Levinson's Theorem Research Articles

Trace formula and Levinson's theorem as an index pairing in the presence of resonances

Trace formula and Levinson's theorem as an index pairing in the presence of resonances

Stability of Krein-von Neumann self-adjoint operator extension under unbounded perturbations

We have considered a fourth order difference operator defined on the Hilbert space of square summable sequences on N. We investigated the stability of existence of Krein-von Neumann self-adjoint extension of difference operators under bounded and unbounded coefficients. Using asymptotic summation based on discretised Levinson's theorem and appropriate smoothness and decay conditions, we have shown that unlike the case of deficiency indices and discrete spectrum, the existence of positive self-adjoint operator extensions is stable under unbounded perturbations. These results now exhaustively characterise the spectral and structural properties of difference operators with unbounded coefficients.

Levinson's theorem as an index pairing

We build on work of Kellendonk, Richard, Tiedra de Aldecoa and others to show that the wave operators for Schrödinger scattering theory on Rn generically have a particular form. As a consequence, Levinson's theorem can be interpreted as the pairing of the K-theory class of the unitary scattering operator and the K-homology class of the generator of the group of dilations.

Levinson theorem for discrete Schrödinger operators on the line with matrix potentials having a first moment

This paper proves new results on spectral and scattering theory for matrix-valued Schrödinger operators on the discrete line with non-compactly supported perturbations whose first moments are assumed to exist. In particular, a Levinson theorem is proved, in which a relation between scattering data and spectral properties (bound and half-bound states) of the corresponding Hamiltonians is derived. The proof is based on stationary scattering theory with prominent use of Jost solutions at complex energies that are controlled by Volterra-type integral equations.

Generalized topological bulk-edge correspondence in bulk-Hermitian continuous systems with non-Hermitian boundary conditions

The bulk-edge correspondence (BEC) is the hallmark of topological systems. In continuous (nonlattice) Hermitian systems with an unbounded wave vector, it was recently shown that the BEC of Chern insulators is modified. How would it be further affected in non-Hermitian systems, experiencing loss and/or gain? In this work, we take the first step in this direction, by studying a bulk-Hermitian continuous system with non-Hermitian boundary conditions. We find in this case that edge modes emerge at the roots of the scattering matrix, as opposed to the Hermitian case, where they emerge at its poles (or, more accurately, coalescence of roots and poles). This entails a nontrivial modification to the relative Levinson's theorem. We then show that the topological structure remains the same as in the Hermitian case, and the generalized BEC holds, provided one employs appropriately modified contours in the wave-vector plane so that the scattering matrix phase winding counts the edge modes correctly. We exemplify all this using a paradigmatic model of waves in a shallow ocean or active systems in the presence of odd viscosity, as well as 2D electron gas with Hall viscosity. We use this opportunity to examine the case of large odd viscosity, where the scattering matrix becomes $2\ifmmode\times\else\texttimes\fi{}2$, which has not been discussed in previous works on the Hermitian generalized BEC.

Topology in Shallow-Water Waves: A Spectral Flow Perspective

In the context of topological insulators, the shallow-water model was recently shown to exhibit an anomalous bulk-edge correspondence. For the model with a boundary, the parameter space involves both longitudinal momentum and boundary conditions, and exhibits a peculiar singularity. We resolve the anomaly in question by defining a new kind of edge index as the spectral flow around this singularity. Crucially, this edge index samples a whole family of boundary conditions, and we interpret it as a boundary-driven quantized pumping. Our edge index is stable due to the topological nature of spectral flow, and we prove its correspondence with the bulk Chern number index using scattering theory and a relative version of Levinson's theorem. The full spectral flow structure of the model is also investigated.

A Borg–Levinson theorem for magnetic Schrödinger operators on a Riemannian manifold

We establish uniqueness and stability results for the inverse spectral problem of recovering the magnetic field and the electric potential in a Riemannian manifold from the knowledge of boundary spectral data of the corresponding magnetic Schrödinger operator with Dirichlet boundary condition. The spectral data consist in the knowledge of asymptotic properties, that we specify hereafter, of the sequence of eigenvalues and Neumann traces of the corresponding sequence of eigenfunctions. We also prove similar results for Schrödinger operators with Neumann boundary conditions. To our knowledge our results are the first ones involving such weak boundary spectral data.

Universal scattering with general dispersion relations

Many synthetic quantum systems allow particles to have dispersion relations that are neither linear nor quadratic functions. Here, we explore single-particle scattering in general spatial dimension $D\ensuremath{\ge}1$ when the density of states diverges at a specific energy. To illustrate the underlying principles in an experimentally relevant setting, we focus on waveguide quantum electrodynamics (QED) problems (i.e., $D=1$) with dispersion relation $\ensuremath{\epsilon}(k)=\ifmmode\pm\else\textpm\fi{}|d|{k}^{m}$, where $m\ensuremath{\ge}2$ is an integer. For a large class of these problems for any positive integer $m$, we rigorously prove that when there are no bright zero-energy eigenstates, the $S$ matrix evaluated at an energy $E\ensuremath{\rightarrow}0$ converges to a universal limit that is only dependent on $m$. We also give a generalization of a key index theorem in quantum scattering theory known as Levinson's theorem---which relates the scattering phases to the number of bound states---to waveguide QED scattering for these more general dispersion relations. We then extend these results to general integer dimensions $D\ensuremath{\ge}1$, dispersion relations $\ensuremath{\epsilon}(\mathbit{k})={|\mathbit{k}|}^{a}$ for a $D$-dimensional momentum vector $\mathbit{k}$ with any real positive $a$, and separable potential scattering.

