On spectral analysis of Dirac operators P. A. Cojuhari AGH - - PowerPoint PPT Presentation
On spectral analysis of Dirac operators P. A. Cojuhari AGH University of Science and Technology Krakw, Poland Operator Theory and Krein Spaces (dedicated to the memory of Hagen Neidhardt) Vienna, December 19-22, 2019 1 / 39 The Dirac
AGH University of Science and Technology Kraków, Poland Operator Theory and Krein Spaces (dedicated to the memory of Hagen Neidhardt) Vienna, December 19-22, 2019
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The Dirac operator (on R3) for a relativistic particle in an external field is formally given by the matrix-valued differential expression h(ψ) = i−1cα · ∇ψ + βmc2ψ + Qψ, where α · ∇ =
3
αj∂xj, x = (x1, x2, x3) ∈ R3, α = (α1, α2, α3), ∇ = (∂x1, ∂x2, ∂x3), ∂xj = ∂/∂xj (j = 1, 2, 3); αj (j = 1, 2, 3) and α4 := β are Dirac matrices, i.e., 4 × 4 Hermitian matrices satisfying the anticommutation relations (or, so-called Clifford’s relations)
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αjαk + αkαj = 2δjkI, j, k = 1, 2, 3, 4, (δjk - Kronecker’s delta and I - the 4 × 4 identity matrix); c is the velocity of light; m is the mass of the particle; is the (reduced) Plank’s constant (we take = 1, in appropriate units ).
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Q ∼ Q(x) (the potential) describes the external field, that in fact represents an operator of multiplication with a 4 × 4 matrix-valued function Q(x) = [qjk(x)], x = (x1, x2, x3) ∈ R3. We assume that Q(x) is a measurable matrix-valued function and, as usually, vanishing at infinity.
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h is defined on four components vector-valued functions (wavefunctions) ψ = (ψ1, ..., ψ4)T. In "standard representation" αj = σj σj
I2 −I2
σj being the Pauli matrices σ1 = 1 1
σ2 = −i i
σ3 = 1 −1
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V.A.Fock, Fundamentals of Quantum Machanics, MIR Publishers, Moscow, 1978. B.Thaller, The Dirac equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. L.I.Schiff, Quantum mechanics, New York-Toronto-London, McGraw-Hill, 1955.
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The Dirac operator H is defined on the Hilbert space L2(R3; C4) with the aid of the differential form h H ∼ H0 + Q - perturbed Dirac operator. The unperturbed operator H0 is the free Dirac operator H0 = −icα · ∇ + βmc2. The free Dirac operator H0 describes the situation in which the relativistic particle is moving freely as it there were no external fields or other particles.
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The potential matrix Q is added to the free Dirac operator H0 to
H ∼ H0 + Q, H0 = −ieα · ∇ + βmc2.
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The free Dirac operator H0 is self-adjoint in L2(R3; C4) with domain D(H0) = W 1
2 (R3; C4) (the Sobolev class).
H0 has purely absolutely continuous spectrum σ(H0) = σac(H0) = (−∞, −mc2] ∪ [mc2, ∞).
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We emphasize from the outset that we shall have to deal mainly with self-adjoint operators, which of course requires special
almost everywhere x ∈ R3, and certain regularity conditions should be required.
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B.Thaller, The Dirac equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. J.Weidman, Linear operators in Hilbert Spaces, Springer-Verlag, Heidelberg, New York, 1980. T.Kato, Perturbation Theory of Linear operators, Springer-Verlag, Heidelberg, New York, 1995. and, also, the references quated there.
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We take the attitude of studying problems in operator - theoretical terms as far as possible and then handling differential operators, in particular, Dirac operators, by applying the obtained results. The
perturbation Q. Among a few ways of expressing H as a perturbation of H0 the so-called factorization scheme will be suitable for our purposes.
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non-selfadjoint operators, Math. Ann., 182(1966), 258-279. S.T.Kuroda. An abstract stationary approach to perturbation
20 (1967)57-117. S.T.Kuroda. Scattering theory for differential operators. I. Operator theory. J. Math. Soc. Japan, 25 (1973)75-104.
