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 Kraków, Poland Operator Theory and Krein Spaces (dedicated to the memory of Hagen Neidhardt) Vienna, December 19-22, 2019 1 / 39

The Dirac operator (on R 3 ) for a relativistic particle in an external field is formally given by the matrix-valued differential expression h ( ψ ) = i − 1 � c α · ∇ ψ + β mc 2 ψ + Q ψ, where 3 � x = ( x 1 , x 2 , x 3 ) ∈ R 3 , α · ∇ = α j ∂ x j , j = 1 α = ( α 1 , α 2 , α 3 ) , ∇ = ( ∂ x 1 , ∂ x 2 , ∂ x 3 ) , ∂ x j = ∂/∂ x j ( j = 1 , 2 , 3 ); α j ( j = 1 , 2 , 3 ) and α 4 := β are Dirac matrices, i.e., 4 × 4 Hermitian matrices satisfying the anticommutation relations (or, so-called Clifford’s relations) 2 / 39

α j α k + α k α j = 2 δ jk I , j , k = 1 , 2 , 3 , 4 , ( δ jk - 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 ). 3 / 39

Q ∼ Q ( x ) (the potential) describes the external field, that in fact represents an operator of multiplication with a 4 × 4 matrix-valued function x = ( x 1 , x 2 , x 3 ) ∈ R 3 . Q ( x ) = [ q jk ( x )] , We assume that Q ( x ) is a measurable matrix-valued function and, as usually, vanishing at infinity. 4 / 39

h is defined on four components vector-valued functions (wavefunctions) ψ = ( ψ 1 , ..., ψ 4 ) T . In "standard representation" � 0 � I 2 � � σ j 0 α j = ( j = 1 , 2 , 3 ) , β = , − I 2 σ j 0 0 σ j being the Pauli matrices � 0 � 0 � 1 � � � 1 − i 0 σ 1 = , σ 2 = , σ 3 = . 1 0 i 0 0 − 1 5 / 39

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. 6 / 39

The Dirac operator H is defined on the Hilbert space L 2 ( R 3 ; C 4 ) with the aid of the differential form h H ∼ H 0 + Q - perturbed Dirac operator. The unperturbed operator H 0 is the free Dirac operator H 0 = − ic α · ∇ + β mc 2 . The free Dirac operator H 0 describes the situation in which the relativistic particle is moving freely as it there were no external fields or other particles. 7 / 39

The potential matrix Q is added to the free Dirac operator H 0 to obtain the Dirac operator H in an external field H 0 = − ie α · ∇ + β mc 2 . H ∼ H 0 + Q , 8 / 39

The free Dirac operator H 0 is self-adjoint in L 2 ( R 3 ; C 4 ) with domain D ( H 0 ) = W 1 2 ( R 3 ; C 4 ) (the Sobolev class). H 0 has purely absolutely continuous spectrum σ ( H 0 ) = σ ac ( H 0 ) = ( −∞ , − mc 2 ] ∪ [ mc 2 , ∞ ) . 9 / 39

We emphasize from the outset that we shall have to deal mainly with self-adjoint operators, which of course requires special treatment. In particular, the matrix Q ( x ) has to be Hermitian for almost everywhere x ∈ R 3 , and certain regularity conditions should be required. 10 / 39

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. 11 / 39

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 operator H is considered to be obtained from the operator H 0 by a perturbation Q . Among a few ways of expressing H as a perturbation of H 0 the so-called factorization scheme will be suitable for our purposes. 12 / 39

T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 182(1966), 258-279. S.T.Kuroda. An abstract stationary approach to perturbation ofcontinuous spectra and scattering theory . J. Analyse Math., 20 (1967)57-117. S.T.Kuroda. Scattering theory for differential operators. I. Operator theory. J. Math. Soc. Japan, 25 (1973)75-104. 13 / 39

To accurately define the perturbed operator H, the following assumptions are required. 14 / 39

(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 , respectively. Furthermore, for z ∈ ρ ( H 0 ) the operator BR ( z ; H 0 ) is bounded, that is, BR ( z ; H 0 ) ∈ B ( H , K ) and the (densely defined) operator R ( z ; H 0 ) A has a (it is unique) bounded extension [ R ( z ; H 0 ) A ] on the whole space K . 15 / 39

(A2). For one, or equivalently all, z ∈ ρ ( H 0 ) the densely defined operator BR ( z ; H 0 ) A has a bounded extension [ BR ( z ; H 0 ) A ] on the whole space K , let it be denoted by T ( z ) , i.e., T ( z ) := [ BR ( z ; H 0 ) A ] , z ∈ ρ ( H 0 ) . 16 / 39

(A3). There exists a regular point z of H 0 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 ( H 0 ) are compact operators. 17 / 39

Remark From the assumptions (A1) and (A2) it follows that the range of [ R ( z ; H 0 ) A ] is contained in the domain D ( B ) of B , i.e., the operator B [ R ( z ; H 0 ) A ] is well defined, and T ( z ) = B [ R ( z ; H 0 ) A ] , z ∈ ρ ( H 0 ) . 18 / 39

Under the assumptions (A1), (A2), and (A3) one can define the operator-valued function R ( z ) := R ( z ; H 0 ) − [ R ( z ; H 0 ) A ]( I + T ( z )) − 1 BR ( z ; H 0 ) for all z ∈ ρ ( H 0 ) whenever I + T ( z ) is invertible as in the assumption (A3). 19 / 39

It turns out that thus defined operator-valued function R ( z ) is the resolvent of a closed operator H which is an extension of H 0 + Q (= H 0 + AB ) . This will be the definition of our perturbed operator H ⊃ H 0 + Q If, in addition, the operator Q is symmetric, then the operator thus obtained H is self-adjoint. 20 / 39

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 on Ω with T ( z ) compact operators in H for z ∈ Ω . Then either 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. 21 / 39

I.C.Gohberg. On linear operators depending analytically on a parameter . DAN, 78( 1951)629-632. 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. 22 / 39

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 , Arch. Rational Mech. Anal., 31 (1968/69), 372-379. M.Reed and B.Simon N.Dunford and J.T.Shwartz, I 23 / 39

Theorem Under the assumptions (A1), (A2) and (A3) the essential spectra of H 0 and H coincide σ ess ( H ) = σ ess ( H 0 ) . 24 / 39

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. 25 / 39

σ d ( H ) − ? M.S. Birman, On the spectrum of Schrödinger and Dirac operators , Dokl. Acad. Nauk 129, no.2 (1959), 239-341. 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 operators , Topics in Math. Physics, vol. 3, Spectral Theory (1969), 43-92. 26 / 39

M.Klaus, On the point spectrum of Dirac operators , Helv. Phys. Acta, 53 (1980), 453-462. A.Berthier and V.Georgescu, On the point spectrum of Dirac operators , J. Func. Anal., 71 (1987), 309-338. 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. 27 / 39

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 of Singular Differential Operators , M., (1963). 28 / 39

σ d ( H ) H 0 = − ic α · ∇ + β mc 2 , H ∼ H 0 + Q , m > 0 σ ess ( H ) = σ ( H 0 ) = ( −∞ , − mc 2 ] ∪ [ mc 2 , + ∞ ) Λ = ( − mc 2 , mc 2 ) - the spectral gap of H 0 Λ ⊂ ρ ( H 0 ) x = ( x 1 , x 2 , x 3 ) ∈ R 3 Q ( x ) = [ q jk ( x )] , 29 / 39

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. Uchebn. Zaved. Mat., 1(1989), 42-50; Translation in Soviet Math. (Iz. VUZ) 33 (1989), no. 1, 48-58. 30 / 39