Weyl Limit Research Articles

Canonical systems with discrete spectrum

We study spectral properties of two-dimensional canonical systems y′(t)=zJH(t)y(t), t∈[a,b), where the Hamiltonian H is locally integrable on [a,b), positive semidefinite, and Weyl's limit point case takes place at b. We answer the following questions explicitly in terms of H: Here asymptotic distribution means summability and limit superior conditions relative to comparison functions growing sufficiently fast. Making an analogy with complex analysis, this corresponds to convergence class and type w.r.t. proximate orders having order larger than 1. It is a surprising fact that these properties depend only on the diagonal entries of H.In 1968 L.de Branges posed the following question as a fundamental problem: We give a complete and explicit answer.

Eigenfunction expansions associated with the one‐dimensional Schroedinger operator

AbstractWe consider the one‐dimensional Schroedinger operator H on L2(−∞, ∞) associated with where q(r) : R → R is locally integrable, λ ∈ R is a spectral parameter, and the differential expression is in Weyl's limit point cases at both endpoints. Starting from the well known Weyl‐Kodaira formulation of the eigenfunction expansion associated with H [1], an alternative formulation can be derived in such a way that the principal features contributing to it, namely a scalar spectral density function, the location and multiplicity of the spectrum, and the spectral types of the generalised eigenfunctions, can be explicitly exhibited in the resulting expansion. The proof is based on classical work of Kac [2] and the theory of subordinacy [3], [4]. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the essential spectrum of a class of singular matrix differential operators. II. Weyl's limit circles for the Hain–Lüst operator whenever quasi-regularity conditions are not satisfied

On the essential spectrum of a class of singular matrix differential operators II. Weyl's limit circles for Hain-Lust operator whenever quasiregularity conditions are not satisfied

Eigenvalues of the Radially Symmetric p ‐Laplacian in R n

For the p-Laplacian p = div:(| |p–2), p>1, the eigenvalue problem –p + q(|x|)||p–2 = ||p–2 in Rn is considered under the assumption of radial symmetry. For a first class of potentials q(r) as r at a sufficiently fast rate, the existence of a sequence of eigenvalues k if k is shown with eigenfunctions belonging to Lp(Rn). In the case p=2, this corresponds to Weyl's limit point theory. For a second class of power-like potentials q(r)– as r at a sufficiently fast rate, it is shown that, under an additional boundary condition at r=, which generalizes the Lagrange bracket, there exists a doubly infinite sequence of eigenvalues k with k ± if k±. In this case, every solution of the initial value problem belongs to Lp(Rn). For p=2, this situation corresponds to Weyl's limit circle theory.

Quantum singularity of Levi-Civita spacetimes

Quantum singularities in general relativistic spacetimes are determined by the behaviour of quantum test particles. A static spacetime is quantum mechanically singular if the spatial portion of the wave operator is not essentially self-adjoint. Here Weyl's limit point–limit circle criterion is used to determine whether a wave operator is essentially self-adjoint. This test is then applied to scalar wave packets in Levi-Civita spacetimes to help elucidate the physical properties of the spacetimes in terms of their metric parameters.

A limit point criterion with applications to nonselfadjoint equations

A unified approach is given which allows one to construct examples of all three cases resulting from a generalization of the classical Weyl limit point limit circle analysis, for differential equations as well as for difference equations.

On Generalized Friedrichs and Krein-von Neumann Extensions and Canonical Systems

Let S be a densely defined and closed symmetric relation in a Hilbert space ℋ︁ with defect numbers (1,1), and let A be some of its canonical selfadjoint extensions. According to Krein's formula, to S and A corresponds a so-called Q-function from the Nevanlinna class N. In this note we show to which subclasses Nγ of N the Q-functions corresponding to S and its canonical selfadjoint extensions belong and specify the Q-functions of the generalized Friedrichs and Krein-von Neumann extensions. A result of L. de Branges implies that to each function Q ∈ N there corresponds a unique Hamiltonian H such that Q is the Titchmarsh-Weyl coefficient of the two-dimensional canonical system Jy′ = —zHy on [0, ∞) where Weyl's limit point case prevails at ∞. Then the boundary condition y(0) = 0 corresponds to a symmetric relation Tmin with defect numbers (1,1) in the Hilbert space L2H, and Q is equal to the Q-function with respect to the extension corresponding to the boundary condition y1(0) = 0. If H satisfies some growth conditions at 0 or ∞, wepresent results on the corresponding Q-functions and show under which conditions the generalized Friedrichs or Krein-von Neumann extension exists.

Dissipative Operators Generated by the Sturm–Liouville Differential Expression in the Weyl Limit Circle Case

In this paper, using Livšic's theorem, we investigate the problem of completeness of the system of eigenfunctions and associated functions of dissipative operators generated by the Sturm–Liouville differential expression on the semi-axis in Weyl's limit-circle case.

Small perturbations of canonical systems

We consider a singular two-dimensional canonical systemJy′=−zHy on [0, ∞) such that at ∞ Weyl's limit point case holds. HereH is a measurable, real and nonnegative definite matrix function, called Hamiltonian. From results of L. de Branges it follows that the correspondence between canonical systems and their Titchmarsh-Weyl coefficients is a bijection between the class of all Hamiltonians with trH=1 and the class of Nevanlinna functions. In this note we show how the HamiltonianH of a canonical system changes if its Titchmarsh-Weyl coefficient or the corresponding spectral measure undergoes certain small perturbations. This generalizes results of H. Dym and N. Kravitsky for so-called vibrating strings, in particular a generalization of a construction principle of I.M. Gelfand and B.M. Levitan can be shown.

Qualitative properties of the Dirac equation in a central potential

The Dirac equation for a massive spin-1/2 field in a central potential V in three dimensions is studied without fixing a priori the functional form of V. The second-order equations for the radial parts of the spinor wavefunction are shown to involve a squared Dirac operator for the free case, whose essential self-adjointness is proved by using the Weyl limit point-limit circle criterion, and a `perturbation' resulting from the potential. One then finds that a potential of Coulomb type in the Dirac equation leads to a potential term in the above second-order equations which is not even infinitesimally form-bounded with respect to the free operator. Moreover, the conditions ensuring essential self-adjointness of the second-order operators in the interacting case are changed with respect to the free case, i.e. they are expressed by a majorization involving the parameter in the Coulomb potential and the angular momentum quantum number. The same methods are applied to the analysis of coupled eigenvalue equations when the anomalous magnetic moment of the electron is not neglected.

Essential self-adjointness in 1-loop quantum cosmology

The quantization of closed cosmologies makes it necessary to study squared Dirac operators on closed intervals and the corresponding quantum amplitudes. This paper shows that the proof of essential self-adjointness of these second-order elliptic operators is related to Weyl's limit point criterion, and to the properties of continuous potentials which are positive near zero and are bounded on the interval .

Indefinite metric in Schur's interpolation problem for analytic functions. V

A theorem on the factorization of the radii of the Weyl limit circle, arising in the Schur interpolation problem, is proved by the methods of the J-theory.

Computation of the regularized trace of the Sturm-Liouville operator in the weyl limit circle case

Computation of the regularized trace of the Sturm-Liouville operator in the weyl limit circle case

20.—On the Limit-point and Limit-circle Theory of Second-order Differential Equations

SynopsisIn this paper the Weyl limit-point and limit-circle theory of second-order differential equations is extended to the case that the weight function is allowed to take on both positive and negative values—the polar case. This extension is achieved using Weyl's limit circle method.