Matrix-valued Potentials Research Articles

Stability via closure relations with applications to dissipative and port-Hamiltonian systems

We consider differential operators A that can be represented by means of a so-called closure relation in terms of a simpler operator Aext\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_{{\ ext {ext}}}$$\\end{document} defined on a larger space. We analyse how the spectral properties of A and Aext\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_{{\ ext {ext}}}$$\\end{document} are related and give sufficient conditions for exponential stability of the semigroup generated by A in terms of the semigroup generated by Aext\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_{{\ ext {ext}}}$$\\end{document}. As applications we study the long-term behaviour of a coupled wave–heat system on an interval, parabolic equations on bounded domains that are coupled by matrix-valued potentials, and of linear infinite-dimensional port-Hamiltonian systems with dissipation on an interval.

Sets of Non-Lyapunov Behaviour for Scalar and Matrix Schrödinger Cocycles

Abstract We discuss the growth of the singular values of symplectic transfer matrices associated with ergodic discrete Schrödinger operators in one dimension, with scalar and matrix-valued potentials. While for an individual value of the spectral parameter the rate of exponential growth is almost surely governed by the Lyapunov exponents, this is not, in general, true simultaneously for all the values of the parameter. The structure of the exceptional sets is interesting in its own right, and is also of importance in the spectral analysis of the operators. We present new results along with amplifications and generalisations of several older ones, and also list a few open questions. Here are two sample results. On the negative side, for any square-summable sequence $p_{n}$ there is a residual set of energies in the spectrum on which the middle singular value (the $W$-th out of $2W$) grows no faster than $p_{n}^{-1}$. On the positive side, for a large class of cocycles including the i.i.d. ones, the set of energies at which the growth of the singular values is not as given by the Lyapunov exponents has zero Hausdorff measure with respect to any gauge function $\rho (t)$ such that $\rho (t)/t$ is integrable at zero. The employed arguments from the theory of subharmonic functions also yield a generalisation of the Thouless formula, possibly of independent interest: for each $k$, the average of the first $k$ Lyapunov exponents is the logarithmic potential of a probability measure.

Discrete approximations to Dirac operators and norm resolvent convergence

We consider continuous Dirac operators defined on \mathbf{R}^d , d\in\{1,2,3\} , together with various discrete versions of them. Both forward-backward and symmetric finite differences are used as approximations to partial derivatives. We also allow a bounded, Hölder continuous, and self-adjoint matrix-valued potential, which in the discrete setting is evaluated on the mesh. Our main goal is to investigate whether the proposed discrete models converge in norm resolvent sense to their continuous counterparts, as the mesh size tends to zero and up to a natural embedding of the discrete space into the continuous one. In dimension one we show that forward-backward differences lead to norm resolvent convergence, while in dimension two and three they do not . The same negative result holds in all dimensions when symmetric differences are used. On the other hand, strong resolvent convergence holds in all these cases. Nevertheless, and quite remarkably, a rather simple but non-standard modification to the discrete models, involving the mass term, ensures norm resolvent convergence in general.

Propagation of Wave Packets for Systems Presenting Codimension One Crossings

We analyze the propagation of wave packets through general Hamiltonian systems presenting codimension one eigenvalue crossings. The class of time-dependent Hamiltonians we consider is of general pseudodifferential form with subquadratic growth. It comprises Schrödinger operators with matrix-valued potential, as they occur in quantum molecular dynamics, but also covers matrix-valued models of solid state physics describing the motion of electrons in a crystal. We calculate precisely the non-adiabatic effects of the crossing in terms of a transition operator, whose action on coherent states can be spelled out explicitly.

High energy bounds on wave operators

The wave operators W±(H1,H0) of two selfadjoint operators H0 and H1 are analyzed at asymptotic spectral values. Sufficient conditions for ∥(W±(H1,H0)−Pac1Pac0)f(H0)∥<∞ are given, where Pacj projects onto the subspace of absolutely continuous spectrum of Hj and f is an unbounded function (f-boundedness), both in the case of trace-class perturbations and in terms of the high-energy behaviour of the boundary values of the resolvent of H0 (smooth method). Examples include f-boundedness for the perturbed polyharmonic operator and for Schr\"odinger operators with matrix-valued potentials. We discuss an application to the problem of quantum backflow.

