non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?

We are considering a matrix A=(aij)i,j=1,…,d∈Rd×d with the following property: aii=−∑j≠iaij, i.e., the matrix is not only weak diagonal-dominant, but its rows also sum up to 0. Note that the matrix is not necessarily symmetric (otherwise positive semi-definiteness would follow immediately from the weak diagonal-dominance). Furthermore, the matrix has the property that there are no … Read more

Eigenspaces of the Laplace operator on a unit ball

I am interested in structures of the eigenspaces of the Laplace operator on the n-dimensional unit ball with Neumann or Dirichlet boundary conditions as representations of the special orthogonal group of dimension n (denoted by SO(n)). My question is: are all of these eigenspaces irreducible representations of SO(n)? If yes, how to prove it? Or … Read more

Gradient of spectral function on noncompact homogeneous space

Let (M,g) be a noncompact Riemannian manifold whose isometry group acts transitively on M, i.e. a (not necessarily normal) homogeneous space. Let eλ(x,y) be the integral kernel of f↦∫λ0dEν(f) where dEν is the spectral measure of the (non-negative) Laplacian associated to (M,g). Is there a relatively simple proof that ∫M|∇xeλ(x,y)|2gdy=∫Meλ(x,y)⋅Δxeλ(x,y)dy ? Note that the differentiation is … Read more

Laplace spectra of “half” grid graph

Let G=(E,V) be a simple graph. The graph Laplacian is given by L=D−A, where D is the degree matrix (diagonal matrix with entries corresponding to the degree of the vertex) and A the adjacency matrix. Denote with Gnm the n×m grid graph Example grid graph G66. We like to label the vertices row by row … Read more

Tight bound on spectral gap of compact homogeneous manifold?

This paper by Peter Li proves a bound on the spectral gap of the Laplacian on a compact homogeneous manifold of diameter d: λ1≥c/d2, where c=π2/4. Can this bound be strengthened in a large number of dimensions, or is it essentially tight? In other words, if we restrict to manifolds of dimension N what is … Read more

System of eigenvalues of partial Laplacians and the Jacquet Langlands correspondence

I’m looking for a precise reference. I could not dig it up in the original paper of Jacquet-Langlands. Let F be a totally real number field with [F:Q]=r, let B1=M2(F) be the split quaternion algebra and let B2/F be a non-split quaternion algebra. Choose orders O1⊆M2(OF) and O2=B2(OF). Assume that there exists coprime and square … Read more

Pointwise convergence of the eigenfunctions expansion of f(x)=1|x|f(x)=\frac{1}{|x|}

Let Ω⊂Rn a bounded domain with smooth boundary, 0<λ1≤λ2≤⋯≤λk≤… the Dirichlet eigenvalues and {wk}+∞k=1 an L2(Ω)-orthonormal basis made of eigenfunctions ordered by the corresponding eigenvalues. Actually, using rigged Hilbert spaces, {wk}+∞k=1 (up to renormalization) is an orthonormal basis in H1(Ω). Hence for any f∈H1(Ω) we have N∑k=1akwk(x)→f(x)inH1(Ω),whereak:=∫Ωwk(x)f(x)dx. Furthermore it is well-known that wk(x)∈C∞(ˉΩ) and is … Read more

Is Δϕ\Delta \phi monotone operator on H1(Rd)H^1(\mathbb{R}^d) for monotone ϕ\phi

Let H1(Rd) be the usual Sobolev space and let ϕ:R→R be a non decreasing Lipschitz function with ϕ(0)=0. Is the operator Δϕ on H1(Rd) monotone? i.e. Do we have ⟨Δϕ(u)−Δϕ(v),u−v⟩≥0( or ≤) for u,v∈H1(Rd)? This is easily verified for ϕ(x)=x, but I’m not able to conclude anything for general ϕ. ⟨Δϕ(u)−Δϕ(v),u−v⟩=−⟨∇ϕ(u)−∇ϕ(v),∇u−∇v⟩=−⟨ϕ′(u)∇u−ϕ′(v)∇v,∇u−∇v⟩ If ϕ′ is constant … Read more

Spectrum of Laplace-Beltrami with piecewise constant coefficients

By the Laplace-Beltrami with piecewise constant coefficients I means the operator −div(f∇.) in the 2-sphere. Where f is a piecewise constant function that takes two values 1 and a>0. The spectrum of such operator is known to be discret. Except for the case a=1, i can’t find any work related to the computation of the … Read more

Is the heat semigroup on a manifold the limit of the heat semigroups associated to a compact exhaustion?

Let M be a paracompact Riemannian manifold, and E→M a Hermitian vector bundle endowed with a Hermitian connection ∇. Write M as an exhaustion ⋃j≥0Uj with relatively compact open subsets with smooth boundaries. If L2(M,E) is the space of square-integrable sections in E, let L:L2(M,E)→L2(M,E) be the (densely defined) Friedrichs extension of ∇∗∇ (with ∇∗ … Read more