Integral estimates for the spectral shift function A. B. Pushnitski Abstract The spectral shift function () is considered for the pair of operators H

Integral estimates for the spectral shift function

^

A. B. Pushnitski

Abstract

The spectral shift function^(^) is considered for the pair of operators^H0,^H0+^V, where ^H0 is the Schrodinger operator with a variable Riemannian metric and an electro-magnetic eld, and^V is the operator of multiplication by the potential^V(^x).

For integrals of the type^{R }(^)^f(^)^d, where^f(^) is a weight, the estimates in terms of the integral characteristics of the potential^V are obtained. These estimates are of an asymptotically \correct" order in^and ^V; they will be used in a subsequent paper in the problem of asymptotics of ^(^) in the large coupling constant limit.

0 Introduction

0.1

Let H⁰ and H be selfadjoint operators in a Hilbert space ^H and let their dierence, V, be a trace-class operator:

V :=H^?H^{0 2}

S

^{1(H): (0.1)}

Then the Lifshits|Krein trace formula holds [21, 19]:

Tr((H)^?(H⁰)) =^{Z 1}

?1

(;H;H⁰)⁰()d: (0.2) Here  is any function of some class and (;H;H⁰) is the spectral shift function (SSF) for the pair H⁰,H, which is given by the Krein formula

(;H;H⁰) = 1^{"lim!+0argdet(}I+V(H^0?(+i")I)^?1); a. e.  ²

R

^{: (0.3)}

The branch of the argument in (0.3) is xed by the condition argdet(I+V(H^0?zI)^?1)^! 0, Imz ^!+¹. A historical review and description of the modern state of the SSF theory can be found in [8, 36]. Among the most important facts of the SSF theory are (see [19]) the Krein inequality ^Z

1

?1

j(;H;H⁰)^jd^kV^kS1 (0.4)

submitted to Algebra i Analiz (St.Petersburg Math. J.)

1

and the monotonicity of the SSF:

V ^0 ^{) }(;H;H⁰)^0: (0.5) It follows from (0.5) that

(;H^0?V^?;H⁰)^(;H⁰+V;H⁰)^(;H⁰+V⁺;H⁰); (0.6) where 2V^=^jV^jV.

0.2

In applications, instead of (0.1), it is usually possible to check the inclusion h(H)^?h(H⁰)²

S

^{1(H); (0.7)}

where h :

R

^!

R

is a smooth enough monotonic function. In this case, the SSF for the pair H⁰,H is dened by the natural formula

(;H;H⁰) := signh⁰
(h();h(H);h(H⁰)): (0.8) The trace formula (0.2) is, of course, still valid; only the class of admissible functions  has to be changed. The relation (0.5) sometimes can be also justied. But this problem is far from being trivial if the SSF is dened via (0.8). In [18] (see also [36, ^x8.10]), (0.5) has been proved for

h() = (^?⁰)^?k; k >0; ⁰ <inf((H⁰)^[(H)) (0.9) for allk, ⁰ specied. This is sucient for most applications.

0.3

^Let

H⁰ =^?4 inL²(

R

^d); d^1; V =V(x); H =H⁰+V: (0.10) If the potentialV decays at innity rapidly enough, then the inclusion (0.7) can be veried for the function h of the type (0.9). This enables one to dene the SSF via (0.8); the relations (0.5), (0.6) are valid in this case due to the results of [18].

In [35], estimates for the SSF for the pairH⁰,H of the form (0.10) were studied. The potentialV was subjected to the condition

jV(x)^j(1 +^jx^j)^?;  > d:

The following estimate (see [35, theorem 4.2]) was obtained:

(;H⁰+V;H⁰)^C(^(d=2)+^(d=2)?1(^jlog^j+ 1)); ⁸ > 0;^c; (0.11) for V ^0 it can be improved up to

(;H⁰+V;H⁰)^C^(d=2)?1(^jlog^j+ 1); ⁸ >0; ^c: (0.12) Here C, c are some positive numbers which do not depend on the spectral parameter  and the coupling constant, but may depend onV (this dependence has not been studied in [35]).

2

Besides the estimates (0.11), (0.12), there are many results concerning the asymptotics of (;H⁰+V;H⁰) for ^!1 and xed (high energy asymptotics | here the initial results were obtained in [9]) and for  ^{! 1} and  ^{! 1} and xed = (semiclassical asymptotics) | see, e.g., [30] and references therein.

