Singular Weyl Research Articles

Singular Weyl’s law with Ricci curvature bounded below

We establish two surprising types of Weyl’s laws for some compact RCD ⁡ ( K , N ) \operatorname {RCD}(K, N) /Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for RCD ⁡ ( K , N ) \operatorname {RCD}(K,N) spaces. Our results depend crucially on analyzing and developing important properties of the examples constructed in Pan and Wei [Geom. Funct. Anal. 32 (2022), pp. 676–685], showing them isometric to the α \alpha -Grushin halfplanes. Of independent interest, this also allows us to provide counterexamples to conjectures in Cheeger and Colding [J. Differential Geom. 46 (1997), pp. 406–480] and Kapovitch, Kell, and Ketterer [Math. Z. 301 (2022), pp. 3469–3502].

Multi-Fold Fan-Shape Surface State Induced by an Isolated Weyl Phonon Beyond No-Go Theorem.

Absence of any surface arc state has been regarded as the fundamental property of singular Weyl points, because they are circumvented from the Nielsen-Ninomiya no-go theorem. In this work, through systematic investigations on topological properties of isolated Weyl phonons (IWPs) surrounded by closed Weyl nodal walls (WNWs), which are located at the Brillouin zone (BZ) boundaries of bosonic systems, it uncovers that a new kind of phononic surface state, that is, the multi-fold fan-shape surface state named by us, is exhibited to connect the projections of IWP and WNWs. Importantly, the number of fan leaves in this surface state is associated with the Chern number of IWP. Moreover, the topological features of charge-two IWP in K2 Mg2 O3 (SG No. 96) and charge-four IWP in Nb3 Al2 N (SG No. 213) confirm further the above fundamental properties of this kind of surface state. The theoretical work not only provides an effective way to seek for IWPs as well as to determine their Chern number in real materials, but also uncovers a new class of surface states in the topological Weyl complex composed of IWPs andWNWs.

Weyl and Majorana Spinors as Pure Goldstone Bosons

General spinors in polar form display a structure that is made up by real scalar degrees of freedom plus six components which can be recognized as Goldstone bosons: in the present paper we show that of all singular spinors, Weyl and Majorana spinors have no real degree of freedom and so that they can be interpreted as pure Goldstone states.

Three-nodal surface phonons in solid-state materials: Theory and material realization

This year, Liu \textit{et al}. [Phys. Rev. B \textbf{104}, L041405 (2021)] proposed a new class of topological phonons (TPs; i.e., one-nodal surface (NS) phonons), which provides an effective route for realizing one-NSs in phonon systems. In this work, based on first-principles calculations and symmetry analysis, we extended the types of NS phonons from one- to three-NS phonons. The existence of three-NS phonons (with NS states on the $k_{i}$ = $\pi$ ($i$ = $x$, $y$, $z$) planes in the three-dimensional Brillouin zone (BZ)) is enforced by the combination of two-fold screw symmetry and time reversal symmetry. We screened all 230 space groups (SGs) and found nine candidate groups (with the SG numbers (Nos.) 19, 61, 62, 92, 96, 198, 205, 212, and 213) hosting three-NS phonons. Interestingly, with the help of first-principles calculations, we identified $P2_{1}$2$_{1}$2$_{1}$-type YCuS$_{2}$ (SG No. 19), $Pbca$-type NiAs$_{2}$ (SG No. 61), $Pnma$-type SrZrO$_{2}$ (SG No. 62), $P4_{1}$2$_{1}$2-type LiAlO$_{2}$ (SG No. 92), $P4_{3}$2$_{1}$2-type ZnP$_{2}$ (SG No. 96), $P2_{1}$3-type NiSbSe (SG No. 198), $Pa\bar{3}$-type As$_{2}$Pt (SG No. 205), $P4_{3}$32-type BaSi$_{2}$ (SG No. 212), and $P4_{1}$32-type CsBe$_{2}$F$_{5}$ (SG No. 213) as realistic materials hosting three-NS phonons. The results of our presented study enrich the class of NS states in phonon systems and provide concrete guidance for searching for three-NS phonons and singular Weyl point phonons in realistic materials.

