Our main result is that if a generic convex domain in $\R^n$ collapses to a domain in $\R^{n-1}$, then the difference between the first two Dirichlet eigenvalues of the Euclidean Laplacian, known as the fundamental gap, diverges. The boundary of the domain need not be smooth, merely Lipschitz continuous. To motivate the general case, we first prove the analogous result for triangular and polygonal domains. In so doing, we prove that the first two eigenvalues of triangular domains cannot be polyhomogeneous on the moduli space of triangles without blowing up a certain point. Our results show that the gap generically diverges under one dimensional collapse and is bounded only if the domain is sufficiently close to a rectangle in two dimensions or a cylinder in higher dimensions.
By introducing a weight function to the Laplace operator, Bakry and Émery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-Émery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $\R^n$ and Neumann eigenvalues of domains in $\R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to gen…
We study the behavior of the spectrum of the Dirac operator together with a symmetric $W^{1, \infty}$-potential on spin manifolds under a collapse of codimension one with bounded sectional curvature and diameter. If there is an induced spin structure on the limit space $N$ then there are convergent eigenvalues which converge to the spectrum of a first order differential operator $D$ on $N$ together with a symmetric $W^{1,\infty}$-potential. If $N$ is orientable and the dimension of the limit space is even then $D$ is the Dirac operator $D^N$ on $N$ and if the dimension of the limit space is od…
We show that for compact Riemannian manifolds of dimension at least $3$ with nonempty boundary, we can modify the manifold by performing surgeries of codimension $2$ or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimensions $3$ and higher. Our result generalizes the 1-dimensional surgery in [FS2] to higher dimensional surgeries and to higher eigenvalues. It is proved in [FS2] that the unit ball does not maximize the first no…
We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on the fact that Hodge Laplacian restricted to the subspace of co-exact two-forms is minus the square of the Beltrami operator, a first-order operator. We prove that for generic metrics the spectrum of the Beltrami operator is simple. Because the Beltrami operator in this setting is …
We consider Robin Laplace operators on a class of two-dimensional domains with cusps. Our main results include the formula for the asymptotic distribution of the eigenvalues of such operators. In particular, we show how the eigenvalue asymptotic depends on the geometry of the cusp and on the boundary conditions.
The paper presents estimates for the number of negative eigenvalues of a two-dimensional Schrödinger operator in terms of $L\log L$ type Orlicz norms of the potential and proves a conjecture by N.N. Khuri, A. Martin and T.T. Wu.
We analyze eigenvalues emerging from thresholds of the essential spectrum of one-dimensional Dirac operators perturbed by complex and non-symmetric potentials. In the general non-self-adjoint setting we establish the existence and asymptotics of weakly coupled eigenvalues and Lieb-Thirring inequalities. As physical applications we investigate the damped wave equation and armchair graphene nanoribbons.
It is natural to consider continuous dependence of the $n$-th eigenvalue on $d$-dimensional ($d\geq2$) Sturm-Liouville problems after the results on $1$-dimensional case by Kong, Wu and Zettl [14]. In this paper, we find all the boundary conditions such that the $n$-th eigenvalue is not continuous, and give complete characterization of asymptotic behavior of the $n$-th eigenvalue. This renders a precise description of the jump phenomena of the $n$-th eigenvalue near such a boundary condition. Furthermore, we divide the space of boundary conditions into $2d+1$ layers and show that the $n$-th ei…
In this paper, we introduce a new family of functions to construct Schrödinger operators with embedded eigenvalues. This particularly allows us to construct discrete Schrödinger operators with arbitrary prescribed sets of eigenvalues.
We revisit an archive submission by P. B. Denton, S. J. Parke, T. Tao, and X. Zhang, arXiv:1908.03795, on $n \times n$ self-adjoint matrices from the point of view of self-adjoint Dirichlet Schrödinger operators on a compact interval.
The celebrated Cwikel-Lieb_Rozenblum inequality gives an upper estimate for the number of negative eigenvalues of Schroedinger operators in dimension three and higher. The situation is much more difficult in the two dimensional case. There has been significant progress in obtaining upper estimates for the number of negative eigenvalues of two dimensional Schroedinger operators on the whole plane. In this thesis, we present estimates of the Cwikel-Lieb_Rozenblum type for the number of eigenvalues (counted with multiplicities) of two dimensional Schroedinger operators lying below the essential s…
We present upper estimates for the number of negative eigenvalues of two-dimensional Schroedinger operators with potentials generated by Ahlfors regular measures of arbitrary dimension $α\in (0, 2]$.The estimates are given in terms of the integrals of the potential with a logarithmic weight and of its L $\log$ L type Orlicz norms. In the case $α= 1$, our estimates are stronger than the known ones about Schroedinger operators with potentials supported by Lipschitz curves.