0.4

The above mentioned results on the pointwise estimates and asymptotics for the SSF of the pair (0.10) are based on the fact that the spectrum of H⁰ is absolutely con- tinuous and V is smooth with respect to H⁰ in some sense. In this paper we follow a dierent approach and consider the SSF entirely in the the framework of the trace class perturbation theory. Starting from the Krein inequality (0.4), we obtain integral estimates for the SSF. As an example, we present the weighted estimates for the SSF of the pair (0.10) for d^3 (see Theorem 6.4(ii), Corollary 6.8(ii) below and (0.6)):

Z

1

0

j(;H⁰+V;H⁰)^jf()d^C^{1Z 1}

0

f()d^Z V^?d=2(x)dx+ +C^{2Z 1}

0

^(d=2)?1f()d^{Z j}V(x)^jdx; (0.13) if V ^0, then, obviously, the rst term in the right hand side vanishes and we obtain

Z

1

0

(;H⁰+V;H⁰)f()d^C^{2Z 1}

0

^(d=2)?1f()d^Z V(x)dx: (0.14) Here f = f() is any nonnegative monotone decreasing function and C¹, C² are the constants which do not depend on , V, f. The precise statements are given in ^x6.2{6.3.

Obviously, the pointwise estimates (0.11), (0.12) are stronger than the integral ones (0.13), (0.14) (up to constants and a logarithmic factor). But our method of deriving the estimates is not \sensible" to the quality of the spectrum ofH⁰. This allows us to consider a wide class ofH⁰'s (Schrodinger operators with a variable metric and an electromagnetic eld | see^x5.2 below) and to avoid imposing too restrictive conditions onV. Besides, the estimates (0.13), (0.14), in contrast to (0.11), (0.12), explicitly indicate the dependence onV. We also obtain some analogues of (0.13), (0.14) forH⁰ = (^?4)^l in L²(

R

^d), d^1;

here l is not necessarily an integer.

Note that in [12], for the quantities ^R0(t;H⁰ + V;H⁰)dt and ^R01e^?t(;H⁰ + V;H⁰)d, related to the pair (0.10) for d = 3, the questions of concavity and subad- ditivity with respect toV were studied.

0.5

The present paper is mostly aimed at the applications to the problem of asymp- totics of the SSF in the large coupling constant limit (the asymptotics itself will be considered in the subsequent paper). To a large extent, this has determined the style of the paper. Let V ^ 0; then for  < inf(H⁰) the SSF (;H^0?V;H⁰) equals minus the quantity N⁺(;H⁰;^pV ;), which is the number of eigenvalues of H^0?V, located on the left from the point  on the real axis (see, e. g., [8]). The question of asymp- totics of N⁺(;H⁰;^pV ;) as  ^{! 1} is well studied in applications | see, e. g., [7]

and references therein. In particular, a description of a class of perturbations V (regular perturbations), such that the leading term of the asymptotics does not depend on , is

3

known. For this reason, in order to study the SSF in the large coupling constant limit for regular perturbationsV ^0, it is natural to consider the quantity

(;H^0?V;H⁰) +N⁺(^?;H⁰;^pV ;); (0.15) for some xed ^? <inf(H⁰). The initial estimates for (;H^0?V;H⁰) are formulated in terms of this quantity (see Theorems 6.6, 6.9 below). From this follow inequalities of the form (0.13) (see Corollary 6.8).

0.6

In order to obtain estimates for the SSF, one can, due to (0.6), consider the case of perturbations of a denite sign. Here we use the following approach. Let a nonnegative operator V be factored as V = G^G. In [25], the following new representation for the SSF was obtained (see Propositions 2.6, 2.7 below):

(;H^0V;H⁰) =^N(;H⁰;G): (0.16) Here ^N(;H⁰;G) is an integral (see (2.4)) of a counting function of eigenvalues of a family of compact operators, related to G and to the resolvent of H⁰. The relation (0.16) is of an abstract character. In applications, it appears that the existence of^N can be established under somewhat more general and natural conditions than the existence of the SSF (see ^x6.1). For this reason, we consider the quantities ^N as a basic object and dene them independently of SSF. To analize ^N, we use a mixed approach: both straightforward analysis of the integral representation for ^N and the equality (0.16) together with the estimate (0.4) for \intermediate" objects are employed. This yields integral estimates for ^N(;H⁰;G). The corresponding inequalities for the SSF are ob- tained by using Propositions 2.6 and 2.7, which give sucient conditions for the validity of (0.16).