Quark Cluster Expansion Model for Interpreting Finite-T Lattice QCD Thermodynamics

In this work, we present a unified approach to the thermodynamics of hadron–quark–gluon matter at finite temperatures on the basis of a quark cluster expansion in the form of a generalized Beth–Uhlenbeck approach with a generic ansatz for the hadronic phase shifts that fulfills the Levinson theorem. The change in the composition of the system from a hadron resonance gas to a quark–gluon plasma takes place in the narrow temperature interval of 150–190 MeV, where the Mott dissociation of hadrons is triggered by the dropping quark mass as a result of the restoration of chiral symmetry. The deconfinement of quark and gluon degrees of freedom is regulated by the Polyakov loop variable that signals the breaking of the Z(3) center symmetry of the color SU(3) group of QCD. We suggest a Polyakov-loop quark–gluon plasma model with O(αs) virial correction and solve the stationarity condition of the thermodynamic potential (gap equation) for the Polyakov loop. The resulting pressure is in excellent agreement with lattice QCD simulations up to high temperatures.

Topology in Shallow-Water Waves: A Violation of Bulk-Edge Correspondence

We study the two-dimensional rotating shallow-water model describing Earth's oceanic layers. It is formally analogue to a Schr\"odinger equation where the tools from topological insulators are relevant. Once regularized at small scale by an odd-viscous term, such a model has a well-defined bulk topological index. However, in presence of a sharp boundary, the number of edge modes depends on the boundary condition, showing an explicit violation of the bulk-edge correspondence. We study a continuous family of boundary conditions with a rich phase diagram, and explain the origin of this mismatch. Our approach relies on scattering theory and Levinson's theorem. The latter does not apply at infinite momentum because of the analytic structure of the scattering amplitude there, ultimately responsible for the violation.

Generalization of the Levinson Theorem on the Asymptotic Value of the Scattering-Amplitude Phase Shift

The Levinson theorem relates the difference of the phase shifts at the threshold and at infinity to the number of bound states. The theorem is modified in view of the fact that Castillejo–Dalitz–Dyson (CDD) poles and Jaffe–Low primitives corresponding to zeros of the $$D$$ function on the unitary cut are present in the scattering amplitude. It is shown that, in general, the difference of the phase shifts at the threshold and at infinity is determined by the number of bound states, the number of CDD poles, and the number of primitives. Some consequences of this theorem that concern the properties of the nucleon–nucleon interaction are discussed.

A unified quark-nuclear matter equation of state from the cluster virial expansion within the generalized Beth\u2013Uhlenbeck approach

We consider a cluster expansion for strongly correlated quark matter where the clusters are baryons with spectral properties that are described within the generalized Beth–Uhlenbeck approach by a medium dependent phase shift. We employ a simple ansatz for the phase shift which describes an on-shell bound state with an effective mass and models the continuum by an anti-bound state located at the mass of the three-quark continuum threshold, so that the Levinson theorem is fulfilled by construction. The quark and baryon interactions are accounted for by the coupling to scalar and vector meson mean fields modelled by density functionals. At increasing density and temperature, due to the different medium-dependence of quark and baryon masses, the Mott dissociation of baryons occurs and its contributions to the thermodynamics vanish. It is demonstrated on this simple example that this unified approach to quark-hadron matter is capable of describing crossover as well as first order phase transition behaviour in the phase diagram with a critical endpoint. Changing the meson mean field, the case of a “crossover all over” in the phase diagram is also obtained.

A partial data inverse problem for the electro-magnetic wave equation and application to the related Borg–Levinson theorem

In this article we study the stability in an inverse problem of recovering the magnetic field and the electric potential in a bounded smooth domain from boundary observation of the corresponding wave equation. We prove that the knowledge of the partial Dirichlet-to-Neumann map measured on arbitrary subset of the boundary determines the electric potential and the magnetic field. Next, we apply this result to prove the uniqueness for the multidimensional Borg–Levinson theorem for the electro-magnetic potential from partial data.

