Safronov, O L

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EXT-2000-05701/09/1999SPECTRAL SHIFT FUNCTION IN THE LARGECOUPLING CONSTANT LIMITO. SAFRONOVAbstract. Given two selfadjoint operators H0and V = V+ V ,we study the motion of the spectrum of the operator H() =H0+ V as  increases. Let  be a real number. We considerthe quantity (;H();H0) dened as a generalization of Krein'sspectral shift function of the pair H(); H0. We study the as-ymptotic behavior of (;H();H0) as  ! 1: Applications todierential operators are given.0. IntroductionLet H0= H0be a selfadjoint operator and let V = Vbe of theformV = V+  V : (1)We putH() = H0+ V;  > 0and denote by (; ) = (;H();H0) the \generalized spectral shiftfunction" of the pairH(); H0which coincides with the Kreins spectralshift function if V is of trace class. For trace class operators V thefunction (; ) 2 L1can be dened by the relationTr[(H())  (H0)] =Z+1 1(; )0()d;called the trace formula. ( Here  is an arbitrary function on R ofa suitable class.) The main object of our study is the leading term(in the power expansion) of the asymptotics of (; ) as !1: Wend conditions on H0; V ensuring the stability of such an asymptoticsunder variations of  2 R. A similar theorem for negative V =  V wasobtained in [7]. We treat the more dicult case of the perturbations ofthe form (1). It should be mentioned also that the idea of the stabilitytheorem is presented in [1], where only the discrete spectrum of H()was studied.In x1 we formulate the problem and describe the main result in detail.It should be noted that if  2 (H0) is a regular point for H0, then(; ) coincides with the dierence between the number of eigenvalues12 O. SAFRONOVof H() having crossed  in each of the two directions. This allowsus to study the asymptotics of (; ) for  2 (H0) with the helpof a special perturbation theory developed in [8], [9]. The stabilitytheorem obtained in this paper reduces the case of an arbitrary  tothat of  2 (H0). In applications to dierential operators, this oersa possibility of using known results.If V < 0,then the function (; ) is monotone decreasing in  (fora xed  = ) and coincides with a certain integral of the distributionfunction of a compact selfadjoint operator. A suitable version of therepresentation formula can be found in [6], [7]. In [7], the said aboveforms a basis of the proof of a stability theorem for V < 0:The study of the perturbations (1) requires, st of all, a new versionof representation formula for (; ). Such a formula is obtained in [4].Moreover, the technique used in [8], [9] can be also simplied with thehelp of this representation.Acknowledgments. The author is grateful to A. Pushnitskii forhis suggestions and remarks to the text.1. Main result (Theorem 1.1)1. Notations Throughout the paper formulae and statements withdouble indices ( or ) are understood independently, as pairs of for-mulae. Below H is a separable Hilbert space. By (T ), (T ) we denoterespectively the resolvent set, the spectrum of a linear operator T . Fora selfadjoint operator T let ET() be the spectral measure of a Borelset   R and2T:= jT j  T:By S1we denote the space of compact operators. For T = T2 S1and s > 0 let n(s; T ) = rankET(s;+1), and for T 2 S1and s > 0let n(s; T ) = n+(s2;TT ). Recall (see, e.g., [2]) that for a pair ofcompact operators the following two inequalities hold:n(s1+ s2; T1+ T2)  n(s1; T1) + n(s2; T2); s1; s2> 0;and (the Horn inequality)n(s1s2; T1T2)  n(s1; T1) + n(s2; T2); s1; s2> 0: (1.1)These inequalities are applicable not only to compact operators but atleast to all bounded normal operators for which the right hand side isnite. Moreover one can write the estimates which are equivalent tothe Weyl inequalities for selfadjoint operators T1, T2:n(s1+ s2; T1+ T2)  n(s1; T1) + n(s2; T2); s1; s2> 0:SPECTRAL SHIFT FUNCTION 3For 0 < p < 1 a class Spis dened as the set of all compactoperators T such that the following functional is nite:kTkpSp:= pZ10sp 