0.7

The paper is organized as follows. In ^x1 the necessary notation and denitions are introduced. In ^x2, the quantities ^N(;H⁰;G) are dened and their basic properties and connections with the SSF and the \counting function"N^(;H⁰;G;) are discussed.

The estimates for ^N are obtained in ^x3 and for the quantity (0.15) | in ^x4. The results of ^xx2{4 are of an abstract character. In ^x5, the Schrodinger and polyharmonic operators (which, in applications, play the part of H⁰) are introduced and some relevant statements on this operators are presented. Finally, in ^x6, on the basis of the results of

x3{4 the integral estimates for^N(;H⁰;^pV) are derived, whereH⁰ is the Schrodinger or polyharmonic operator andV is the operator of multiplication by the potentialV(x)^0.

0.8 Acknowledgments

The author is deeply grateful to his scientic advisor M. Sh. Birman, who introduced him into the SSF theory, for many valuable discussions and remarks. The author is grateful to V. S. Buslaev, who attracted his attention to the question of integral estimates for the SSF. The author is grateful to A. Laptev and G. Rozenblioum for consultations and to N. D. Filonov for useful remarks on the text of the paper. Finally, the author expresses his gratitude to the Department of Mathemat- ics of the Royal Institute of Technology, Stockholm, for the hospitality and to the ISF foundation for the nancial support.

4

1 Notation and preliminaries

1.1 Notation

1) Integral without the domain of integration explicitly indicated implies integration over

R

^d. We denote !d := vol^fx²

R

^{d jj}x^j1^g. Statements with double indices (^and

) should be considered as a pair of statements, in one of which all the indices take the upper values, and in another | the lower ones. In the statements which involve upper estimates we suppose all the quantities (norms and integrals) in the right hand side to be nite. A constant which appears for the rst time in formula (i:j) is denoted by Ci:j.

2) Functions. The spaces Lp(

R

^{d), Lp;loc(}

R

^d) are dened in a usual way. The space l¹(

Z

^d;L²(

Q

^d)) consists of functions u²L²(

R

^d) such that the functional

ku^kl¹⁽L²⁾ := ^X

j^2Zd

Z

Q d

+j^ju^j2dx^1=2;

Q

^d = (0;1)^{d }

R

^d;

is nite. For a real-valued functionF we put 2F^ :=^jF^jF. The characteristic function of a set M is denoted by M.

3) Operators. Below ^H, ^H1, ^H2 are separable Hilbert spaces. By DomA, RanA and KerA we denote the domain, range and kernel of a linear operator A; ^jA^j:=^pA^A;I is the identity operator. For a selfadjoint operator A the symbols (A), (A) =

C

ⁿ(A) and EA() denote respectively spectrum, resolvent set and a spectral measure of a Borel set  ^

R

^{; 2A := jAj  A}. Resolvent of a selfadjoint operator H⁰ is denoted by R⁰(z) = (H^0?zI)^?1.

By

S

^1(H1;^H2) we denote the space of compact operators acting from ^H1 into ^H2;

S

^{1(H) :=}

S

^{1(H;H). For T = T 2}

S

^{1(H) and s >} 0 we denote n^(s;T) :=

dimRanET^(s;+¹), and for T ²

S

¹(^H1;^H2) we put n(s;T) := n⁺(s²;T^T). For com- pact selfadjoint operators T¹,T² the following estimates hold (see, e.g., [3]):

n^(s¹+s²;T¹+T²)^n^(s¹;T¹) +n^(s²;T²); s¹;s² >0; (1.1) which can be written as

n^(s¹;T¹+T²)^n^(s¹+s²;T¹)^?n^(s²;T²); s¹;s²>0: (1.2) For 1^p <¹ the Neumann-Schatten class

S

p(^H1;^H2)^

S

¹(^H1;^H2) is dened as a set of operators V such that the norm

kT^kSp :=^p^{Z 1}

0

s^p?1n(s;T)ds^1=p

is nite. In particular,

S

¹ is the trace class,

S

² is the Hilbert{Schmidt class.

