Related papers: Critical metrics of eigenvalue functionals via Cla…
In this paper, we investigate critical points of the Laplacian's eigenvalues considered as functionals on the space of Riemmannian metrics or a conformal class of metrics on a compact manifold. We obtain necessary and sufficient conditions…
We construct a Riemannian metric on the $ 2 $-dimensional torus, such that for infinitely many eigenvalues of the Laplace-Beltrami operator, a corresponding eigenfunction has infinitely many isolated critical points. A minor modification of…
Consider the first nontrivial eigenvalue of the Laplacian on a closed surface as a functional on the space of Riemannian metrics of unit area. N. Nadirashvili has discovered a remarkable connection between critical points of this functional…
We consider critical points of a class of functionals on compact four-dimensional manifolds arising from Regularized Determinants for conformally covariant operators, whose explicit form was derived in [10], extending Polyakov's formula.…
Let $M$ be a compact Riemannian manifold with or without boundary, and let $-\Delta $ be its Laplace-Beltrami operator. For any bounded scalar potential $q$, we denote by $\lambda\_i(q)$ the $i$-th eigenvalue of the Schr\"{o}dinger type…
It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In…
We define a number of natural (from geometric and combinatorial points of view) deformation spaces of valuations on finite graphs, and study functions over these deformation spaces. These functions include both direct metric invariants…
Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite…
Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures - symplectic,…
Critical quantum metrology relies on the extreme sensitivity of a system's eigenstates near the critical point of a quantum phase transition to Hamiltonian perturbations. This means that these eigenstates are extremely sensitive to all the…
We consider a compact Riemannian manifold with boundary and a metric that is singular at the boundary. The associated Laplace-Beltrami operator is of the form of a Grushin operator plus a singular potential. In a supercritical parameter…
We study the eigenvalues of the Kohn Laplacian on a closed embedded strictly pseudoconvex CR manifold as functionals on the set of positive oriented pseudohermitian structures $\mathcal{P}_{+}$. We show that the functionals are continuous…
We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with a Kirchhoff-type coefficient and involving a critical nonlinearity term in the sense of Sobolev embeddings. Our approach is of variational and…
We introduce a general method for transforming the equations of motion following from a Das-Jevicki-Sakita Hamiltonian, with boundary conditions, into a boundary value problem in one-dimensional quantum mechanics. For the particular case of…
We study critical metrics of higher-order curvature functionals on compact Riemannian $n$-manifolds $(M,g)$. For an integer $k$ with $2 \leq 2k \leq n$, let $R^k$ denote the $k$-th exterior power of the Riemann curvature tensor. We…
On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of k distinct sets. We show that the k-th eigenvalues of the metric Laplacian gives exponential…
We consider a compact Riemannian manifold with boundary with a certain class of critical singular Riemannian metrics that are singular at the boundary. The corresponding Laplace-Beltrami operator can be seen as a Grushin-type operator plus…
The theory of finite-rank perturbations allows for the determination of spectral information for broad classes of operators using the tools of analytic function theory. In this work, finite-rank perturbations are applied to powers of the…
We review a recent new approach to the study of critical points of Laplacian eigenfunctions. Its core novelty is a non-standard variational principle for the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on…
On finite metric graphs we consider Laplace operators, subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices. We investigate spectral properties, existence of a Riesz basis of projectors and similarity…