Unique Self-adjoint Extension Research Articles

Dirac–Coulomb operators with general charge distribution I. Distinguished extension and min-max formulas

This paper is the first of a series where we study the spectral properties of Dirac operators with the Coulomb potential generated by any finite signed charge distribution μ. We show here that the operator has a unique distinguished self-adjoint extension under the sole condition that μ has no atom of weight larger than or equal to one. Then we discuss the case of a positive measure and characterize the domain using a quadratic form associated with the upper spinor, following earlier works [EL07, EL08] by Esteban and Loss. This allows us to provide min-max formulas for the eigenvalues in the gap. In the event that some eigenvalues have dived into the negative continuum, the min-max formulas remain valid for the remaining ones. At the end of the paper we also discuss the case of multi-center Dirac–Coulomb operators corresponding to μ being a finite sum of deltas.

Self-adjoint indefinite Laplacians

Let $\Omega_-$ and $\Omega_+$ be two bounded smooth domains in $\mathbb{R}^n$, $n\ge 2$, separated by a hypersurface $\Sigma$. For $\mu>0$, consider the function $h_\mu=1_{\Omega_-}-\mu 1_{\Omega_+}$. We discuss self-adjoint realizations of the operator $L_{\mu}=-\nabla\cdot h_\mu \nabla$ in $L^2(\Omega_-\cup\Omega_+)$ with the Dirichlet condition at the exterior boundary. We show that $L_\mu$ is always essentially self-adjoint on the natural domain (corresponding to transmission-type boundary conditions at the interface $\Sigma$) and study some properties of its unique self-adjoint extension $\mathcal{L}_\mu:=\overline{L_\mu}$. If $\mu\ne 1$, then $\mathcal{L}_\mu$ simply coincides with $L_\mu$ and has compact resolvent. If $n=2$, then $\mathcal{L}_1$ has a non-empty essential spectrum, $\sigma_\mathrm{ess}(\mathcal{L}_{1})=\{0\}$. If $n\ge 3$, the spectral properties of $\mathcal{L}_1$ depend on the geometry of $\Sigma$. In particular, it has compact resolvent if $\Sigma$ is the union of disjoint strictly convex hypersurfaces, but can have a non-empty essential spectrum if a part of $\Sigma$ is flat. Our construction features the method of boundary triplets, and the problem is reduced to finding the self-adjoint extensions of a pseudodifferential operator on $\Sigma$. We discuss some links between the resulting self-adjoint operator $\mathcal{L}_\mu$ and some effects observed in negative-index materials.

Hyperspherical [formula omitted] potentials

The spherically symmetric potential aδ(r−r0)+bδ′(r−r0) is generalised for the d-dimensional space as a characterisation of a unique selfadjoint extension of the free Hamiltonian. For this extension of the Dirac delta, the spectrum of negative, zero and positive energy states is studied in d≥2, providing numerical results for the expectation value of the radius as a function of the free parameters of the potential. Remarkably, only if d=2 the δ-δ′ potential for arbitrary a>0 admits a bound state with zero angular momentum.

Quantum particle probe of the Kerr naked singularity

We investigate Kerr’s timelike naked singularity within the framework of quantum mechanics. A quantum particle in the form of a massive boson is sent in the plane \(\theta = \pi/2\) to the naked ring singularity of Kerr which develops for the overspinning case \((a>M)\) to test it from a quantum picture. This singularity is analysed in two different coordinate systems. We show that the spatial operator of the Klein-Gordon equation both in Boyer-Lindquist and in the dragging coordinate systems has a unique self-adjoint extension. As a result, the classical Kerr’s ring singularity becomes quantum regular, if it is probed with massive bosonic particles obeying the Klein-Gordon equation.

Anisotropic models are unitary: A rejuvenation of standard quantum cosmology

The present work proves that the folklore of the pathology of non-conservation of probability in quantum anisotropic models is wrong. It is shown in full generality that all operator ordering can lead to a Hamiltonian with a self-adjoint extension as long as it is constructed as a symmetric operator. It is indicated that the self-adjoint extension, however, is not unique and this non-uniqueness is suspected not to be a feature of anisotropic models only, in the sense that there exists operator orderings such that Hamiltonian for an isotropic homogeneous cosmological model does not have unique self-adjoint extension. For isotropic model, there is a special unique extension associated with quadratic form of Hamiltonian, i.e., a Friedrich’s extension. Details of calculations are carried out for a Bianchi III model as an example.