1.2 Auxiliary facts of the perturbation theory

Let^Hbe a \basic" and^K| an \auxiliary" Hibert space, H⁰ be a selfadjoint operator in ^H and G be a linear operator acting from ^H into^K. Assume that

G is closed; DomG^Dom^jH^0j1=2; G(^jH^0j+I)^{?1=2 2}

S

^{1(H;K): (1.3)}

5

Forz ²(H⁰) dene the operator

T(z;H⁰;G) := (G(^jH^0j+I)^?1=2)(^jH^0j+I)R⁰(z)(G(^jH^0j+I)^?1=2)^; (1.4) which is compact in ^K. We shall write T(z) instead ofT(z;H⁰;G) if the choice of H⁰, G is clear from the context. On the dense in^Kset DomG^ the operatorT(z) can be dened by the expression

T(z) =GR⁰(z)G^ ; ²DomG^: (1.5) It is easy to check that for all z ²

C

ⁿ

R

the operator I ^T(z;H⁰;G) has a bounded inverse. LetV :=G^G. Below we introduce the operator which corresponds to the formal sum H^0V. For some z ²

C

ⁿ

R

dene a bounded operator R^(z) in^H by

R^(z) =R⁰(z)^(GR⁰(z))^(I^T(z;H⁰;G))^?1(GR⁰(z)): (1.6)

Proposition 1.1

(see [14] or [36]) The operator R^(z) coincides with the resolvent of a selfadjoint in ^H operator H^ =H^(H⁰;G) taken at the point z. This operator does not depend onz. For allz ²(H⁰)^\(H^) the operatorI^T(z;H⁰;G) has a bounded inverse and the equality (1.6) holds true for R^(z) = (H^?zI)^?1. If H⁰ is semibounded from below, thenH^ coincides with the sumH^0V in the form sense. IfV isH⁰-bounded with the relative bound <1, then H^ =H^0V in the sense of the Kato-Rellich theorem.

It follows from (1.6) and (1.3) that the dierenceR^(z)^?R⁰(z) is compact; thus, the essential spectra ofH^andH⁰coincide. It is easy to check thatH^(H⁰;G) =H^(H⁰;^jG^j).

Proposition 1.2

[17, Lemma 1] Let the conditions (1.3) hold true. Then, for every

²(H⁰)^\

R

^,

dimKer(H^(H⁰;G)^?I) = dimKer(T(;H⁰;G)^I): (1.7) Proposition 1.2, in particular, implies that the eigenvalues of H^(H⁰;^pG),  > 0, are monotonic functions of . For  ² (H⁰) ^\

R

and  > 0 the \counting func- tion" N^(;H⁰;G;) is dened as the number of eigenvalues (counting multiplicities) of H^(H⁰;^ptG), which cross the point  as t grows monotonically in the interval (0;).

From Proposition 1.2 follows the equality which is known as the Birman-Schwinger prin- ciple:

N^(;H⁰;G;) =n^(^?1;T(;H⁰;G)); ²(H⁰)^\

R

^{: (1.8)}

In this paper, the scope of applications is restricted to the semibounded from below operators H⁰ (Schrodinger and polyharmonic operator). In this case, H^ is dened as a form sum, which simplies the arguments. However, in the abstract part of the paper we do not suppose H⁰ to be semibounded from below (with the exception of ^x4, where it is dictated by the nature of the question).

Note that the condition of closedness ofGis introduced only for the sake of simplicity.

The contents of the present section, as well as that of ^x2{4, can be reformulated for the case of non-closable G's. It is only important that that the second and the third of the conditions (1.3) hold true.

6

2 The functions

^N

2.1 Denition of the functions

^N

In this and the next two sections ^H is a \basic" and ^K | an \auxiliary" Hilbert space,H⁰ is a selfadjoint operator in^H and G| the operator acting from^H into ^Kand satisfying conditions (1.3). Forz ²(H⁰) dene compact in^K operatorsT(z;H⁰;G) (see (1.4)) and put

A(z;H⁰;G) := ReT(z;H⁰;G); K(z;H⁰;G) := ImT(z;H⁰;G): (2.1) We shall write A(z), K(z) instead of A(z;H⁰;G), K(z;H⁰;G) if the choice of H⁰, G is clear from the context. Suppose that for some  ²

R

for a pair of operators H⁰, G the following condition holds.

Condition 2.1

^{The limit}

"lim^!+0T(+i";H⁰;G) =: T(+i0;H⁰;G); (2.2) exists in the operator norm and

K(+i0;H⁰;G)²

S

^{1(K): (2.3)}

Then dene

N

(;H⁰;G) := 1

Z

1

?1

1 +dtt²n^(1;A(+i0) +tK(+i0)): (2.4) It is easy to see that under the condition (2.3) the integral (2.4) converges.

Lemma 2.2

Fix an open interval ^

R

and a number ² and put G :=GEH⁰().

