The Erwin Schrodinger International Pasteurgasse 6/7. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Pasteurgasse 6/7 Institute for Mathematical Physics A-1090 Wien, Austria On the Lieb{Thirring Estimates for the Pauli Operator Alexander V. Sobolev Vienna, Preprint ESI 212 (1995) March 2, 1995 Supported by Federal Ministry of Science and Research, Austria Available via

2 ON THE LIEB-THIRRING ESTIMATES FOR THE PAULI OPERATOR A.V. Sobolev 1 Abstract. We establish Lieb-Thirring type estimates for the sums P of the negative eigenvalues of the two-dimensional Pauli operator with a non-homogeneous magnetic eld perturbed by a decreasing electric potential. 1. Introduction The aim of the paper is to establish some spectral properties of the Pauli operator, that is of the operator describing the motion of a particle with spin in a magnetic eld. It acts in L 2 (R d ) C 2 with d = 2 or d = 3 and has the form where H (d) a H (d) P auli = H (d) a I? B; I= = (?ir? a) 2 is the usual spinless Schrodinger operator with the magnetic vector-potential a = fa 1 ; : : : ; a d g, B = r a is the eld and stands for the vector 1 ; 2 ; 3 of 2 2 Pauli matrices (see []). Suppose that the eld B is pointed along the x 3?axis, i.e. a = (a 1 ; a 2 ; 0) with a = a (x 1 ; x 2 ) (which is always true for d = 2). In this case B = (0; 0; B); B 1 a 2 a 1 and H (d) P auli loos especially simple: H (2) P auli = A+ 0 ; A 0 A = H (2)? H (3) P auli = H (2) P auli +?@ ?@ 2 3 ; B; (1.1) a : (1.2) Though this operator does not seem to be non-negative, the entries A can be rewritten as A = Q Q with the operators Q = 1 i 2 ; =?i@? a ; = 1; 2; (1.3) which allows one to dene H (2) P auli as a non-negative self-adoint operator (see Sect. 2 below). A remarable property of H (2) P auli is that the point = 0 belongs to its Mathematics Subect Classication: 35J10, 35P15 1 Author supported by EPSRC under grant B/9/AF/1793 Typeset by AMS-TE 1

3 2 A.V. SOBOLEV spectrum. This assertion was proved under fairly broad conditions on the magnetic eld B (see [1], [6] and also [],[8]). If the operator (1.1) or (1.2) is perturbed by a real-valued function (electric potential) V = V, decaying at innity, then it can have some discrete spectrum below = 0. Denote by = (a; V ); 2 N; the negative eigenvalues of H (d) P auli + V I enumerated in the non-decreasing order counting multiplicity. We study the sums M (a; V ) = ; 0: It is well-nown that, without any magnetic eld, M satises the following estimate 2 : M (0; V ) C d; V? (x) + d 2 dx; + d=2 > 1; (1.) which is usually referred to as the Lieb{Thirring inequality if > 0 and the Rosenblum{Lieb{Cwicel inequality if = 0. The same estimate holds for the negative spectrum of the spinless operator H (d) a + V with a 6= 0. The crucial technical reason for that is the so-called diamagnetic inequality (see [2]). It means, loosely speaing, that the magnetic eld "pushes the spectrum upwards", which leads to (1.) for H (d) a + V as well. This type of argument does not wor for the Pauli operator and as a consequence, the standard Lieb-Thirring estimate (1.) no longer holds. A suitable replacement for (1.) for a homogeneous eld B(x 1 ; x 2 ) = const and = 1 was found in [10] (see also short communication [11]) and [12]: M 1 (a; V ) C 0 d V? (x) 1+ d 2 dx + C 00 db V? (x) d 2 dx; d = 2; 3: A natural generalization of this result for non-homogeneous elds and arbitrary would be as follows: M (a; V ) C 0 ;d V? (x) + d 2 dx + C 00 ;d B(x 1 ; x 2 )V? (x) + d 2?1 dx; + d=2 > 2: (1.5) The validity of this conecture was thoroughly investigated in [7] ( see also [8] for more details and further references) for d = 3. In particular, (1.5) was established for the magnetic elds satisfying the lower bound B B 0 > 0 under some additional constraints on the behaviour at innity. Besides, the author constructed a counterexample showing that the bound (1.5) cannot hold for arbitrary B. In this connection it was conectured in [7] on the grounds of physical considerations, that in order to save (1.5), one has to replace the magnetic eld in (1.5) with a suitable \smeared" modication of B. In the present paper we give a simple proof of (1.5) in the case d = 2; > 1 under fairly general conditions on B. The results on d = 2; = 1 and d = 3 will be published elsewhere. We nd out that (1.5) holds with B replaced by an \eective" magnetic eld b(x) (see Sect. 2 for precise denition). The eld b coincides with the 2 Here and in what follows we denote by C and c (with or without indices) various positive constants whose precise value is of no importance.