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To accurately define the perturbed operator H, the following assumptions are required.
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(A1). The perturbation operator Q can be written formally as Q = AB, where A and B are closed densely defined operators acting B from H to another space K and A from K to H ,
bounded, that is, BR(z; H0) ∈ B(H, K) and the (densely defined)
[R(z; H0)A] on the whole space K.
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(A2). For one, or equivalently all, z ∈ ρ(H0) the densely defined
whole space K, let it be denoted by T(z), i.e., T(z) := [BR(z; H0)A], z ∈ ρ(H0).
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(A3). There exists a regular point z of H0 such that the operator I + T(z) (I denotes the identity operator) is invertible, with its inverse belonging to B(K). (A3’). The values T(z), z ∈ ρ(H0) are compact operators.
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Remark From the assumptions (A1) and (A2) it follows that the range of [R(z; H0)A] is contained in the domain D(B) of B, i.e., the
T(z) = B[R(z; H0)A], z ∈ ρ(H0).
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Under the assumptions (A1), (A2), and (A3) one can define the
R(z) := R(z; H0) − [R(z; H0)A](I + T(z))−1BR(z; H0) for all z ∈ ρ(H0) whenever I + T(z) is invertible as in the assumption (A3).
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It turns out that thus defined operator-valued function R(z) is the resolvent of a closed operator H which is an extension of H0 + Q(= H0 + AB). This will be the definition of our perturbed
H ⊃ H0 + Q If, in addition, the operator Q is symmetric, then the operator thus
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Theorem (Analytic Fredholm Theorem). Let H be a Hilbert space, let Ω be an open and connected domain in the complex plane, and suppose that T(·) is an analytic operator-valued function defined
I + T(z) is nowhere invertible in Ω or else the inverse (I + T(z))−1 exists for all z ∈ Ω except at a countable number of isolated points.
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I.C.Gohberg. On linear operators depending analytically on a
I.C.Gohberg and M. G. Krein. Introduction to the theory of linear non-self-adjoint operators. Translations of Mathematical Monographs, Vol.18. AMS, Providence, R.I., 1969.
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J.D.Tamarkin, On Fredholm’s integral equations, whose kernels are analytic in a parameter, Ann. of Math. (2), 28 (1926/27), 127-152. S.Steinberg, Meromorphic families of compact operators,
M.Reed and B.Simon N.Dunford and J.T.Shwartz, I
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Theorem Under the assumptions (A1), (A2) and (A3) the essential spectra of H0 and H coincide σess(H) = σess(H0).
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Our aim to describe conditions under which the discrete spectrum in a spectral gap of the unperturbed operator is only finite; to estimate the number of the set of perturbed eigenvalues to prove the discreteness of the singular spectrum (modulo a finite number of accumulations points); to obtain conditions for the absence of eigenvalues embedded in the continuous spectrum; to study the problem of spectral asymptotics for Dirac and Pauli operators.
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σd(H)−? M.S. Birman, On the spectrum of Schrödinger and Dirac
M.S. Birman, On the spectrum of singular boundary-value problems, Math. Sb. 55, (1961), 125-174. O.I.Kurbenin, The discrete spectra of the Dirac and Pauli
(1969), 43-92.
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M.Klaus, On the point spectrum of Dirac operators, Helv.
A.Berthier and V.Georgescu, On the point spectrum of Dirac
A.Berthier and V.Georgescu, C. R. Acad. Sci. Paris, Sc. A 291 (1980), 603; 35 (1983). M.S. Birman and A.Laptev, Discrete spectrum of the perturbed Dirac operator, Arh. Mat. 32 (1994), 13-32.
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B.Thaller, The Dirac equation, Texts and Monographs Physiscs, Springer-Verlag, Berlin, (1992). J.Weidman, Linear operators in Hilbert Spaces, Springer-Verlag, Heidelberg, New York, (1980). I.M.Glazman, Direct Methods of Qualitative Spectral Analysis
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σd(H) H ∼ H0 + Q, H0 = −icα · ∇ + βmc2, m > 0 σess(H) = σ(H0) = (−∞, −mc2] ∪ [mc2, +∞) Λ = (−mc2, mc2) - the spectral gap of H0 Λ ⊂ ρ(H0) Q(x) = [qjk(x)], x = (x1, x2, x3) ∈ R3
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P.A.Cojuhari, On the finiteness of the discrete spectrum of the Dirac operator, Reports on Math. Physics, 57 (2006), 333-341. P.A.Cojuhari, On the finiteness of the discrete spectrum of some matrix pseudodifferential operators, Math. Vyssh.