Then Condition 2.1 at the point  is satised (or not satised) for the pairs H⁰, G and H⁰, G simultaneously.

Proof

Denote ~G :=GEH⁰(

R

ⁿ). Obviously, for every z ²(H⁰):

T(z;H⁰;G) = T(z;H⁰;G) +T(z;H⁰;G^~):

It is clear that the limit T( + i0;H⁰;G^~) exists in the operator norm and K( + i0;H⁰;G^~) = 0. From here follows the assertion of the lemma. ²

Lemma 2.3

^{For any  2}

R

Condition 2.1 is satised (or not satised) for the pairs H⁰, G and H⁰, ^jG^j simultaneously and

N

(;H⁰;G) =^N(;H⁰;^jG^j): (2.5)

Proof

Let G = ^jG^j be a polar decomposition of G;  acts unitarily from Ran^jG^j into RanG. It is easy to see that

T(z;H⁰;^jG^j) =T(z;H⁰;G); ⁸z ²(H⁰): 7

From here follows the rst assertion of the lemma and the equality

n^(1;A(+i0;H⁰;G)+tK(+i0;H⁰;G)) =n^(1;A(+i0;H⁰;^jG^j)+tK(+i0;H⁰;^jG^j)): Integrating the last equality over t with the weight ^?1(1 +t²)^?1, we arrive at (2.5). ²

Lemma 2.4

^{Let V1, V2} be selfadjoint operators in ^H such that 0 ^ V^{1 } V² and suppose that for some  ²

R

Condition 2.1 holds for the pair H⁰, V²¹⁼². Then it also holds for the pair H⁰, V¹¹⁼² and

N

(;H⁰;V¹¹⁼²)^N(;H⁰;V²¹⁼²):

Proof

^{Write V1 = (BV21=2)BV21=2, where kBk1. Then}

T(z;H⁰;V¹¹⁼²) =BT(z;H⁰;V²¹⁼²)B^; ⁸z ²(H⁰):

From here it follows that Condition 2.1 is satised for the pair H⁰,V¹¹⁼² and n^(1;A(+i0;H⁰;V¹¹⁼²) +tK(+i0;H⁰;V¹¹⁼²))^n^(1;A(+i0;H⁰;V²¹⁼²) +

+tK(+i0;H⁰;V²¹⁼²)):

Integrating the last inequality over t with the weight ^?1(1 +t²)^?1, we get the desired result. ²

2.2 Connection between

^N

and the \counting function"

N^

Let²(H⁰)^\

R

. Comparing (1.8) and (2.4) and observing thatK(+i0;H⁰;G) = 0 and A(+i0;H⁰;G) = T(+i0;H⁰;G), we arrive at the following

Proposition 2.5

^For ²(H⁰)^\

R

N

(;H⁰;G) =N^(;H⁰;G;1): (2.6)

2.3 Connection between

^N

and the SSF

Here we formulate some results of [25] which will be used below.

1) Trace class perturbations.

^{Let H0} be a selfadjoint operator in ^H and G ²

S

²(^H;^K). It is well-known that under these conditions for a.e.  ²

R

the limits (2.2) exist in the Hilbert-Schmidt norm and the inclusion (2.3) holds (see [2] or [36]).

Proposition 2.6

^{[25] Let H0} be a selfadjoint operator in ^H and G²

S

^{2(H;K); then}

(;H^0G^G;H⁰) =^N(;H⁰;G); a. e. ²

R

: (2.7)

2) Relatively trace class perturbations.

LetH⁰be a selfadjoint in^Hsemibounded from below operator:

H⁰ =H^0 in ^H; ^?1<inf(H⁰): (2.8) Let the operatorG:^H!K satisfy the conditions (1.3) and for some m >0

GR^m0 ()²

S

^{2(H;K);  <inf(H0): (2.9)}

8

Then for a. e.  ²

R

Condition 2.1 holds for the pair H⁰, G (see, e. g., Corollary 3.8 below). Let H^ = H^(H⁰;G) (see Proposition 1.1). In order to dene the SSF for the pair H⁰,H^, assume that for some k >0 and ⁰ <inf((H⁰)^[(H^)) the inclusion

(H^?⁰I)^?k?(H^0?⁰I)^{?k 2}

S

^{1(H) (2:10)}

holds. The inclusion (2.10) enables one to dene the SSF via (0.8) for h() = (^?⁰)^?k.