We consider Dirichlet Laplacian in a thin curved three-dimensional rod. The rod is finite. Its cross-section is constant and small, and rotates along the reference curve in an arbitrary way. We find a two-parametric set of the eigenvalues of such operator and construct their complete asymptotic expansions. We show that this two-parametric set contains any prescribed number of the first eigenvalues of the considered operator. We obtain the complete asymptotic expansions for the eigenfunctions associated with these first eigenvalues.
We prove dynamical upper bounds for discrete one-dimensional Schroedinger operators in terms of various spacing properties of the eigenvalues of finite volume approximations. We demonstrate the applicability of our approach by a study of the Fibonacci Hamiltonian.
We prove that the number of negative eigenvalues of two-dimensional magnetic Schroedinger operators is bounded from above by the strength of the corresponding electric potential. Such estimates fail in the absence of a magnetic field. We also show how the corresponding upper bounds depend on the properties of the magnetic field and discuss their connection with Hardy-type inequalities.
We study one-dimensional Schrödinger operators with complex measures as potentials and present an improved criterion for absence of eigenvalues which involves a weak local periodicity condition. The criterion leads to sharp quantitative bounds on the eigenvalues. We apply our result to quasiperiodic measures as potentials.
We study the Schroedinger operator with a constant magnetic field in the exterior of a compact domain in $\mathbb{R}^{2d}$, $d\geq 1$. The spectrum of this operator consists of clusters of eigenvalues around the Landau levels. We give asymptotic formulas for the rate of accumulation of eigenvalues in these clusters. When the compact is a Reinhart domain we are able to show a more precise asymptotic formula.
A generalized two-dimensional periodic Dirac operator is considered, with $L^{\infty}$-matrix-valued coefficients of the first order derivatives and with complex matrix-valued potential. It is proved that if the matrix-valued potential has zero bound relative to the free Dirac operator, then the spectrum of the operator in question contains no eigenvalues.
In this paper, we determine, in the case of the Laplacian on the flat two-dimensional torus (R/Z) 2 , all the eigenvalues having an eigenfunction which satisfies Courant's theorem with equality (Courant-sharp situation). Following the strategy o A. Pleijel (1956), the proof is a combination of a lower bound a la Weyl) of the counting function, with an explicit remainder term, and of a Faber--Krahn inequality for domains on the torus (deduced as in B{é}rard-Meyer from an isoperimetric inequality), with an explicit upper bound on the area.
A fundamental result of Solomyak says that the number of negative eigenvalues of a Schrödinger operator on a two-dimensional domain is bounded from above by a constant times a certain Orlicz norm of the potential. Here we show that in the case of Dirichlet boundary conditions the constant in this bound can be chosen independently of the domain.
I present an example of a discrete Schr"odinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than exponentially. This settles a conjecture of Simon (in the negative). The potential is of von Neumann-Wigner type, with careful navigation around a previously identified borderline situation.
We prove the absence of eigenvaues of the three-dimensional Dirac operator with non-Hermitian potentials in unbounded regions of the complex plane under smallness conditions on the potentials in Lebesgue spaces. Our sufficient conditions are quantitative and easily checkable.
By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding two-dimensional Schroedinger operator possesses no point spectrum. The settings of complex-valued electric potentials and singular magnetic potentials of Aharonov-Bohm field are also covered.
In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus $(\mathbb{R}/\mathbb{Z})^3$, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domains theorem with equality (Courant-sharp situation). Following the strategy of Å. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber-Krahn-type inequality for domains on the torus, deduced as, in the work of P. Bérard and D. Meyer (1982), from an isoperimetric inequality. This inequality relies on the work of L. Hauswirth, J. Perez, P. R…
This paper is a continuation of the paper \cite{W} by the third author, which studied quantum walks with special long-range perturbations of the coin operator. In this paper, we consider general long-range perturbations of the coin operator and prove the non-existence of a singular continuous spectrum and embedded eigenvalues. The proof relies on the construction of generalized eigenfunctions (Jost solutions) which was studied in the short-range case in \cite{MSSSSdis}.
