Spectral Theory

For a bounded corner domain $\Omega$, we consider the Robin Laplacian in $\Omega$ with large Robin parameter. Exploiting multiscale analysis and a recursive procedure, we have a precise description of the mechanism giving the ground state…

In this paper we discuss the problem of decomposition for unbounded $2\times 2$ operator matrices by a pair of complementary invariant graph subspaces. Under mild additional assumptions, we show that such a pair of subspaces decomposes the…

We consider a family $\{\Omega^\varepsilon\}_{\varepsilon>0}$ of periodic domains in $\mathbb{R}^2$ with waveguide geometry and analyse spectral properties of the Neumann Laplacian $-\Delta_{\Omega^\varepsilon}$ on $\Omega^\varepsilon$. The…

We establish an isoperimetric inequality for the fundamental tone (first nonzero eigenvalue) of the free plate of a given area, proving the ball is maximal. Given $\tau>0$, the free plate eigenvalues $\omega$ and eigenfunctions $u$ are…

It has been shown recently that spectral flow admits a natural integer-valued extension to essential spectrum. This extension admits four different interpretations; two of them are singular spectral shift function and total resonance index.…

We establish a partial generalization of a prior isoperimetric inequality for the fundamental tone (first nonzero eigenvalue) of the free plate to plates of nonzero Poisson's ratio.

We provide explicit upper bounds for the eigenvalues of the Laplacian on a finite metric tree subject to standard vertex conditions. The results include estimates depending on the average length of the edges or the diameter. In particular,…

The variation of spectral subspaces for linear self-adjoint operators under an additive bounded off-diagonal perturbation is studied. To this end, the optimization approach for general perturbations in [J. Anal. Math., to appear;…

In this paper, we study the operator equation $AB=\lambda BA$ for a bounded operator $A,B$ on a complex Hilbert space. We focus on algebraic relations between different operators that include normal, $M$-hyponormal, quasi $*$-paranormal and…

We establish a sharp estimate on the size of the spectral clusters of the Landau Hamiltonian with $L^p$ potentials in two dimensions as the cluster index tends to infinity. In three dimensions, we prove a new limiting absorption principle…

Canonical Polyadic Decomposition (CPD) of a third-order tensor is a minimal decomposition into a sum of rank-$1$ tensors. We find new mild deterministic conditions for the uniqueness of individual rank-$1$ tensors in CPD and present an…

We provide a quantitative version of the isoperimetric inequality for the fundamental tone of a biharmonic Neumann problem. Such an inequality has been recently established by Chasman adapting Weinberger's argument for the corresponding…

We prove that heat content determines planar triangles.

We study resolvents and spectral projections for quadratic differential operators under an assumption of partial ellipticity. We establish exponential-type resolvent bounds for these operators, including Kramers-Fokker-Planck operators with…

We present asymptotically sharp inequalities for the eigenvalues $\mu_k$ of the Laplacian on a domain with Neumann boundary conditions, using the averaged variational principle introduced in \cite{HaSt14}. For the Riesz mean $R_1(z)$ of the…

We recall criteria on the spectrum of Jacobi matrices such that the corresponding isospectral torus consists of periodic operators. Motivated by those known results for Jacobi matrices, we define a new class of operators called GMP…

We show that spectral problems for periodic operators on lattices with embedded defects of lower dimensions can be solved with the help of matrix-valued integral continued fractions. While these continued fractions are usual in the…

Let $\mu$ be a probability measure with an infinite compact support on $\mathbb{R}$. Let us further assume that $(F_n)_{n=1}^\infty$ is a sequence of orthogonal polynomials for $\mu$ where $(f_n)_{n=1}^\infty$ is a sequence of nonlinear…

Let $K(\gamma)$ be the weakly equilibrium Cantor type set introduced in [10]. It is proven that the monic orthogonal polynomials $Q_{2^s}$ with respect to the equilibrium measure of $K(\gamma)$ coincide with the Chebyshev polynomials of the…

We study the one-dimensional Schr\"odinger operators $$ S(q)u:=-u"+q(x)u,\quad u\in \mathrm{Dom}\left(S(q)\right), $$ with $1$-periodic real-valued singular potentials $q(x)\in H_{\operatorname{per}}^{-1}(\mathbb{R},\mathbb{R})$ on the…