Proposition 2.7

[25] Let the conditions (1.3), (2.8){(2.10) hold true. Then

(;H^;H⁰) =^N(;H⁰;G); a. e.  ²

R

^{: (2.11)}

Finally, we need two sucient conditions for the inclusion (2.10).

Proposition 2.8

(i) Assume the conditions (1.3) and (2.9) for m = 1. Then, for all

^{0 2}(H⁰)^\(H^) and k= 1 the inclusion (2.10) holds.

(ii) Assume the conditions (1.3), (2.8) and for some m > 0 let

(GR⁰¹⁼²())^(GR^m0 ())²

S

¹(^H);  <inf(H⁰): (2.12) Then for all integersk > m+¹² and all large enough⁰ <inf((H⁰)^[(H^)) the inclusion (2.10) holds.

(i) follows directly from the denition (1.6). (ii) has been proved in [28, Theorem XI.12].

3 Estimates for

^N

In this section H⁰ is a selfadjoint operator in ^H and G is an operator which acts from^H into ^Kand satises the conditions (1.3).

3.1 Monotonicity estimates

For ²

R

let

E() :=EH⁰(^?1;); G^?() :=GE(); G⁺() :=GEH⁰(;+¹): (3.1)

Lemma 3.1

Let numbers , ^ be such that ^(^?^)> 0. Then Condition 2.1 is satised (or not satised) for the pairs H⁰, G^(^) and H⁰, G simultaneously and the following estimate holds:

N

(;H⁰;G)^N(;H⁰;G^(^)): (3:2^)

Proof

The rst assertion follows from Lemma 2.2. To be denite, let us prove (3:2^?). It is easy to see that

A(+i0;H⁰;G)^A(+i0;H⁰;G^?(^?)); K(+i0;H⁰;G) =K(+i0;H⁰;G^?(^?)): This gives (3:2^?). ²

9

Corollary 3.2

^{For any} ²

R

the following estimate¹ holds:

Z

(t^?^)0N(t;H⁰;G)dt^kG^()^k2S2: (3:3^)

Proof

Using Lemma 3.1, Proposition 2.6 and inequality (0.4), we obtain the esti- mates:

Z

(t^?^)0N(t;H⁰;G)dt^Z

(t^?^)0N(t;H⁰;G^())dt

=^Z

(t^?^)0j(t;H^0G^()G^();H⁰)^jdt^{Z 1}

?1

j(t;H^0G^()G^();H⁰)^jdt

kG^()G^()^kS1 =^kG^()^k2S2: ²

Lemma 3.3

Let the function f :

R

^![0;1) be such that ^f is nondecreasing. Then the following inequality holds:

Z

1

?1 N

(;H⁰;G)f()d^kG^qf(H⁰)^k2S2: (3:4^)

Proof

To be denite, let us consider the case of the lower signs. First assume that f()^!0 as ^!+¹. Denote

max := sup suppf ^+¹ (3.5)

Since G^qf(H⁰)²

S

^{2, one has G?()2}

S

^{2 for  < }max. Fix some numbers R¹, R² such that R¹ < R² < max (from what follows it is easy to see that for a bounded f, one can take R¹ =^?1). For  ² (R¹;R²) put S() :=^R_R^1N?(t;H⁰;G)dt (here the integral converges due to the inclusion G^?() ²

S

² and the inequality (3:3^?)). Integrating by parts, we get:

Z R²

R^{1 N?}(;H⁰;G)(f()^?f(R²))d=^Z_R^R2

1

(f()^?f(R²))dS()

=^?Z_R^R2

1

S()df()^?Z_R^R2

1

kG^?()^k2S2df(); (3.6) where on the last step we have used the inequality (3:3^?). Next, integrating by parts again, we nd:

? Z R²

R^{1 k}G^?()^k2S2df() =^?Tr(G^?(R²)^Z_R^R2

1

E()d(f()^?f(R²))G^?(R²))

= Tr(G^?(R²)(f(R¹)^?f(R²))E(R¹)G^?(R²) +Tr(G^?(R²)(f(H⁰)^?f(R²)I)EH⁰(R¹;R²)G^?(R²))^Tr(G^?(R²)f(H⁰)E(R¹)G^?(R²))

+Tr(G^?(R²)f(H⁰)G^?(R²))^kG^qf(H⁰)E(R¹)^k2S2 +^kG^qf(H⁰)^k2S2: (3.7)

1We remind that in the estimates from above all the quantities in the right hand side are supposed to be nite.