1n(s; T )ds <1:Functional k  kSpis a norm for p  1 and a quasinorm for p < 1. For0 < p <1 a class pis dened as the set of all compact operators Tsuch that the following functional (which is a quasinorm) is nite:kTkpp:= sups>0spn(s; T ) <1:2. Let R = Rbe a bounded operator and let the spectrum of R inthe interval  = (a; b) is disctrete or empty. Then for every selfadjointcompact operator K and  2  we introduce the number(;R+K;R) := ind(ER+K( 1; ); ER( 1; ));whereind(P;Q) = dim Ker(P  Q  I)  dim Ker(P  Q+ I):Some properties of the function  should be mentioned here. For ex-ample we have the \addition property":(;R+K1+K2; R) == (;R+K1+K2; R+K1) + (;R +K1; R);which holds for Kj= Kj2 S1, j = 1; 2: And for signdenite pertur-bations K the following Birman-Schwinger principle holds true.Proposition 1.1. Let K  0 and  2 (R). Then(;R +K;R) =  lims!10n(s; T ); (1.2)whereT =qjKj(R   I) 1qjKj:Proof. In fact Proposition 1.1 is a slight modication of Corollary 4.8of [4]. In particular it was shown that under the conditions K  0,K 2 S1,  2 (R + K) the value of (;R + K;R) coincides withthe number of eigenvalues of R + tK which pass  as t grows from 0to 1. Thus, according to the classical Birman-Schwinger principle (seefor example [1]) the relation (1.2) holds true for K 2 S1. Moreover itwas proved in [4] that if  2 (R+K), then the left hand side of (1.2)is continuous with respect to small perturbations of K in the operatornorm. Therefore for general operators K 2 S1;  2 (R + K), theequality (1.2) is obtained by approximation K by operators from S1.Now note that both sides of (1.2) are left continuous with respect to, therefore we get rid of the assumption  2 (R +K). 24 O. SAFRONOV3. Let H0; V be selfadjoint operators in the Hilbert space H andJ = Jbe the sign of V . SupposejV j1=2jH0+ iIj 1=22 S1:The family of selfadjoint operatorsH() = H0+ V;  > 0;is understood in the sense of quadratic forms (see [10]). For a realvalued function f() (dened on R) such thatf(1) > 0;we introduce the operatorVf= f(J)jV jand dene the family of perturbed operatorsHf() = H0+ Vf;  > 0:The main question of our paper is how to describe the motion of thespectrum of H() as  grows. The family of operators Hf() playsonly an auxiliary role. We shall always assume that the limitX:= limz!+i0qjV j(H0  zI) 1qjV j (1.3)exists in the operator norm for a.e.  and for these B:= ImX2 S1: (1.4)Below we also use the notationA:= ReX;  2 R:For a.e.  2 R; we dene the \generalized spectral shift function" forthe pair of selfadjoint operators H();H0as follows(;H();H0) ==Z+1 1(0;J + (A+ tB); J)dw(t); (1.5)dw(t) =  1(1 + t2) 1dt:Note that for trace class perturbations V the function  coincides withKrein's spectral shift function (see [4]).If the limit (1.3) exists and the relation (1.4) is fullled only for a.e.  2 , where  is a measurable subset of R then we say that thegeneralized spectral shift function (1.5) is dened on .SPECTRAL SHIFT FUNCTION 54. Here we present our main result which requires some additionalconditions. First of all we shall assume that there exists a point  = 2 (H0) and a number p > 0 such that(;Hf();H0)  (C+jf(1)jp  C jf( 1)jp)p; (1.6)!1; f(1) > 0;where the constants Cdo not depend on f (but may depend on V).Moreover, denotingQ = Q(t) := (A+ tB X);we also require thatZ+1 1n(;Q(t))(1 + t2) 1dt = o(p); !1; 8 > 0:(1.7)Theorem 1.1. Let  =  2 (H0) and let the conditions (1.6),(1.7) be fullled. Then for the function () := (;H();H0)  (;H();H0)the following relation holds for a.e.  2 R () = o(p); !1:2. Proof of Theorem 1.11. We start with the following auxiliaryProposition 2.1. Assume that f( 1) < 0 and f(1) > 0. Then(;Hf();H0) ==Z+1 1(0; f(J) 1+ (A+ tB); f(J) 1)dw(t); (2.8)dw(t) =  1(1 + t2) 1dt:Proof. For every pair of selfadjoint bounded operators R;K, suchthat 0 2 (R) and K 2 S1we introduce the operatorsRs: = SRS; Ks= SKS;where S = Sis a bounded invertible operator. Then in order toestablish (2.8) it is sucient to prove that(0;Rs+Ks; Rs) = (0;R+K;R): (2.9)Moreover the substantiation of (2.9) can be reduced to the cases K  0and K  0. But if K is of denite sign, the quantity (0;R +K;R)coincides with the number of eigenvalues of the operator R+ tK which6 O. SAFRONOVpass point 0 as t grows from 0 to 1. Therefore (2.9) follows fromequivalence of the two statements:0 2 (R+ tK), 0 2 (Rs+ tKs):Now to complete the proof let us take S = jf(J)j 1=2; R = signf(J)and Ks= (A+ tB) in (2.9). 