arXiv subjects

Joachim Stubbe

Publications and source records attributed to Joachim Stubbe.

17 recordsLinked to original sources

arXiv2026

Sum rules and a second order Feynman-Hellman theorem for abstract operators with applications

We discuss the role of the Feynman-Hellmann theorem for abstract one-parameter families of Hamiltonians in sum rules and trace identities of Harrell and the author and its application to spectral theory. In particular, we derive a sum rule for the second derivative of eigenvalues of a one-parameter family of Hamiltonians extending thereby concepts of second order perturbation theory. We present applications to semiclassical eigenvalue bounds of Schrodinger operators as Lieb-Thirring inequalities, zeros of Bessel functions, eigenvalue inequalities for sums of matrices and trace inequalities.

math.SP↗

arXiv2025

Semiclassical eigenvalue bounds for compact homogeneous irreducible Riemannian manifolds

We exploit an identity for the gradients of Laplacian eigenfunctions on compact homogeneous Riemannian manifolds with irreducible linear isotropy group to obtain asymptotically sharp universal eigenvalue inequalities and sharp Weyl bounds on Riesz means. The approach is non variational and is based on identities for spectral quantities in the form of sum rules.

math.SP↗

arXiv2023

Semiclassical estimates for eigenvalue means of Laplacians on spheres

We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means we prove upper and lower bounds involving asymptotically sharp shift terms, and we extend them to domains of $\mathbb S^d$. We also prove a Berezin-Li-Yau inequality for domains contained in the hemisphere $\mathbb S^2_+$. Moreover, we consider polyharmonic operators for which we prove analogous results that highlight the role of dimension for Pólya-type inequalities. Finally, we provide sum rules for Laplacian eigenvalues on spheres and compact two-point homogeneous spaces.

math.SP↗

arXiv2022

The first Grushin eigenvalue on cartesian product domains

In this paper we consider the first eigenvalue $λ_1(Ω)$ of the Grushin operator $Δ_G:=Δ_{x_1}+|x_1|^{2s}Δ_{x_2}$ with Dirichlet boundary conditions on a bounded domain $Ω$ of $\mathbb{R}^d= \mathbb{R}^{d_1+d_2}$. We prove that $λ_1(Ω)$ admits a unique minimizer in the class of domains with prescribed finite volume which are the cartesian product of a set in $\mathbb{R}^{d_1}$ and a set in $\mathbb{R}^{d_2}$, and that the minimizer is the product of two balls $Ω^*_1 \subseteq \mathbb{R}^{d_1}$ and $Ω_2^* \subseteq \mathbb{R}^{d_2}$. Moreover, we provide a lower bound for $|Ω^*_1|$ and for $λ_1(Ω_1^*\timesΩ_2^*)$. Finally, we consider the limiting problem as $s$ tends to $0$ and to $+\infty$.

math.AP↗

arXiv2021

Semiclassical bounds for spectra of biharmonic operators

We provide complementary semiclassical bounds for the Riesz means $R_1(z)$ of the eigenvalues of various biharmonic operators, with a second term in the expected power of $z$. The method we discuss makes use of the averaged variational principle (AVP), and yields two-sided bounds for individual eigenvalues, which are semiclassically sharp. The AVP also yields comparisons with Riesz means of different operators, in particular Laplacians.

math.SP↗

arXiv2021

On the spectral asymptotics for the buckling problem

We provide a direct proof of Weyl's law for the buckling eigenvalues of the biharmonic operator on a wide class of domains of $\mathbb R^d$ including bounded Lipschitz domains. The proof relies on asymptotically sharp lower and upper bounds that we develop for the Riesz mean $R_2(z)$. Lower bounds are obtained by making use of the so-called "averaged variational principle". Upper bounds are obtained in the spirit of Berezin-Li-Yau. Moreover, we state a conjecture for the second term in Weyl's law and prove its correctness in two special cases: balls in $\mathbb R^d$ and bounded intervals in $\mathbb R$.

math.SP↗

arXiv2019

Complementary asymptotically sharp estimates for eigenvalue means of Laplacians

We present asymptotically sharp inequalities, containing a second term, for the Dirichlet and Neumann eigenvalues of the Laplacian on a domain, which are complementary to the familiar Berezin-Li-Yau and Kröger inequalities in the limit as the eigenvalues tend to infinity. We accomplish this in the framework of the Riesz mean $R_1(z)$ of the eigenvalues by applying the averaged variational principle with families of test functions that have been corrected for boundary behaviour.

math.SP↗

arXiv2016

Weyl-type bounds for Steklov eigenvalues

We present upper and lower bounds for Steklov eigenvalues for domains in $\mathbb{R}^{N+1}$ with $C^2$ boundary compatible with the Weyl asymptotics. In particular, we obtain sharp upper bounds on Riesz-means and the trace of corresponding Steklov heat kernel. The key result is a comparison of Steklov eigenvalues and Laplacian eigenvalues on the boundary of the domain by applying Pohozaev-type identities on an appropriate tubular neigborhood of the boundary and the min-max principle. Asymptotically sharp bounds then follow from bounds for Riesz-means of Laplacian eigenvalues.