10

Combining (3.6) and (3.7), we see that

Z R²

R^{1 N?}(;H⁰;G)(f()^?f(R²))d ^kG^qf(H⁰)^k2S2 +^kG^qf(H⁰)E(R¹)^k2S2: TakingR^{1 !?1} and R^{2 !}max in the last inequality, we arrive at (3:4^?).

Now suppose lim^!+1f() =: f(¹) ⁶= 0. Then G^qf(H⁰) ²

S

^{2 implies G 2}

S

^2.

Hence, by Proposition 2.6 and inequality (0.4),

Z

1

?1 N

?(;H⁰;G)d ^kG^G^kS1 =^kG^k2S2: (3.8) Next, from the rst part of the proof it follows that

Z

1

?1 N

?(;H⁰;G)(f()^?f(¹))d^kG^qf(H⁰)^?f(¹)I^k2S2

= Tr(G(f(H⁰)^?f(¹)I)G^) =^kG^qf(H⁰)^{k2S2 ?}f(¹)^kG^k2S2: Combining the last inequality with (3.8), we arrive at (3:4^?). ²

The following result of [4] is very close to Lemma 3.3.

Proposition 3.4

^Let H⁰ = H^0 in ^H, V = V^{ 2}

S

¹(^H) and H = H⁰ + V. Let f(),  ²

R

be a continuous² nonnegative nonincreasing function. Then the following inequalities are valid:

Tr(f(H)V)^{Z 1}

?1

(;H;H⁰)f()d^Tr(f(H⁰)V):

It is easy to see that for trace-class perturbations of a denite sign and bounded functions f Proposition 3.4 and Lemma 3.3 are equivalent.

In some important special case the estimates of the type (3:4^?) for the SSF can be proved straightforwardly, without using^N?.

Proposition 3.5

^Let H^{0 }0 be a selfadjoint operator in ^H, G satisfy the condition (1.3) and GR⁰(^?1) ²

S

^{2(H;K). Put H+ =H+(H0;G}). Then for k = 1 and any ⁰ < 0 the inclusion (2:10+) holds; thus, the SSF for the pair H⁰, H⁺ is correctly dened. The following estimate holds:

Z

1

?1

(^?⁰)^?2(;H⁺;H⁰)d^kGR⁰(⁰)^kS22 ; ⁸⁰ <0: (3.9)

Proof

The inclusion (2:10+) (for k= 1) follows from (1.6). The SSF (;H⁺;H⁰) is dened on the basis of (2:10+) via (0.8) (withH =H⁺ andh() = (^?⁰)^?1). Applying (0.4) and taking into account the fact that T(⁰;H⁰;G)^0, we nd

Z

1

?1

(^?⁰)^?2(;H⁺;H⁰)d^k(H^+?⁰I)^?1?(H^0?⁰I)^?1kS1

=^k(GR⁰(⁰))^(I+T(⁰;H⁰;G))^?1(GR⁰(⁰))^{kS1 k}GR⁰(⁰)^k2S2: ²

2It is easy to see that the condition of continuity of^f can be dropped

11

3.2 Partition of spectrum

Let the Borel sets ¹;^{2 }

R

, be such that (H⁰) ^^1[². Introduce the notations Gj =GEH⁰(j), j = 1;2.

Lemma 3.6

Let for some  ²

R

Condition 2.1 be satised for the pair of operators H⁰, Gj, j = 1;2. Then it is satised for the pair H⁰, G and the following estimates hold:

N

(;H⁰;G)^N(;H⁰;(1^?)^?1=2G¹) +^N(;H⁰;^?1=2G²); ⁸ ²(0;1); (3:10^)

N

(;H⁰;G)^N(;H⁰;(1 +)^?1=2G¹)^?N(;H⁰;^?1=2G²); ⁸ > 0: (3:11^)

Proof

1. First suppose ^1\² =^;. Clearly, for all z ²(H⁰), T(z;H⁰;G) =T(z;H⁰;G¹) +T(z;H⁰;G²); A(z;H⁰;G) =A(z;H⁰;G¹) +A(z;H⁰;G²);

K(z;H⁰;G) =K(z;H⁰;G¹) +K(z;H⁰;G²): ^(3.12) From here follows the rst assertion of the lemma. Using (3.12) and (1.1), we obtain

n^(1;A(+i0;H⁰;G) +tK(+i0;H⁰;G))^n^(;A(+i0;H⁰;G¹) +tK(+i0;H⁰;G¹)) +n^(1^?;A(+i0;H⁰;G²) +tK(+i0;H⁰;G²)): Integrating the last inequality with respect to t with the weight ^?1(1 +t²)^?1, we arrive at (3.10). Similarly (using (1.2) instead of (1.1)) one proves (3.11).

