Bottom Of Spectrum Research Articles

Normal Covering Spaces with Maximal Bottom of Spectrum

We study the property of spectral-tightness of Riemannian manifolds, which means that the bottom of the spectrum of the Laplacian separates the universal covering space from any other normal covering space of a Riemannian manifold. We prove that spectral-tightness of a closed Riemannian manifold is a topological property characterized by its fundamental group. As an application, we show that a non-positively curved, closed Riemannian manifold is spectrally-tight if and only if the dimension of its Euclidean local de Rham factor is zero. In their general form, our results extend the state of the art results on the bottom of the spectrum under Riemannian coverings.

The spectrum of the Laplacian and volume growth of proper minimal submanifolds

We give upper bounds for the bottom of the essential spectrum of properly immersed minimal submanifolds of \(\mathbb {R}^{n}\) in terms of their volume growth. Our result can be viewed as an extrinsic version of Brooks’s essential spectrum estimate (Brooks, Math Z 178(4): 501–508, 1981, Thm. 1) and it gives a fairly general answer to a question of Yau (Asian J Math 4(1): 235–278, 2000) about upper bounds for the first eigenvalue (bottom of the spectrum) of immersed minimal surfaces of \(\mathbb {R}^{3}.\)

Spectral Theory of Infinite Quantum Graphs

We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney-type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.

Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry

Li and Wang [J Differ Geom 58:501–534 (2001), J Differ Geom 62:143–162 (2002)] proved a splitting theorem for an n-dimensional Riemannian manifold with $$Ric \ge -(n-1)$$ and the bottom of spectrum $$\lambda _0(M)=\frac{(n-1)^2}{4}$$ . For an n-dimensional compact manifold M with $$Ric\ge -(n-1)$$ with the volume entropy $$h(M)=n-1$$ , Ledrappier and Wang (J Differ Geom 85:461–477, 2010) proved that the universal cover $$\widetilde{M}$$ is isometric to the hyperbolic space $$\mathbb {H}^n$$ . We will prove analogue theorems for Alexandrov spaces.

Surface Localization in Impurity Band with Random Displacements and Long-Range Interactions

We study the random Schrödinger operators in a Euclidean space with the disorder generated by two complementary mechanisms: random substitution in a lower-dimensional layer and random displacements in the bulk, without additional assumptions regarding the reflection symmetry of the site potentials. The latter are assumed to be bounded and have a power-law decay. Complementing earlier results obtained in the strong disorder regime, we establish spectral and strong dynamical localization in the impurity zone near the bottom of spectrum for arbitrarily weak amplitudes of the random displacements, provided the concentration of impurities is sufficiently small.

Ground state and orbital stability for the NLS equation on a general starlike graph with potentials

We consider a nonlinear Schrödinger equation (NLS) posed on a graph (or network) composed of a generic compact part to which a finite number of half-lines are attached. We call this structure a starlike graph. At the vertices of the graph interactions of δ-type can be present and an overall external potential is admitted. Under general assumptions on the potential, we prove that the NLS is globally well-posed in the energy domain. We are interested in minimizing the energy of the system on the manifold of constant mass (L2-norm). When existing, the minimizer is called ground state and it is the profile of an orbitally stable standing wave for the NLS evolution. We prove that a ground state exists for sufficiently small masses whenever the quadratic part of the energy admits a simple isolated eigenvalue at the bottom of the spectrum (the linear ground state). This is a wide generalization of a result previously obtained for a star-graph with a single vertex. The main part of the proof is devoted to prove the concentration compactness principle for starlike structures; this is non trivial due to the lack of translation invariance of the domain. Then we show that a minimizing, bounded, H1 sequence for the constrained NLS energy with external linear potentials is in fact convergent if its mass is small enough. Moreover we show that the ground state bifurcates from the vanishing solution at the bottom of the linear spectrum. Examples are provided with a discussion of the hypotheses on the linear part.