math.SP↗

arXiv2016

Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian

We present asymptotically sharp inequalities for the eigenvalues $μ_k$ 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 eigenvalues we improve the known sharp semiclassical bound in terms of the volume of the domain with a second term with the best possible expected power of $z$. In addition, we obtain two-sided bounds for individual $μ_k$, which are semiclassically sharp. In a final section, we remark upon the Dirichlet case with the same methods.

math.SP↗

arXiv2015

On sums of eigenvalues of elliptic operators on manifolds

We use the averaged variational principle introduced in a recent article on graph spectra [7] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of Kr{ö}ger 's bound for Neumann spectra of Laplacians on Euclidean domains [12]. Among the operators we consider are the Laplace-Beltrami operator on compact subdomains of manifolds. These estimates become more explicit and asymptotically sharp when the manifold is conformal to homogeneous spaces (here extending a result of Strichartz [21] with a simplified proof). In addition we obtain results for the Witten Laplacian on the same sorts of domains and for Schr{ö}dinger operators with confining potentials on infinite Euclidean domains. Our bounds have the sharp asymptotic form expected from the Weyl law or classical phase-space analysis. Similarly sharp bounds for the trace of the heat kernel follow as corollaries.

math.MG↗

arXiv2013

On sums of graph eigenvalues

We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both the standard combinatorial Laplacian and the renormalized Laplacian. We also provide upper bounds for sums of squares of eigenvalues of these three matrices. Among our results, we generalize an inequality of Fiedler for the extreme eigenvalues of the graph Laplacian to a bound on the sums of the smallest (or largest) k such eigenvalues, k < n. Furthermore, if lambda_j are the eigenvalues of the positive graph Laplacian H, in increasing order, on a finite graph with |V| vertices and |E| edges which is isomorphic to a subgraph of the ν-dimensional infinite cubic lattice, then the spectral sums obey a Weyl-type upper bound, a simplification of which reads $$ \sum_{j=1}^{k-1}{λ_j} \le \frac{π^2 |\mathcal{E}|}{3} \left(\frac{k}{|\mathcal{V}|} \right)^{1+\frac{2}ν} $$ for each k < |V|. This and related estimates for sums of lambda_j^2 provide a family of necessary conditions for the embeddability of the graph in a lattice of dimension νor less.

math.SP↗

arXiv2008

Bounds states of the Schrödinger-Newton model in low dimensions

We prove the existence of quasi-stationary symmetric solutions with exactly n>=0 zeros and uniqueness for n=0 for the Schrödinger-Newton model in one dimension and in two dimensions along with an angular momentum m>=0. Our result is based on an analysis of the corresponding system of second-order differential equations.

math-ph↗

arXiv2008

Universal Monotonicity of Eigenvalue Moments and Sharp Lieb-Thirring Inequalities

We show that phase space bounds on the eigenvalues of Schrödinger operators can be derived from universal bounds recently obtained by E. M. Harrell and the author via a monotonicity property with respect to coupling constants. In particular, we provide a new proof of sharp Lieb-Thirring inequalities.

math-ph↗

arXiv2008

Universal bounds and semiclassical estimates for eigenvalues of abstract Schroedinger operators

We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues lambda_k that are analogous to those known for Schroedinger operators and the Dirichlet Laplacian, on which the operators of interest are modeled. In addition we produce inequalities that are new even in the model case. These include a family of differential inequalities for generalized Riesz means and theorems stating that arithmetic means of lambda_k^p for p <= 3 are universally bounded from above by multiples of the geometric mean of the lambda_k. For Schroedinger operators and the Dirichlet Laplacian these bounds are Weyl-sharp, i.e., saturated by the standard semiclassical estimates for lambda_k at large k.

math.SP↗

arXiv2008

Bound states of two-dimensional Schrödinger-Newton equations

We prove an existence and uniqueness result for ground states and for purely angular excitations of two-dimensional Schrödinger-Newton equations. From the minimization problem for ground states we obtain a sharp version of a logarithmic Hardy-Littlewood-Sobolev type inequality.

math-ph↗

arXiv2008

Stationary solutions of the Schrödinger-Newton model - An ODE approach

We prove the existence and uniqueness of stationary spherically symmetric positive solutions for the Schrödinger-Newton model in any space dimension $d$. Our result is based on an analysis of the corresponding system of second order differential equations. It turns out that $d=6$ is critical for the existence of finite energy solutions and the equations for positive spherically symmetric solutions reduce to a Lane-Emden equation for all $d\geq 6$. Our result implies in particular the existence of stationary solutions for two-dimensional self-gravitating particles and closes the gap between the variational proofs in $d=1$ and $d=3$.

math-ph↗

arXiv1995

Spacings of Quarkonium Levels with the Same Principal Quantum Number

The spacings between bound-state levels of the Schrödinger equation with the same principal quantum number $N$ but orbital angular momenta $\ell$ differing by unity are found to be nearly equal for a wide range of power potentials $V = λr^ν$, with $E_{N \ell} \approx F(ν, N) - G(ν,N) \ell$. Semiclassical approximations are in accord with this behavior. The result is applied to estimates of masses for quarkonium levels which have not yet been observed, including the 2P $c \bar c$ states and the 1D $b \bar b$ states.

hep-ph↗