22. The rest of the section is devoted to the proof of Theorem 1.1.Consider the function(0;J + (A+ tB); J):In order to compare this function with (0;J + X; J);  2 (H0);we use its additivity :() := (0;J + (A+ tB); J)  (0;J + X; J) == (0;J + (A+ tB); J + X):Denoting Y := f 2 R+: 0 2 (J + X)g and using the Birman-Schwinger principle (see Proposition 1.1) we obtain n (1   0; T(+))  ()  n+(1; T( ));  2 Y;whereT():=qQ(J + X) 1qQandQ = Q(t) = (A+ tB X):By the Horn inequality (1.1),n(1; T())  2n(;Q) + n( 1; (J + X) 1):SinceZ+1 1n(;Q(t))(1 + t2) 1dt = o(p); !1;it is sucient to establish thatlim sup!1 pn( 1; (J + X) 1)  C()andlim!0C() = 0:First of all note thatn( 1; (J + X) 1) = rankE(J+X)( ; )  (;J + X; J)  ( ;J + X; J) =: m():It is easy to see from the denition of the function , thatm() = (0; (f+(J) + X); f+(J))  (0; (f (J) + X); f (J));SPECTRAL SHIFT FUNCTION 7wheref(J) = J  I:DenotingH() = H0+ f(J) 1jV jand using the representation for (;H();H0) (see (2.8)) we obtainm() = (;H+();H0)  (;H ();H0):Thereforem()  (C++ C )[(1  ) p  (1 + ) p]p; !1;andlim!0lim sup!1 pm() = 0:The proof is complete.3. Limits of compact operatorsHere we present some conditions on V and H0which ensure exis-tence of (1.3) and guarantee (1.4). We begin with the statement whichimmediately follows from the results of [5]. For 0 < q  1 we dene qq= q; if q < 1;q> 1; any number; if q = 1:Proposition 3.1. Assume that for every bounded open interval  R the following inclusion holds:jV j1=2EH0() 2 S2q; 0 < q  1: (3.10)Then for a.e.  2 R the limitX(; ) := limz!+i0qjV jEH0()(H0  zI) 1qjV jexists in the Sq-norm andImX(; ) 2 Sq:The following statement is a direct consequence of Proposition 3.1.Proposition 3.2. [7] Assume that for every bounded interval  RjV j1=2EH0() 2 S2: (3.11)Then for a.e.  2 R the limit (1.3) exists and the condition (1.4) isfullled.The proof of the following statement is a repeat of the proof of The-orem 4.1 in [7].8 O. SAFRONOVProposition 3.3. Let the condition (3.10) be satised andp  q; if q < 1;p > 1; if q = 1:Assume thatjV j1=2jH0+ iIj 1=22 2p(3.12)Then for a.e.  2 R the condition (1.7) holds.4. Regular points of H0Here we present some sucient for (1.6) conditions. For a regularpoint  2 (H0) we introduce the operatorsX:=qV(H0  I) 1qVandX0:=qV+(H0  I) 1qV :Theorem 4.1. Let  =  2 (H0). Assume that there exist con-stants C0 ; C0+such thatn(s;X)  C0s p; s! 0; p > 0; (4.13)andn(s;X0) = o(s p); s! 0: (4.14)Then (1.6) is fullled with C= C0, in particular(;H();H0)  (C0+ C0 )p; !1: (4.15)Proof. It is sucient to establish (4.15). In our case(;H();H0) = (0;J + X; J):We are going to compare this function with(0;J + L; J); L := X++X :We use its additivity :0() := (0;J + X; J)  (0;J + L; J) == (0;J + X; J + L):Denoting Y0:= f 2 R+: 0 2 (J + L)g and using the Birman-Schwinger principle (see Proposition 1.1) we obtain n (1  0; T(+)0)  0()  n+(1; T( )0);  2 Y0;whereT()0:=qQ0(J + L) 1qQ0SPECTRAL SHIFT FUNCTION 9andQ0:= (X  L):By the Horn inequality (1.1),n(1; T())  2n(;Q0) + n( 1; (J + L) 1):Since by (4.14)n(;Q0) = o(p); !1;it is sucient to establish thatlim sup!1 pn( 1; (J + L) 1)  C()andlim!0C() = 0:First of all note thatn( 1; (J + L) 1) = rankE(J+L)( ; )  (;J + L; J)  ( ;J + L; J) =: m0():It is easy to see from the denition of the function , thatm0() = (0; (f+(J) + L); f+(J))  (0; (f (J) + L); f (J));wheref(J) = J  I:It follows from (4.13) that(0; f(J) + L; f(J))  (C0+jf(1)j p  C0 jf( 1)j p); !1;for real f such that f( 1) < 0 and f(1) > 0. Thereforem0()  (C0++ C0 )[(1  ) p  (1 + ) p]p; !1;andlim!0lim