On Stability of Bases Made from Perturbed Exponential Systems in Morrey Type Spaces

Perturbed exponential system {eiλkχ}keZ (where {λn} is some sequence of real numbers) isconsidered in Morrey spaces Lp,α (0, π) These spaces arenon-separable (except for exceptional cases), and thereforethe above system is not complete in them. Based on theshift operator, we define the subspace Mp,a (0, π)C Lp,α (0, π) where continuous functions aredense. We find a condition on the sequence {λn} which issufficient for the above system to form a basis for thesubspace Mp,a (0, π). Our results are the analogues ofthose obtained earlier for the Lebesgue spaces Lp. Wealso establish an analogue of classical Levinson theorem onthe completeness of above system in the spaces Lp,1 <= p <=+∞

Explicit formula for Schrödinger wave operators on the half-line for potentials up to optimal decay

We give an explicit formula for the wave operators for perturbations of the Dirichlet Laplacian by a potential on the half-line. The potential is assumed to decay strictly faster than the polynomial of degree minus two. The formula consists of the main term given by the scattering operator and a function of the generator of the dilation group, and a Hilbert-Schmidt remainder term. Our method is based on the elementary construction of the generalized Fourier transforms in terms of the solutions of the Volterra integral equations. The Hilbert-Schmidt property of the remainder term follows from a decay estimate for the Jost solution, which is established by performing a perturbation expansion of sufficiently high order. As a corollary, a topological interpretation of Levinson's theorem is established via an index theorem approach.

Spectral density, Levinson's theorem, and the extra term in the second virial coefficient for the one-dimensional δ -function potential

In contrast with the 3D result, the Beth-Uhlenbeck (BU) formula in 1D contains an extra -1/2 term. The origin of this -1/2 term is explained using a spectral density approach. To be explicit, a delta-function potential is used to show that the correction term arises from a pole of the density of states at zero energy. The spectral density method shows that this term is actually an artifact of the non-normalizability of the scattering states and an infrared cutoff regularization scheme has to be used to get the correct result in 1D. The formal derivation of the BU formula would miss this term since it ignores the effects of the boundary terms. While the result is shown for the delta-function potential, the method and result are valid for more general potentials. Additionally, the 1D Levinson's theorem can be extracted from the spectral density method using the asymptotic form of general potentials. The importance of the result lies in the fact that all these correction terms in 1D have a universal source: a pole at zero energy. Similar calculations using quantum field theoretical approaches (without explicit infrared cutoff regularization schemes) also show the same subtleties with the correction term originating from the zero energy scattering states (appendix A).

Thermodynamics of a generalized graphene-motivated (2+1) D Gross–Neveu model beyond the mean field within the Beth–Uhlenbeck approach

Abstract We investigate the thermodynamics at finite density of a generalized $(2 + 1)$D Gross–Neveu model of $N$ fermion species with various types of four-fermion interactions. The motivation for considering such a generalized schematic model arises from taking the Fierz transformation of an effective Coulomb current–current interaction and certain symmetry-breaking interaction terms, as considered for graphene-type models in Ref. [29]. We then apply path-integral bosonization techniques, based on the large-$N$ limit, to derive the thermodynamic potential. This includes the leading-order mean-field (saddle point) contribution as well as the next-order contribution of Gaussian fluctuations of exciton fields. The main focus of the paper is then the investigation of the thermodynamic properties of the resulting fermion–exciton plasma. In particular, we derive an extended Beth–Uhlenbeck form of the thermodynamic potential, and discuss the Levinson theorem and the decomposition of the phase of the exciton correlation into resonant and scattering parts.

Effective lambda-proton and lambda-neutron potentials from subthreshold inverse scattering

Potentials are constructed for the lambda-nucleon interaction in the and channels. These potentials are recovered from scattering phases below the inelastic threshold through Gel’fand-Levitan-Marchenko theory. Experimental data with good statistics is not available for lambda-nucleon scattering. This leaves theoretical scattering phases as the only option through which the rigorous theory of quantum inverse scattering can be used in probing the lambda-nucleon force. Using rational-function interpolations on the theoretical scattering data, the kernels of the Gel’fand-Levitan-Marchenko integral equation become degenerate, resulting in a closed-form solution. The new potentials restored, which are shown to be unique through the Levinson theorem, bear the expected features of short-range repulsion and intermediate-range attraction. Charge symmetry breaking, which is perceptible in the scattering phases, is preserved in the new potentials. The lambda-nucleon force in the channel is observed to be stronger than in the channel, as expected. In addition, the potentials bear certain distinctive features whose effects on hypernuclear systems can be explored through Schrödinger calculations.

Spectral density, Levinson's theorem, and the extra term in the second virial coefficient for the one-dimensional δ-function potential

Spectral density, Levinson's theorem, and the extra term in the second virial coefficient for the one-dimensional δ-function potential

Multidimensional Borg–Levinson theorems for unbounded potentials

We prove that the Dirichlet eigenvalues and Neumann boundary data of the corresponding eigenfunctions of the operator $-\Delta + q$, determine the potential $q$, when $q \in L^{n/2}(\Omega,\mathbb{R})$ and $n \geq 3$. We also consider the case of incomplete spectral data, in the sense that the above spectral data is unknown for some finite number of eigenvalues. In this case we prove that the potential $q$ is uniquely determined for $q \in L^p(\Omega,\mathbb{R})$ with $p=n/2$, for $n\geq4$ and $p>n/2$, for $n=3$.