2. The general case (^{1 \}^{2 6}= ^;) can be reduced to the one considered above by using Lemma 2.4. Indeed, writing the estimate (3.10) for the pair of sets ¹, ^{2 n}¹ and observing that 0 ^(GEH⁰(²ⁿ¹))^(GEH⁰(^{2 n}¹))^(GEH⁰(²))^(GEH⁰(²)), we nd

N

(;H⁰;G)^N(;H⁰;(1^?)^?1=2G¹) +^N(;H⁰;^?1=2GEH⁰(^{2 n}¹))

 N

(;H⁰;(1^?)^?1=2G¹) +^N(;H⁰;^?1=2G²): Similarly,

N

(;H⁰;G)^N(;H⁰;(1 +)^?1=2G¹)^?N(;H⁰;^?1=2GEH⁰(^{2 n}¹))

 N

(;H⁰;(1 +)^?1=2G¹)^?N(;H⁰;^?1=2G²): ²

Remark 3.7

Let us sketch another possible proof of Lemma 3.6, which claries the role of the fact that the perturbation has a denite sign. As above, suppose that^1\² =^;. In order to prove (3.10), we rst observe that, as it is easy to check, for any ²(0;1)

V =G^G^(1^?)^?1G^1G¹+^?1G^2G² = ~G^G;^~

where the operator ~G = (1^? )^?1=2G^{1 }^?1=2G² acts from ^H into ^KK. Here the condition V ^ 0 is essential. Next, use Lemma 2.4 (for V¹ =V, V² = ~G^G^~) and Lemma 2.3:

N

(;H⁰;G)^N(;H⁰;( ~G^G^~)¹⁼²) =^N(;H⁰;G^~)

=^N(;H⁰;(1^?)^?1=2G¹) +^N(;H⁰;^?1=2G²): 12

The proof of (3.11) is similar. Note also that Lemma 3.6 can be reformulated for the case of partition of the spectrum into an arbitrary nite number of sets j.

Corollary 3.8

Suppose that for some open interval  ^

R

the following inclusion holds:

GEH⁰()²

S

^2(H;^K): (3.13) Then for a. e. ² for the pair H⁰, G Condition 2.1 holds and

N

(;H⁰;G)²L¹;^loc(): (3.14) In particular, if (3.13) holds for any bounded interval  ^

R

^{, then}

N

(;H⁰;G)²L¹;^loc(

R

):

Proof

To be denite, consider the case of upper signs. In (3:10+) take = 1=2, ¹ =,

² =

R

^n. Due to (3.13), Proposition 2.6 and the inequality (0.4), one has

N

+(;H⁰;^p2G¹) =^?(;H^0?2G^1G¹;H⁰)²L¹(

R

):

At the same time, it is easy to see that T( + i0;H⁰;G²) is selfadjoint and

N

+(;H⁰;^p2G²) = n⁺(1;2T(+i0;H⁰;G²)) is a nonincreasing

Z

⁺{valued function of

² , possibly with a singularity at sup. From here it follows that ^N+(;H⁰;^p2G²)² L¹;^loc(), and we arrive at (3.14). ²

Note that the statement, analogous to Corollary 3.8, is false for the SSF. Namely, one can construct such operators H⁰, G and an interval , that the hypothesis of Corollary 3.8 holds true, but for some  ² C⁰¹() the dierence (H^)^?(H⁰) ⁶²

S

¹. Thus, the SSF for the pair H^, H⁰ cannot be dened on the interval  in any reasonable sense.

4 Estimates for

^{N+ ? N+}

The contents of this section is aimed at the applications to the asymptotics of SSF in the large coupling constant limit. Operator H⁰ is supposed to be nonnegative; we are interested in ^N+(;H⁰;G). For  < inf(H⁰) the question of asymptotics of

N

+(;H⁰;^ptG) = N⁺(;H⁰;G;t) as t ^{! 1} is well studied in applications. For this reason, we estimate the dierence

Q(;^?;) :=^N+(;H⁰;G)^?N⁺(^?;H⁰;G;);

where ^?<inf(H⁰) is xed and >0 is some auxiliary parameter. We dene Q^(;^?;) := (Q(;^?;))^:

We also use the notations (3.1) and set ~E() =I^?E().

4.1 Local estimates

13