains, we show how a locality…
Consider a finite system of diffusing particles coupled through a reactive boundary. Each particle is reflected, but may react with the boundary according to a killing mechanism which depends on the current reactivity of the boundary and…
We study convergence modes as well as their respective rates for the resolvent difference of Robin and Dirichlet Laplacian on bounded smooth domains in the large coupling limit. Asymptotic expansions for the resolvent, the eigenprojections…
These lecture notes focus on the bound state sector of QCD. Motivated by data which suggests that the strong coupling \alpha_s(Q) freezes at low Q, and by similarities between the spectra of hadrons and atoms, I discuss if and how QCD bound…
In this article, we discuss the efficient ways of implementing the transparent boundary condition (TBC) and its various approximations for the free Schr\"{o}dinger equation on a hyperrectangular computational domain in $\field{R}^d$ with…
We study spectra of surface states in 2D topological insulators (TIs) based on HgTe/(Hg,Cd)Te quantum wells and 3D Bi$_2$Se$_3$-type compounds by constructing a class of feasible time-reversal invariant boundary conditions (BCs) for an…
We analyze an approximation of a Laplacian subject to non-local interface conditions of a $\delta'$-type by Neumann Laplacians on a family of Riemannian manifolds with a sieve-like structure. We establish a (kind of) resolvent convergence…
We consider the Laplacian on a class of smooth domains $\Omega\subset \mathbb{R}^{\nu}$, $\nu\ge 2$, with attractive Robin boundary conditions: \[ Q^\Omega_\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text{ on }…
A basic problem in operator theory is to estimate how a small perturbation effects the eigenspaces of a self-adjoint compact operator. In this paper, we prove upper bounds for the subspace distance, taylored for structured random…
We consider a model of leaky quantum wire in three dimensions. The Hamiltonian is a singular perturbation of the Laplacian supported by a line with the coupling which is bounded and periodically modulated along the line. We demonstrate that…
We consider small perturbations of the Laplace operator in a multi-dimensional cylindrical domain by second order differential operators with periodic coefficients. We show that under certain non-degeneracy conditions such perturbations can…
The sigma models on projective superspaces CP^{N+M-1|N} with topological angle theta=pi mod 2pi flow to non-unitary, logarithmic conformal field theories in the low-energy limit. In this paper, we determine the exact spectrum of these…
We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $\mathbb{R}^N$. We prove that the symmetric functions of the eigenvalues depend real analytically…
On a bounded Lipschitz domain we consider two selfadjoint operator realizations of the same second order elliptic differential expression subject to Robin boundary conditions, where the coefficients in the boundary conditions are functions.…
We give a variational proof of the existence of infinitely many bound states below the continuous spectrum for spin-orbit Hamiltonians (including the Rashba and Dresselhaus cases) perturbed by measure potentials thus extending the results…
We prove that a spectral gap-filling phenomenon occurs whenever a Hamiltonian operator encounters a coarse index obstruction upon compression to a domain with boundary. Furthermore, the gap-filling spectra contribute to quantised current…
In this expository article some spectral properties of self-adjoint differential operators are investigated. The main objective is to illustrate and (partly) review how one can construct domains or potentials such that the essential or…
We study the $L^2$ spectral gap of a large system of strongly coupled diffusions on unbounded state space and subject to a double-well potential. This system can be seen as a spatially discrete approximation of the stochastic Allen-Cahn…
In this paper we investigate the existence, uniqueness and stability of weak solutions of the initial boundary value problem with the Dirichlet boundary conditions for a parabolic equation with a drift $b\in L_2$. We prove $L_1$-stability…