Observation of a singular Weyl point surrounded by charged nodal walls in PtGa

Constrained by the Nielsen-Ninomiya no-go theorem, in all so-far experimentally determined Weyl semimetals (WSMs) the Weyl points (WPs) always appear in pairs in the momentum space with no exception. As a consequence, Fermi arcs occur on surfaces which connect the projections of the WPs with opposite chiral charges. However, this situation can be circumvented in the case of unpaired WP, without relevant surface Fermi arc connecting its surface projection, appearing singularly, while its Berry curvature field is absorbed by nontrivial charged nodal walls. Here, combining angle-resolved photoemission spectroscopy with density functional theory calculations, we show experimentally that a singular Weyl point emerges in PtGa at the center of the Brillouin zone (BZ), which is surrounded by closed Weyl nodal walls located at the BZ boundaries and there is no Fermi arc connecting its surface projection. Our results reveal that nontrivial band crossings of different dimensionalities can emerge concomitantly in condensed matter, while their coexistence ensures the net topological charge of different dimensional topological objects to be zero. Our observation extends the applicable range of the original Nielsen-Ninomiya no-go theorem which was derived from zero dimensional paired WPs with opposite chirality.

On self-adjoint boundary conditions for singular Sturm–Liouville operators bounded from below

We extend the classical boundary values(0.1)g(a)=−W(ua(λ0,⋅),g)(a)=limx↓a⁡g(x)uˆa(λ0,x),g[1](a)=(pg′)(a)=W(uˆa(λ0,⋅),g)(a)=limx↓a⁡g(x)−g(a)uˆa(λ0,x)ua(λ0,x), for regular Sturm–Liouville operators associated with differential expressions of the type τ=r(x)−1[−(d/dx)p(x)(d/dx)+q(x)] for a.e. x∈(a,b)⊂R, to the case where τ is singular on (a,b)⊆R and the associated minimal operator Tmin is bounded from below. Here ua(λ0,⋅) and uˆa(λ0,⋅) denote suitably normalized principal and nonprincipal solutions of τu=λ0u for appropriate λ0∈R, respectively.Our approach to deriving the analog of (0.1) in the singular context employing principal and nonprincipal solutions of τu=λ0u is closely related to a seminal 1992 paper by Niessen and Zettl [58]. We also recall the well-known fact that the analog of the boundary values in (0.1) characterizes all self-adjoint extensions of Tmin in the singular case in a manner familiar from the regular case.We briefly discuss the singular Weyl–Titchmarsh–Kodaira m-function and finally illustrate the theory in some detail with the examples of the Bessel, Legendre, and Kummer (resp., Laguerre) operators.

Completeness Theorem for Eigenparameter Dependent Dissipative Dirac Operator with General Transfer Conditions

This paper deals with a singular (Weyl’s limit circle case) non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition and finite general transfer conditions. Using the equivalence between Lax-Phillips scattering matrix and Sz.-Nagy-Foiaş characteristic function, the completeness of the eigenfunctions and associated functions of this dissipative operator is discussed.

On spectral deformations and singular Weyl functions for one-dimensional Dirac operators

We investigate the connection between singular Weyl–Titchmarsh–Kodaira theory and the double commutation method for one-dimensional Dirac operators. In particular, we compute the singular Weyl function of the commuted operator in terms of the data from the original operator. These results are then applied to radial Dirac operators in order to show that the singular Weyl function of such an operator belongs to a generalized Nevanlinna class Nκ0 with κ0=κ+12, where κ ∈ ℝ is the corresponding angular momentum.