The spectrum of the sub-Laplacian on the Heisenberg group

We give a complete analysis of the spectrum of the unique self-adjoint extension of the sub-Laplacian on the one-dimensional Heisenberg group.

Degenerate elliptic operators in one dimension

Let H be the symmetric second-order differential operator on L 2(R) with domain \({C_c^\infty({\bf R})}\) and action \({H\varphi=-(c\,\varphi^{\prime})^{\prime}}\) where \({ c\in W^{1,2}_{\rm loc}({\bf R})}\) is a real function that is strictly positive on \({{\bf R}\backslash\{0\}}\) but with c(0) = 0. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of H. In particular if \({\nu=\nu_+\vee\nu_-}\) where \({\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}}\) then H has a unique self-adjoint extension if and only if \({\nu\not\in L_2(0,1)}\) and a unique submarkovian extension if and only if \({\nu\not\in L_\infty(0,1)}\). In both cases, the corresponding semigroup leaves L 2(0,∞) and L 2(−∞,0) invariant. In addition, we prove that for a general non-negative \({ c\in W^{1,\infty}_{\rm loc}({\bf R})}\) the corresponding operator H has a unique submarkovian extension.

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

We study spectral properties for H K , Ω , the Krein–von Neumann extension of the perturbed Laplacian − Δ + V defined on C 0 ∞ ( Ω ) , where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ R n belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C 1 , r , r > 1 / 2 . In particular, in the aforementioned context we establish the Weyl asymptotic formula # { j ∈ N | λ K , Ω , j ⩽ λ } = ( 2 π ) − n v n | Ω | λ n / 2 + O ( λ ( n − ( 1 / 2 ) ) / 2 ) as λ → ∞ , where v n = π n / 2 / Γ ( ( n / 2 ) + 1 ) denotes the volume of the unit ball in R n , and λ K , Ω , j , j ∈ N , are the non-zero eigenvalues of H K , Ω , listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein–von Neumann extension of − Δ + V defined on C 0 ∞ ( Ω ) ) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980s. Our work builds on that of Grubb in the early 1980s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting. We also study certain exterior-type domains Ω = R n ∖ K , n ⩾ 3 , with K ⊂ R n compact and vanishing Bessel capacity B 2 , 2 ( K ) = 0 , to prove equality of Friedrichs and Krein Laplacians in L 2 ( Ω ; d n x ) , that is, − Δ | C 0 ∞ ( Ω ) has a unique nonnegative self-adjoint extension in L 2 ( Ω ; d n x ) .

The initial value problem for linearized gravitational perturbations of the Schwarzschild naked singularity

The coupled equations for the scalar modes of the linearized Einstein equations around Schwarzschild's spacetime were reduced by Zerilli to a (1+1) wave equation , where is the Zerilli ‘Hamiltonian’ and x is the tortoise radial coordinate. From its definition, for smooth metric perturbations the field Ψz is singular at rs = −6M/(ℓ − 1)(ℓ +2), with ℓ being the mode harmonic number. The equation Ψz obeys is also singular, since V has a second-order pole at rs. This is irrelevant to the black hole exterior stability problem, where r > 2M > 0, and rs < 0, but it introduces a non-trivial problem in the naked singular case where M < 0, then rs > 0, and the singularity appears in the relevant range of r (0 < r < ∞). We solve this problem by developing a new approach to the evolution of the even mode, based on a new gauge invariant function, , that is a regular function of the metric perturbation for any value of M. The relation of to Ψz is provided by an intertwiner operator. The spatial pieces of the (1 + 1) wave equations that and Ψz obey are related as a supersymmetric pair of quantum Hamiltonians and . For has a regular potential and a unique self-adjoint extension in a domain defined by a physically motivated boundary condition at r = 0. This allows us to address the issue of evolution of gravitational perturbations in this non-globally hyperbolic background. This formulation is used to complete the proof of the linear instability of the Schwarzschild naked singularity, by showing that a previously found unstable mode belongs to a complete basis of in , and thus is excitable by generic initial data. This is further illustrated by numerically solving the linearized equations for suitably chosen initial data.