Absence of Eigenvalues of Dirac and Pauli Hamiltonians via the Method of Multipliers

By developing the method of multipliers, we establish sufficient conditions on the magnetic field and the complex, matrix-valued electric potential, which guarantee that the corresponding system of Schrödinger operators has no point spectrum. In particular, this allows us to prove analogous results for Pauli operators under the same electromagnetic conditions and, in turn, as a consequence of the supersymmetric structure, also for magnetic Dirac operators.

A Sturm–Liouville theorem for quadratic operator pencils

We establish a Sturm–Liouville theorem for quadratic operator pencils with matrix-valued potentials counting their unstable real roots, with applications to stability of waves. Such pencils arise, for example, in reduction of eigenvalue systems to higher-order scalar problems.

A Hardy-type inequality and some spectral characterizations for the Dirac–Coulomb operator

We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix-valued potentials \({\mathbf {V}}\) of Coulomb type: we characterise its eigenvalues in terms of the Birman–Schwinger principle and we bound its discrete spectrum from below, showing that the ground-state energy is reached if and only if \({\mathbf {V}}\) verifies some rigidity conditions. In the particular case of an electrostatic potential, these imply that \({\mathbf {V}}\) is the Coulomb potential.

Resolvent Convergence to Dirac Operators on Planar Domains

Consider a Dirac operator defined on the whole plane with a mass term of size m supported outside a domain Omega. We give a simple proof for the norm resolvent convergence, as m goes to infinity, of this operator to a Dirac operator defined on Omega with infinite mass boundary conditions. The result is valid for bounded and unbounded domains and gives estimates on the speed of convergence. Moreover, the method easily extends when adding external matrix-valued potentials.

Localization for Gapped Dirac Hamiltonians with Random Perturbations: Application to Graphene Antidot Lattices

In this paper we study random perturbations of first order elliptic operators with periodic potentials. We are mostly interested in Hamiltonians modeling graphene antidot lattices with impurities. The unperturbed operator H_0 := D_S + V_0 is the sum of a Dirac-like operator D_S plus a periodic matrix-valued potential V_0 , and is assumed to have an open gap. The random potential V_\omega is of Anderson-type with independent, identically distributed coupling constants and moving centers, with absolutely continuous probability distributions. We prove band edge localization, namely that there exists an interval of energies in the unperturbed gap where the almost sure spectrum of the family H_\omega := H_0 +V_\omega is dense pure point, with exponentially decaying eigenfunctions, that give rise to dynamical localization.

The Morse and Maslov indices for Schrödinger operators

We study the spectrum of Schrodinger operators with matrixvalued potentials, utilizing tools from infinite-dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrodinger operators with θ-periodic, Dirichlet, and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for multi-dimensional Schrodinger operators with periodic potentials. For quasi-convex domains in Rn, we recast the results, connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.

From spinning conformal blocks to matrix Calogero-Sutherland models

In this paper we develop further the relation between conformal four-point blocks involving external spinning fields and Calogero-Sutherland quantum mechanics with matrix-valued potentials. To this end, the analysis of [1] is extended to arbitrary dimensions and to the case of boundary two-point functions. In particular, we construct the potential for any set of external tensor fields. Some of the resulting Schrödinger equations are mapped explicitly to the known Casimir equations for 4-dimensional seed conformal blocks. Our approach furnishes solutions of Casimir equations for external fields of arbitrary spin and dimension in terms of functions on the conformal group. This allows us to reinterpret standard operations on conformal blocks in terms of group-theoretic objects. In particular, we shall discuss the relation between the construction of spinning blocks in any dimension through differential operators acting on seed blocks and the action of left/right invariant vector fields on the conformal group.

A necessary and sufficient condition for a matrix equilibrium state to be mixing

Since the 1970s there has been a rich theory of equilibrium states over shift spaces associated to Hölder-continuous real-valued potentials. The construction of equilibrium states associated to matrix-valued potentials is much more recent, with a complete description of such equilibrium states being achieved by Feng and Käenmäki [Equilibrium states of the pressure function for products of matrices.Discrete Contin. Dyn. Syst.30(3) (2011), 699–708]. In a recent article [Ergodic properties of matrix equilibrium states.Ergod. Th. & Dynam. Sys.(2017), to appear] the author investigated the ergodic-theoretic properties of these matrix equilibrium states, attempting in particular to give necessary and sufficient conditions for mixing, positive entropy, and the property of being a Bernoulli measure with respect to the natural partition, in terms of the algebraic properties of the semigroup generated by the matrices. Necessary and sufficient conditions were successfully established for the latter two properties, but only a sufficient condition for mixing was given. The purpose of this note is to complete that investigation by giving a necessary and sufficient condition for a matrix equilibrium state to be mixing.

