Mathematics

In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let P: L 2 (R)→ L 2 (R) have the form P:=− d 2 d x 2 +W, where W is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, 1 (−∞, λ 2 ] (P) has a full asymptotic expansion in powers of λ . In particular, our class of potentials W is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, it includes certain potentials with dense pure point spectrum. The proof combines the gauge transform methods of Parnovski-Shterenberg and Sobolev with Melrose's scattering calculus.

Euclidean distance matrices corresponding to an arithmetic progression have rich spectral and structural properties. We exploit those properties to develop completely positive factorizations of translations of those matrices. We show that the minimal translation that makes such a matrix positive semidefinite results in a completely positive matrix. We also discuss completely positive factorizations of such matrices over the integers. Methods developed in the paper can be used to find completely positive factorizations of other matrices with similar properties.

The completeness of the system of eigenfunctions of the complex Schrödinger operator L c =??d 2 /d x 2 +c x α on the semi-axis with Dirichlet boundary conditions is proved for α??0,2) and |argc|<2?α/(α+2)+?t(α) with some ?t(α)>0 .

Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ , with genus γ and b boundary components, we maximize σ j (Σ,g) L(∂Σ,g) over a class of smooth metrics, g , where σ j (Σ,g) is the j -th nonzero Steklov eigenvalue and L(∂Σ,g) is the length of ∂Σ . Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus γ=0 and b=2,…,9,12,15,20 boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with b=2 boundary components and for the third and fifth eigenvalues with b=3 boundary components.

The question of whether it is possible to compute scattering resonances of Schrödinger operators - independently of the particular potential - is addressed. A positive answer is given, and it is shown that the only information required to be known a priori is the size of the support of the potential. The potential itself is merely required to be C 1 . The proof is constructive, providing a universal algorithm which only needs to access the values of the potential at any requested point.

Spectral measures arise in numerous applications such as quantum mechanics, signal processing, resonances, and fluid stability. Similarly, spectral decompositions (pure point, absolutely continuous and singular continuous) often characterise relevant physical properties such as long-time dynamics of quantum systems. Despite new results on computing spectra, there remains no general method able to compute spectral measures or spectral decompositions of infinite-dimensional normal operators. Previous efforts focus on specific examples where analytical formulae are available (or perturbations thereof) or on classes of operators with a lot of structure. Hence the general computational problem is predominantly open. We solve this problem by providing the first set of general algorithms that compute spectral measures and decompositions of a wide class of operators. Given a matrix representation of a self-adjoint or unitary operator, such that each column decays at infinity at a known asymptotic rate, we show how to compute spectral measures and decompositions. We discuss how these methods allow the computation of objects such as the functional calculus, and how they generalise to a large class of partial differential operators, allowing, for example, solutions to evolution PDEs such as Schrödinger equations on L 2 ( R d ) . Computational spectral problems in infinite dimensions have led to the SCI hierarchy, which classifies the difficulty of computational problems. We classify computation of measures, measure decompositions, types of spectra, functional calculus, and Radon--Nikodym derivatives in the SCI hierarchy. The new algorithms are demonstrated to be efficient on examples taken from OPs on the real line and the unit circle (e.g. giving computational realisations of Favard's theorem and Verblunsky's theorem), and are applied to evolution equations on a 2D quasicrystal.

The question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is C 2 . The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point.

Let ( S 2 ,g) be a convex surface of revolution and H⊂ S 2 the unique rotationally invariant geodesic. Let φ ℓ m be the orthonormal basis of joint eigenfunctions of Δ g and ∂ θ , the generator of the rotation action. The main result is an explicit formula for the weak-* limit of the normalized empirical measures, Σ ℓ m=−ℓ || φ ℓ m | | 2 L 2 (H) δ m ℓ (c) on [−1,1] . The explicit formula shows that, asymptotically, the L 2 norms of restricted eigenfunctions are minimal for the zonal eigenfunction m=0 , maximal for Gaussian beams m=±1 , and exhibit a (1− c 2 ) − 1 2 type singularity at the endpoints. For a pseudo-differential operator B we also compute the limits of the normalized measures ∑ ℓ m=−ℓ ⟨B φ ℓ m , φ ℓ m ⟩ δ m ℓ (c) .

Trace formulas play a central role in the study of spectral geometry and in particular of quantum graphs. The basis of our work is the result by Kurasov which links the Euler characteristic χ of metric graphs to the spectrum of their standard Laplacian. These ideas were shown to be applicable even in an experimental context where only a finite number of eigenvalues from a physical realization of quantum graph can be measured. In the present work we analyse sufficient hypotheses which guarantee the successful recovery of χ . We also study how to improve the efficiency of the method and in particular how to minimise the number of eigenvalues required. Finally, we compare our findings with numerical examples---surprisingly, just a few dozens of eigenvalues can be enough.

This work is a continuation of our previos paper, where for the Schrödinger operator $H=-\Delta+ V(\e)\cdot$ $(V(\e)\ge 0)$, acting in the space $L_2(\R^d)\,(d\ge 3)$, some sufficient conditions for discreteness of its spectrum have been obtained on the base of well known Mazya -Shubin criterion and an optimization problem for a set function, which is an infinite-dimensional generalization of a binary linear programming problem. A sufficient condition for discreteness of the spectrum is formulated in terms of the non-increasing rearrangement of the potential $V(\e)$. Using the method of Lagrangian relaxation for this optimization problem, we obtain a sufficient condition for discreteness of the spectrum in terms of expectation and deviation of the potential. By means of suitable perturbations of the potential we obtain conditions for discreteness of the spectrum, covering potentials which tend to infinity only on subsets of cubes, whose Lebesgue measures tend to zero when the cubes go to infinity. Also the case where the operator H is defined in the space L 2 (Ω) is considered ( Ω is an open domain in $\R^d$).