On the essential spectrum of the relativistic magnetic Schrödinger operator

where λ ( ξ) = (|ξ − ( )|2 + 1)1/2. For simplicity we suppose here that the mass of the particle is equal to 1. All the differential and pseudodifferential operators considered in this paper are, possibly unbounded, operators in 2(R ), defined on the Schwartz space S of rapidly decreasing smooth functions on R . In [14] it was proved that if the derivatives of any positive order of ∈ ∞(R ; R ) are bounded, then (λ ) is essentially selfadjoint on S. Let be its unique selfadjoint extension. In [16] the authors proved that if itself is bounded and if all its derivatives converge to zero at infinity, then the essential spectrum of is equal to the essential spectrum of √ + 1, where is the quantum nonrelativistic magnetic Hamiltonian with vector potential , i.e. the selfadjoint operator generated by the differential operator ( − ( ))2. We shall prove in this paper that the essential spectra of and of √ + 1 are still equal if we drop the condition of boundedness of . Thus, vector potentials which behave at infinity as | |1−e, e positive and arbitrary small, are allowed. More precisely, the main result of the paper is the following theorem.

Limit points of eigenvalues of truncated tridiagonal operators

Let T be the tridiagonal operator Te n = a n e n+1 + a n−1 e n−1 + b n e n , Te 1= a 1 e 2+ b 1 e 1, acting on a fixed orthonormal basis { e n }, n=1,2,…, of a Hilbert space H. Let P N be the orthogonal projection on the finite-dimensional space H N spanned by the elements { e 1, e 2,…, e N } and let T N be the truncated operator T N = P N TP N . If T has a unique self-adjoint extension then the set Λ(T)={λ : there exists a sequence of eigenvalues λ N of T N with the property λ N→λ} contains the spectrum σ( T) of T and examples show that, in general, σ( T)≠ Λ( T). For many reasons, the knowledge of the equality σ( T)= Λ( T) is important. In this paper sufficient conditions are presented such that σ( T)= Λ( T).

Functional integration on two-dimensional Regge geometries

By adopting the standard definition of diffeomorphisms for a Regge surface we give an exact expression of the Liouville action both for the sphere and the torus topology in the discretized case. The results are obtained in a general way by choosing the unique self-adjoint extension of the Lichnerowicz operator satisfying the Riemann-Roch relation. We also give the explicit form of the integration measure for the conformal factor. For the sphere topology the theory is exactly invariant under the SL(2, C ) transformations, while for the torus topology we have exact translational and modular invariance. In the continuum limit the results flow into the well-known expressions.

On the Schrödinger functional in QCD

In a series of publications (Nucl. Phys. B384 (1994) 168; B389 (1994) 247), Lüscher et al. have demonstrated the usefulness of the Schrödinger functional in pure SU(2) and SU(3) gauge theory. In this paper, it is shown how their formalism can be extended to include fermions. In the framework of Wilson's lattice QCD, we define the Schrödinger functional by making use of the transfer matrix formalism. Boundary conditions for the fermions arise naturally. We then take the naive continuum limit of the action and show that no lattice peculiarities are left over. The corresponding free Dirac operator has a unique self-adjoint extension with purely discrete spectrum and no zero-modes.

Boundary conditions from path integrals.

A particle moving in two dimensions in a repulsive ${r}^{\ensuremath{-}2}$ potential, and in the presence of a magnetic flux line, is examined as an example of a system for which the Schr\odinger Hamiltonian is not essentially self-adjoint. The path-integral propagator is, nevertheless, shown to exist, and to define a unique self-adjoint extension of the Hamiltonian, corresponding to specific boundary conditions at the origin. Physical implications are discussed.

An example of a Schroedinger equation with almost periodic potential and nowhere dense spectrum

with a continuous periodic potential q(x) = q(x + l) with period l > 0 is given by an infinite or finite sequence of intervals extending to +o0 In the latter case one of the intervals extends to infinity. Here we consider the unique selfadjoint extension of the above operator considered in Comp(R), the space of twice continuously differential functions on the real line R with compact support. Such a selfadjoint extension, with dense domain in L2(R), is unique since this problem is in the limit point Little is known about the spectrum of such operators if q(x) is a continuous almost periodic function, since one has too little information about the solutions of the differential equation L& = h~b in this case. One knows examples of some hyperelliptic functions q(x), which are almost periodic, even quasi-periodic for which the spectrum again consists of a finite number of intervals, one of which extends to +oo. These hyperelliptic functions are, in fact, meromorphic on the Riemann surface constructed from two copies of the complex plane slit along the intervals which constitute the spectrum. For these considerations see Dubrovin,