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(i) If lim
|x|→∞ |x|2qjk(x) = 0 (j, k = 1, 2),
lim
|x|→∞ qjk(x) = 0 (j, k = 3, 4),
lim
|x|→∞ |x|qjk(x) = 0 (j = 1, 2; k = 3, 4 and j = 3, 4; k = 1, 2),
then the spectrum of the perturbed Dirac operator H on the spectral gap Λ = (−mc2, mc2) is only discrete and mc2 is not a point of accumulation of σ(H) ∩ Λ.
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(ii) If lim
|x|→∞ qjk(x) = 0 (j, k = 1, 2),
lim
|x|→∞ |x|2qjk(x) = 0 (j, k = 3, 4),
lim
|x|→∞ |x|qjk(x) = 0 (j = 1, 2; k = 3, 4 and j = 3, 4; k = 1, 2),
then −mc2 is not a point of accumulation of σ(H) ∩ Λ.
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If lim
|x|→∞ |x|2qjk(x) = 0 (j, k = 1, 2 and j, k = 3, 4),
lim
|x|→∞ qjk(x) = 0 (j = 1, 2; k = 3, 4 and j = 3, 4; k = 1, 2),
then the spectrum of the Dirac operator H on the interval (−mc2, mc2) is finity.
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The case of a Dirac particle in an electromagnetic field H = cα · (D − e c A) + βmc2 + q(x) D = (D1, D2, D3), Dj = −i∂/∂xj (j = 1, 2, 3); A(x) = (A1(x), A2(x), A3(x)) - a magnetic potential; q(x) - an electrostatic potential (scalar function); e - charge of the particle Q = −eα · A + q(x)
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If lim
|x|→∞ |x|2q(x) = 0,
lim
|x|→∞ |x|Aj(x) = 0 (j, k = 1, 2, 3),
then the spectrum of H on the interval (−mc2, mc2) is finite.
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σp(H)−? Theorem Let H0 be a self-adjoint operator in a Hilbert space H. There exists a self-adjoint operator B from the Schmidt class B2(H) with arbitrarily small norm such that H + B has a pure point spectrum.
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Theorem Let H0 and B be symmetric operators in a space H and let the
defined on Λ such that (i) T(λ) ∈ B∞(H), λ ∈ Λ; (ii) T(λ) is continuous on Λ in the uniform norm topology; (iii) for each λ ∈ Λ and for each u ∈ Dom(B) such that Bu ∈ Ran(H0 − λI) there holds the following inequality (A − λI)−1Bu ≤ T(λ)u. Then the point spectrum of perturbed operator H = H0 + B on the interval Λ consists only of finite number of eigenvalues of finite multiplicity.
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H =
n
αkDk + α0 + Q, Dk = −i∂/∂xk (k = 1, ..., n), αk (k = 0, 1, ..., n) are m × m Hermitian matrices which satisfy anticommutation relations (Clifford’s relations) αjαk + αkαj = 2δjk (j, k = 0, 1, ..., n), m = 2n/2 for n even and m = 2(n+1)/2 for n odd. Q is considered as a perturbation of the free Dirac operator H0 =
n
αkDk + α0, and represents the operator of multiplication by a given Hermitian matrix-valued function Q(x), x ∈ Rn. The operators H0 and H are considered in the space L2(Rn; Cm).
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Theorem (i) If the entries qjk(x) of Q(x) are such that lim
|x|→∞ |x|qjk(x) = 0
(j, k = 1, ..., m) the the point spectrum of the operator H has only ±1 as accumulation points. Each eigenvalue can be only of a finite multiplicity. If qjk(x) ∼ |x|1+δ, |x| → ∞ (δ > 0), then σsc(H) = ∅.
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