4 LIEB-THIRRING ESTIMATES 3 initial eld B, if the latter obeys some regularity condition, which, in particular, does not allow B to decay at innity too rapidly. This is quite consistent with restrictions on B under which (1.5) was obtained in [7]. On the contrary, if the decay of B is too fast, the eective eld b will be considerably dierent from B. The distinction is especially spectacular for a compactly supported B. This point is discussed in detail in Sect. 2. One should stress that the method of the proof is essentially dierent from that used in [7]. Instead of the technically involved path integration approach adopted in [7], we apply only elementary methods of the spectral theory for Schrodinger operators: the Birman-Schwinger principle, the diamagnetic inequality and Cwicel type estimates. We point out that until recently there have been only a few mathematically rigorous results on the Pauli operator in a non-homogeneous magnetic eld. In the physical literature most of its properties were assumed without proof. For instance, it was unclear under which conditions on the potential V the operator H (2) P auli + V I can be dened as a self-adoint operator. A partial answer to this question was given in [8]. In the present paper we obtain a more general criterion, which guarantees self-adointness of the perturbed operator (see Theorem 2.3). Another important property which required ustication is the coincidence of the non-zero spectra of the entries A in (1.1) (see []): (A + ) n f0g = (A? ) n f0g: This fact follows from the formal relations A + = Q +Q + and A? = Q + Q +, which, strictly speaing, are fullled only if Q = Q. The latter equality is not true in general. In the appendix we nd out that under some conditions on the magnetic potential a and the eld B this equality does tae place. Notation. For any measurable real-valued function f we denote by f + and f? its positive and negative parts respectively: f = (f f)=2. This convention does not apply to operators (cf. (1.1)). The notation f L q stands for the norm of f in the space L q (R 2 ); q 1. If q = 2 we simply write f. For a self-adoint operator T, R(z; T ) = (H? z)?1 denotes its resolvent. If T is semi-bounded, then T [; ] denotes the closed form associated to T, with the domain D[T ]. Notation h; i stands for the inner product in L 2 (R 2 ). 2. Result, discussion 1. Basic denitions and notation. For the operator (1.1) is diagonal, we can study its entries A + ; A? individually. To state results simultaneously for "+" and "-" we use, as a rule, the double subscript: "". In this case each statement must be undestood separately for the upper subscript and the lower one. Let a = (a 1 ; a 2 ) 2 L 2 loc (R2 ) be a magnetic vector-potential with real-valued components. The operators Q ; dened in (1.3) are closable on C 1 0 (R2 ), since are symmetric and Q Q. We use the same letters Q + ; Q? ; for their closures. Dene A = Q Q. This operator can be also interpreted as that associated with the closed quadratic form A [u; u] = Q u 2 with the domain D[A ] = D(Q ). In the same way we dene the usual Schrodinger operator H a with the magnetic eld a: as an operator associated with the form H a [u; v] =