Self-adjointness of certain second order differential operators on Riemannian manifolds

where u E C,“(R”) and Dj = (a/a X, ib,(x)). The problem of its essential self-adjointness (i.e., “when does L have a unique self-adjoint extension ?“) has been widely investigated. When the context is changed from Euclidean space to a more general differentiable manifold, however, there is a paucity of results. The salient theorems in the latter setting are those of Gaffney [7] in 1951 on the Laplace-Beltrami operator on a complete Riemannian manifold, and Cordes [5] in 1972 on its powers. Most recently Chernoff [4] has obtained a result on a type of general second order elliptic operator with singular potential. The reason that self-adjointness theorems on manifolds are hard to come by is that is usually no single set of coefficient functions that describes the operator. One is only given such functions locally. Generally speaking this difficulty must be handled in one of two ways. The first is by means of global (i.e., coordinate free) invariants associated with the operator. The second is to argue and estimate locally in such a way ultimately things can be “pieced together” into a global and workable result. Using each of these approaches we extend to complete Riemannian manifolds two recent results on Euclidean space of A. Devinatz [6]. In the first, the potential is assumed to be in Li,, and bounded below locally with decay to minus infinity tied to a quantity whose Euclidean formulation is 1 x i -2 x a,,(~)lr,x, . In the manifold setting this quantity becomes a(~; dr(x)), where a(x, V) is the symbol of the operator and dr(zc) is the differential of a general distance function. The key to the proof is Proposition 3.1 in which we establish certain global smoothness

The essential self-adjointness of Schrödinger-type operators

The paper discusses conditions under which the formally self-adjoint elliptic differential operator in R m given by 1 $$\tau {\text{ }}u = \sum\limits_{j,{\text{ }}k = 1}^m {[i\partial _j + b_j (x)]} {\text{ }}a_{jk} (x){\text{ }}[i\partial _k + b_k (x)]{\text{ }}u + q(x){\text{ }}u$$ has a unique self-adjoint extension. The novel feature is that the major conditions on the coefficients have to be imposed only in an increasing sequence of shell-like regions surrounding the origin. On the other hand it is shown that if these shells are broken so as to allow a tube extending to infinity in which the conditions on the coefficients are too weak, then, regardless of the coefficients elsewhere, there may not be a unique self-adjoint extension. The mathematical theorems are linked to the quantum-mechanical interpretation of essential self-adjointness (in the case that τ is the Schrodinger operator), that there is a unique self-adjoint extension if the particle cannot escape to infinity in a finite time.

On the Strong and Weak Limit-Point Classification of Second-Order Differential Expressions†

Abstract : The concept of limit-point for a formally self-adjoint ordinary differential operator of the second order is associated with the existence of a unique self-adjoint extension. A concept of strongly limit-point has also been defined, and this paper considers the circumstances under which the two concepts are or are not equivalent. (Author)

On the Unique Self-Adjoint Extension of the Dirac Operator and the Existence of the Green Matrix

On the Unique Self-Adjoint Extension of the Dirac Operator and the Existence of the Green Matrix

Some Converging Examples of the Perturbation Series in the Quantum Field Theory

Properties of the Hamiltonian operator of the quantized field is studied in the framework of the theory of Hilbert space. The fixed source theory and the boson-fermion interaction are mainly investigated in the cases of both discrete and continuous spectra. The total Hamiltonian operator is defined first in a domain dense in the Hilbert space, under the con· clition that the interaction form factor in the momentum space is square integrable. Then it is shown that the total Hamiltonian as a self-adjoint operator can be determined in terms of the perturbation series for every finite value of the coupling constant if the boson mass is not zero and that it is the unique self-adjoint extension of the symmetric operator defined initially. For the boson-fermion interaction with continuous spectrum, some additional con· clition on the interaction form factor is needed. Further the perturbation series of the S-matrix element for scattering is discussed in the framework of the wave packet for· mulation. In the quantum :field theory the problem of the convergence of the pertur­ bation series expressing the solution of the fundamental equation is of both practical and theoretical importance and has been studied by many authors from various points of view. The difficulty of the diverging integral which appears in each term of the perturbation series was successfully removed by the method of renormalization. · In this note we study the other important problem, that is, the summability of the power series, introducing from the beginning a so-called form factor in the interaction Hamiltonian and making each term of the series finite without using the renormalization technique. Some attempts have been made in this direction. For scalar boson :field if;(x) with self-interaction J.q}, it was shown 1 > that the Green function or the S-matrix can never be written in a power series of )., because the number of the Feynman diagrams in the n-th order is roughly proportional to n! and the lower bound of the absolute value of each matrix element corresponding to the Feynman diagram in that order is an with a being some positive number independent of n, and a fabulous cancellation of the contributions within the same order does not occur. For the boson field interacting with the fermion current, a considerable cancellation between the various terms seems to happen/> al