Equivariant spectral asymptotics for h-pseudodifferential operators

We prove equivariant spectral asymptotics for h-pseudodifferential operators for compact orthogonal group actions generalizing results of El Houakmi and Helffer [“Comportement semi-classique en présence de symétries: Action d'un groupe de Lie compact,” Asymp. Anal. 5(2), 91–113 (1991)] and Cassanas [“Reduced Gutzwiller formula with symmetry: Case of a Lie group,” J. Math. Pures Appl. 85(6), 719–742 (2006)]. Using recent results for certain oscillatory integrals with singular critical sets [P. Ramacher, “Singular equivariant asymptotics and Weyl's law: On the distribution of eigenvalues of an invariant elliptic operator,” J. Reine Angew. Math. (Crelles J.) (to be published)], we can deduce a weak equivariant Weyl law. Furthermore, we can prove a complete asymptotic expansion for the Gutzwiller trace formula without any additional condition on the group action by a suitable generalization of the dynamical assumptions on the Hamilton flow.

Spectral analysis of non-compact symmetrizable operators on Hilbert spaces

This paper revisits and complement in different directions the classical work by W. T. Reid on symmetrizable completely continuous transformations in Hilbert spaces and a more recent paper by one of the authors. More precisely, we deal with spectral properties of % non-compact operators G on a complex Hilbert space H such that SG is self-adjoint where S is a (not necessarily injective) nonnegative operator. We study the isolated eigenvalues of G outside its essential spectral interval and provide variational characterization of them as well as stability estimates. We compare them also to spectral objects of SG. Finally, we characterize the Schechter essential spectrum of strongly symmetrizable operators in terms singular Weyl sequences; in particular, we complement J. I. Nieto's paper on the essential spectrum of symmetrizable. Copyright © 2014 John Wiley & Sons, Ltd.

Spectrum of the linear water model for a two‐layer liquid with cuspidal geometries at the interface

We show that the linear water wave problem in a bounded liquid domain may have continuous spectrum, if the interface of a two‐layer liquid touches the basin walls at zero angle. The reason for this phenomenon is the appearance of cuspidal geometries of the liquid phases. We calculate the exact position of the continuous spectrum. We also discuss the physical background of wave propagation processes, which are enabled by the continuous spectrum. Our approach and methods include constructions of a parametrix for the problem operator and singular Weyl sequences.

Singular equivariant asymptotics and Weyl’s law. On the distribution of eigenvalues of an invariant elliptic operator

Abstract We study the spectrum of an invariant, elliptic, classical pseudodifferential operator on a closed Riemannian manifold M carrying an effective and isometric action of a compact, connected Lie group G. Using resolution of singularities, we determine the asymptotic distribution of eigenvalues along the isotypic components, and relate it to the reduction of the corresponding Hamiltonian flow, proving that the reduced spectral counting function satisfies Weyl’s law, together with an estimate for the remainder.

Singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators

We develop singular Weyl-Titchmarsh-Kodaira theory for one-dimensional Dirac operators. In particular, we establish existence of a spectral transformation as well as local Borg-Marchenko and Hochstadt-Liebermann type uniqueness results. Finally, we give some applications to the case of radial Dirac operators.

Spectral Asymptotics for Perturbed Spherical Schrödinger Operators and Applications to Quantum Scattering

We find the high energy asymptotics for the singular Weyl--Titchmarsh m-functions and the associated spectral measures of perturbed spherical Schr\"odinger operators (also known as Bessel operators). We apply this result to establish an improved local Borg-Marchenko theorem for Bessel operators as well as uniqueness theorems for the radial quantum scattering problem with nontrivial angular momentum.

Direct and inverse spectral theory of singular left-definite Sturm–Liouville operators

We discuss direct and inverse spectral theory of self-adjoint Sturm–Liouville relations with separate boundary conditions in the left-definite setting. In particular, we develop singular Weyl–Titchmarsh theory for these relations. Consequently, we apply de Brangesʼ subspace ordering theorem to obtain inverse uniqueness results for the associated spectral measure. The results can be applied to solve the inverse spectral problem associated with the Camassa–Holm equation.