5 A.V. SOBOLEV h u; vi (summation over repeating indices is assumed). We omit the superscript "(2)" from notation (cf. (1.1)) without any ris of confusion, for we study only the case d = 2. It follows from the inequality A [u; u] = H a [u; u]i? h 2 u; 1 ui?h 1 u; 2 ui 2H a [u; u]; 8u 2 C 1 0 (R 2 ); (2.1) that D[H a ] D[A ]. As a rule below we impose Assumption 2.1. The magnetic potential a 2 L 2 loc (R2 ) is such that the magnetic eld B(x) =?i[ 1 ; 2 ] = rot a(x) 1 a 2 2 a 1 (x); dened in the sense of distributions, belongs to L 1 loc (R2 ). It is easy to chec that for any u; v 2 C 1 0 (R2 ) which implies that A [u; v] = H a [u; v] hbu; vi; (2.2) A +? A? =?2B(x); (2.3) in the sense of sesqui-linear forms on C 1 0 (R 2 ). Observe that if B 2 L 1 (R 2 ), then (2.2) entails the estimate A [u; u] H a [u; u]? B L 1u 2 ; which, along with (2.1), implies that D[A ] = D[H a ]. Without the condition B 2 L 1 (R 2 ), we can only claim that the form domains of the operators A + ; A? ; H a coincide locally. Precisely, the following lemma holds: Lemma 2.2. Let the magnetic potential a obey Assumption 2.1. If a function u belongs to one of the domains D[A + ]; D[A? ] or D[H a ], then the function u; 2 C 1 0 (R 2 ), belongs to all of them. This lemma follows from the obvious commutator relation for the operators (1.3): [Q ; ] = [?i@ 2 ; ] = (?i@ 2 ) = ; 8 2 C 1 0 (R2 ); (2.) and Assumption 2.1. >From now on we impose the following conditions on the magnetic eld. Let ` 2 C(R 2 ) be a positive function such that `(x)? `(y) %x? y; 0 % < 1; 8x; y 2 R 2 : (2.5) Sometimes we call this function slowly varying. Denote D(x) = fy 2 R 2 : x? y < `(x)g: (2.6) We assume that there exists a positive function b 2 L 1 loc (R2 ) such that B (x) b(x); a.a. x 2 R 2 ; (2.7) C 1 b(x) b(y) C 2 b(x); a.a. y 2 D(x); a.a. x 2 R 2 ; (2.8) b(x)`(x) 2 c; a.a. x 2 R 2 : (2.9) The next theorem species conditions on V under which the operator P = A +V can be dened as a form sum on D[A ].

6 LIEB-THIRRING ESTIMATES 5 Theorem 2.3. Let B obey the conditions (2.7) { (2.9) with some functions `(x) and b(x). Let V satisfy for some p > 1 the estimate v-sup x V (y) p? b(y) + 1 dy < 1: D(x) Then the form P [; ] = A [; ] + V [; ] is closed on D[P ] = D[A ]. We point out that if B 2 L 1 (R 2 ), then Theorem 2.3 with b(x) = v-sup B (x), `(x) = 1 guarantees that D[P ] = D[A ] under the condition v-sup x V (y) p dy < 1: x?y1 Note that this condition is nown to be sucient for the equality D[H a +V ] = D[H a ] (see []). This fact agrees with the observation made above, that D[A ] = D[H a ], if B 2 L 1 (R 2 ). By denition the operator A is non-negative. Below we shall impose on the potential V the constraints, which will guarantee that the negative spectrum of the operator P associated with the form P [; ] is discrete. Let () ; 2 N; be negative eigenvalues of P, enumerated in the non-decreasing order counting multiplicity. We study the following quantities: M () = () ; N () = sup () ; 0: (2.10) M () Note that N (). When necessary we reect the dependence of various obects on the elds a; V : for example, () = () (a; V ); M () = M () (a; V ). Next Theorem establishes the main result of the paper. Theorem 2.. Let the conditions (2.7) { (2.9) be fullled. Suppose that V obeys the conditions of Theorem 2.3 and V? 2 L +1 (R 2 ); V?b 2 L 1 (R 2 ) for some 1. Then the negative spectrum of P is discrete and M () C 1; V? (x) +1 dx + C 2; V? (x) b(x)dx; > 1; (2.11) N () 1 C 1;1 V? (x) 2 dx + C 2;1 V? (x)b(x)dx; = 1; (2.12) The constants C 1; ; C 2; can be calculated explicitly, but apparently their values are far from being optimal and for this reason we do not give them. 2. Discussion of Theorem 2.. Let us replace the vector-potential a with a where > 0 is a parameter measuring intensity of the magnetic eld. It is clear that the conditions (2.7){(2.9) for B and b imply (2.7){(2.9) for the elds B and b, if 1. Hence by Theorem 2. the sum M () M () (a; V ) C 1; V? (x) +1 dx + C 2; (a; V ) obeys the bound V? (x) b(x)dx; > 1; 1: (2.13)