Spectral analysis of θ-periodic Schrödinger operators and applications to periodic waves

Assuming a symmetric matrix-valued potential, we consider the associated θ-periodic Schrödinger operators Hθ on intervals [0,P], where in our applications P denotes the period of a stationary periodic solution to a nonlinear evolutionary PDE. We relate the Morse index of Hθ to certain Maslov indices, and apply our theory to operators obtained when Allen–Cahn equations and systems are linearized about stationary periodic solutions.

Semiclassical Trace Formula and Spectral Shift Function for Systems via a Stationary Approach

We establish a semiclassical trace formula in a general framework of microhyperbolic hermitian systems of $h$-pseudodifferential operators, and apply it to the study of the spectral shift function associated to a pair of selfadjoint Schrodinger operators with matrix-valued potentials. We give Weyl type semiclassical asymptotics with sharp remainder estimate for the spectral shift function, and, under the existence of a scalar escape function, a full asymptotic expansion in the strong sense for its derivative. A time-independent approach enables us to treat certain potentials with energy-level crossings.

An Egorov Theorem for Avoided Crossings of Eigenvalue Surfaces

We study nuclear propagation through avoided crossings of electron energy levels. We construct a surface hopping semigroup, which gives an Egorov-type description of the dynamics. The underlying time-dependent Schroedinger equation has a two-by-two matrix-valued potential, whose eigenvalue surfaces have an avoided crossing. Using microlocal normal forms reminiscent of the Landau-Zener problem, we prove convergence to the true solution in the semi-classical limit.

Limits of spiked random matrices II

The top eigenvalues of rank $r$ spiked real Wishart matrices and additively perturbed Gaussian orthogonal ensembles are known to exhibit a phase transition in the large size limit. We show that they have limiting distributions for near-critical perturbations, fully resolving the conjecture of Baik, Ben Arous and P\'{e}ch\'{e} [Duke Math. J. (2006) 133 205-235]. The starting point is a new $(2r+1)$-diagonal form that is algebraically natural to the problem; for both models it converges to a certain random Schr\"{o}dinger operator on the half-line with $r\times r$ matrix-valued potential. The perturbation determines the boundary condition and the low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. We treat the real, complex and quaternion ($\beta=1,2,4$) cases simultaneously. We further characterize the limit laws in terms of a diffusion related to Dyson's Brownian motion, or alternatively a linear parabolic PDE; here $\beta$ appears simply as a parameter. At $\beta=2$, the PDE appears to reconcile with known Painlev\'{e} formulas for these $r$-parameter deformations of the GUE Tracy-Widom law.

Spectral analysis near the low ground energy of magnetic Pauli operators

We are interested in 3-D magnetic Pauli operators perturbed by a 2×2 Hermitian matrix-valued potential V=V(x), x∈R3. We extend to the Pauli case the Breit–Wigner-type approximation and trace formula results obtained for the 3-D Schrödinger operator near the Landau levels. Hence, we give a link between the resonances and the spectral shift function near the low ground energy of the operators.

The Morse and Maslov indices for multidimensional Schrödinger operators with matrix-valued potentials

We study the Schrodinger operator L = −∆ +V on a star-shaped domain Ω in Rd with Lipschitz boundary ∂Ω. The operator is equipped with quite general Dirichletor Robin-type boundary conditions induced by operators between H1/2(∂Ω) and H−1/2(∂Ω), and the potential takes values in the set of symmetric N ×N matrices. By shrinking the domain and rescaling the operator we obtain a path in the Fredholm–Lagrangian Grassmannian of the subspace of H1/2(∂Ω)×H−1/2(∂Ω) corresponding to the given boundary condition. The path is formed by computing the Dirichlet and Neumann traces of weak solutions to the rescaled eigenvalue equation. We prove a formula relating the number of negative eigenvalues of L (the Morse index), the signed crossings of the path (the Maslov index), the number of negative eigenvalues of the potential matrix evaluated at the center of the domain, and the number of negative eigenvalues of a bilinear form related to the boundary operator.

Inverse spectral problems for Schrödinger-type operators with distributional matrix-valued potentials

The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schrödinger-type operators on a half-line from the underlying Weyl--Titchmarsh function.