sup!1 pm0() = 0:The proof is complete.10 O. SAFRONOV5. ApplicationsIn this section we present some applications of Theorem 1.1 to thestudy of dierential operators.1. Below we writeR:=RRdand denote Dj=  i@@xj; D =  ir =(D1; : : : ;Dd):In the rst example we deal with the Dirac operator perturbed bya decreasing electric potential. Let g = (g1; g2; g3) and g0be (4  4)-Dirac matrices; 1 denotes the unit matrix. The Dirac matrices satisfythe relationsgjgk+ gkgj= jk1; j; k = 0; 1; 2; 3: (5.16)Let us consider the unperturbed Dirac operator in H = L2(R3; C4)H0= g D + g0;g D =  i3Xj=1gj@@xj;and perturb the operator by a real potentialH() = H0+ V;  > 0; (5.17)V 2 L3(R3); V = V: (5.18)The operator (5.17) needs to be correctly dened. Under the condi-tion (5.18) it is impossible to introduce the operator as the dierenceof two operators, but it can be understood in a sense of the sum ofthe sesquilinear forms. This denition could be used not only for semi-bounded operators but for general ones, too. The corresponding schemefor non-semibounded operators is given in [10].The spectrum of the operator H0is absolutely continuous and coversthe complement of the interval  = ( 1; 1). The essential spectrumof the operator A() coincides with the spectrum of H0. Besides, theoperator H() has a discrete spectrum in the gap . It is clear thatthe generalized spectral shift function of the pair H(); H0exists onthe interval  in the sense of (1.5).Theorem 5.1. Let H0be the Dirac operator and  2 . Under thecondition (5.18) the following asymptotics holds(;H();H0)  (5.19) (3) 23(ZV3+dx ZV3 dx);!1:SPECTRAL SHIFT FUNCTION 11Proof. It is sucient to note that the condition (4.13), (4.14) arefullled andC0= (3) 2ZV3dx:For the reference concerning (4.13) see [3]. The relation (4.14) is ob-tained in [9]. In fact the proof of (5.19) can be found also in [9]. 2Now we are going to apply our abstract theorem to the Dirac oper-ator. Note that the condition (3.11) is fullled if and only ifV 2 L1(R3):The inclusion (3.12) follows from the results of [3], but it can be alsofound in [9]. Combining Theorem 1.1 with Proposition 3.3 and Theo-rem 5.1, we obtainTheorem 5.2. Let H0be the Dirac operator. Under the conditionV 2 L3(R3) \ L1(R3) (5.20)the following asymptotics holds for a.e.  2 R(;H();H0)  (5.21) (3) 23(ZV3+dx ZV3 dx);!1:References[1] M. Sh. Birman: Discrete spectrum in the gaps of a continuous one for pertur-bations with large coupling constants. Adv. Sov. Math. 7 (1991), 57-73.[2] M.Sh. Birman, M.Z. Solomyak: Spectral theory of self-adjoint operators inHilbert space. Mathematics and Its Applications, D. Reidel Publishing Com-pany, 1987.[3] M.Sh. Birman, A.A. Laptev: Discrete spectrum of the perturbed Dirac oper-ator. Ark. Mat., 32 13-32, (1994).[4] F. Gesztesy K.A. Makarov, : The  operator and its relation to Krein's spectralshift function, preprint.[5] S. N. Naboko, On boundary problems for analytic operator valued functionswith positive imaginary part, Zap. Nauchn. Seminarov LOMI (Russian) 157(1987), 55{69.[6] A. Pushnitski, Representation for the spectral shift function for perturbationsof a denite sign, St.Petersburg Math. J. 9, No. 6 (1998).[7] A. Pushnitski: Spectral shift function of the Schrodinger operator in the largecoupling constant limit, to appear in Comm. in PDE.[8] O. Safronov: The discrete spectrum in gaps of the continuous one for non-signdenite perturbations with a large coupling constant St Petersburg Math.Journal 8, No 2 (1997).[9] O. Safronov: The discrete spectrum of selfadjoint operators under non-signdenite perturbations with a large coupling constant, Sfb 288 preprintNo 387, Germany.12 O. SAFRONOV[10] D. Yafaev: Mathematical Scattering Theory, Transl. of Math. Monographs105 , AMS.Universitat Potsdam, Institut fur Mathematik, PF 60 15 53, 144 15Potsdam, Germany, e-mail: safronov@rz.uni-potsdam.de

Spectral shift function in the large coupling constant limit

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