Commutation methods for Schrödinger operators with strongly singular potentials

AbstractWe explore the connections between singular Weyl–Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in terms of the original operator. We apply the results to spherical Schrödinger operators (also known as Bessel operators). We also investigate the connections with the generalized Bäcklund–Darboux transformation.

Weyl-Titchmarsh Theory for Schrodinger Operators with Strongly Singular Potentials

We develop Weyl-Titchmarsh theory for Schroedinger operators with strongly singular potentials such as perturbed spherical Schroedinger operators (also known as Bessel operators). It is known that in such situations one can still define a corresponding singular Weyl m-function and it was recently shown that there is also an associated spectral transformation. Here we will give a general criterion when the singular Weyl function can be analytically extended to the upper half plane. We will derive an integral representation for this singular Weyl function and give a criterion when it is a generalized Nevanlinna function. Moreover, we will show how essential supports for the Lebesgue decomposition of the spectral measure can be obtained from the boundary behavior of the singular Weyl function. Finally, we will prove a local Borg-Marchenko type uniqueness result. Our criteria will in particular cover the aforementioned case of perturbed spherical Schroedinger operators.

Gravitational self-force in a radiation gauge

In this, the first of two companion papers, we present a method for finding the gravitational self-force in a modified radiation gauge for a particle moving on a geodesic in a Schwarzschild or Kerr spacetime. An extension of an earlier result by Wald is used to show the spin weight $\ifmmode\pm\else\textpm\fi{}2$ perturbed Weyl scalar (${\ensuremath{\psi}}_{0}$ or ${\ensuremath{\psi}}_{4}$) determines the metric perturbation outside the particle up to a gauge transformation and an infinitesimal change in mass and angular momentum. A Hertz potential is used to construct the part of the retarded metric perturbation that involves no change in mass or angular momentum from ${\ensuremath{\psi}}_{0}$ in a radiation gauge. The metric perturbation is completed by adding changes in the mass and angular momentum of the background spacetime outside the radial coordinate ${r}_{0}$ of the particle in any convenient gauge. The resulting metric perturbation is singular only on the trajectory of the particle. A mode-sum method is then used to renormalize the self-force. Gralla shows that the renormalized self-force can be used to find the correction to a geodesic orbit in a gauge for which the leading, $O({\ensuremath{\rho}}^{\ensuremath{-}1})$, term in the metric perturbation has spatial components even under a parity transformation orthogonal to the particle trajectory, and we verify that the metric perturbation in a radiation gauge satisfies that condition. We show that the singular behavior of the metric perturbation and the expression for the bare self-force have the same power-law behavior in $L=\ensuremath{\ell}+1/2$ as in a Lorenz gauge (with different coefficients). We explicitly compute the singular Weyl scalar and its mode-sum decomposition to subleading order in $L$ for a particle in circular orbit in a Schwarzschild geometry and obtain the renormalized field. Because the singular field can be defined as this mode sum, the coefficients of each angular harmonic in the sum must agree with the large $L$ limit of the corresponding coefficients of the retarded field. One may therefore compute the singular field by numerically matching the retarded field to a power series in $L$. To check the accuracy of the numerical method, we analytically compute leading and subleading terms in the singular expansion of ${\ensuremath{\psi}}_{0}$ and compare the numerical and analytic values of the renormalization constants, finding agreement to high precision. Details of the numerical computation of the perturbed metric, the self-force, and the quantity ${h}_{\ensuremath{\alpha}\ensuremath{\beta}}{u}^{\ensuremath{\alpha}}{u}^{\ensuremath{\beta}}$ (gauge invariant under helically symmetric gauge transformations) are presented for this test case in the companion paper.

On the singular Weyl–Titchmarsh function of perturbed spherical Schrödinger operators

We investigate the singular Weyl–Titchmarsh m-function of perturbed spherical Schrödinger operators (also known as Bessel operators) under the assumption that the perturbation q ( x ) satisfies x q ( x ) ∈ L 1 ( 0 , 1 ) . We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular m-function belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.