7 6 A.V. SOBOLEV On the contrary, if! 0, then the condition (2.9) for b may be violated and Theorem 2. will be no longer applicable. Theorem 2. allows one to estimate M () for the Pauli operator containing the Planc constant ~: To that end observe that ~P (~) = (?i~r? a) 2 ~B + V; ~ 2 (0; 1]: ~P (~) = ~ 2 P ( ~?1 a; ~?2 V ); which means that the sum of negative eigenvalues ( P ~ ) raised to the power equals ~ 2 M () ( ~?1 a; ~?2 V ): Using for the r.h.s. the estimate (2.13) with = ~?1 1, one obtains that ( P ~ ) C 1; ~?2 V? (x) +1 dx + C 2; ~?1 V? (x) b(x)dx; > 1: Let us consider two examples of Theorem 2.. Example 1. Suppose that B(x) > 0; B 2 C 1 (R 2 ) and rb(x) CB(x) 3 2 ; 8x 2 R 2 : (2.1) For B? = 0, it is clear that M (?) C 1 V? (x) +1 dx; > 1: We claim that M (+) satises (2.11) with b = B. Indeed, dene `(x) = &B(x)?1=2 ; b(x) = B(x) with some & > 0. Clearly, (2.7) and (2.9) are fullled. Let us verify the conditions (2.5) and (2.8). Due to (2.1) r`(x) & 2 B(x)? 3 2 rb(x) C & 2 ; and consequently, for suciently small & the function `(x) obeys (2.5) with % = C&=2 < 1. Furthermore, (2.5) provides the bound (1 + %)?2 B(x) B(y) (1? %)?2 B(x); 8y 2 D(x); which ensures (2.8). Applying Theorem 2., we obtain (2.11) with b = B. Note that the condition (2.1) admits a quic growth of B at innity. instance, any positive function B 2 C 1 (R 2 ) which equals exp(x m ); m > 0; for x R > 0, obeys (2.1). On the contrary, (2.1) does not allow B(x) to decrease as x! 1 too rapidly. In fact, if B(x) = x? for x r > 0; then the condition (2.1) is not fullled if > 2. For

8 LIEB-THIRRING ESTIMATES 7 Example 2. Compactly supported magnetic eld. Suppose that B(x) = B; x R and B(x) = 0; x > R for some B > 0; R > 0. As in example 1, we loo only at M (+). It is easy to see that the functions `(x) = % p 1 + x 2 ; 0 < % < 1; b(x) = 8 < : B ; x R; 2B 1 + x 2 R ; x > R; (2.15)?2 satisfy the conditions (2.5), (2.7) { (2.9), which leads to (2.11) and (2.12). Note that the eective eld b is not only non-compactly supported, but even non-integrable! Let us explain why one cannot nd another function b satisfying all the conditions of Theorem 2. and decreasing quicer than the function in (2.15). The thing is that the choice of b(x) should be made simultaneously with that of the slowly varying function `(x). In view of (2.9), in order to minimize b(x), one should maximize `(x). Clearly, `(x) s %x is the fastest growing (as x! 1) function which obeys (2.5). Now the condition (2.9) ensures that b(x) Cx?2. Explicit calculations for this example were carried out in [8]. They show that the compactly supported magnetic eld creates a non-integrable "tail" behaviour of a relevant quantity (ground state density), whith decay at innity as Bx?2 (log x)?1. The function b(x) in (2.15) provides quite a precise estimate for such a tail. 3. Auxiliary information. Here we provide well-nown facts to be used in the sequel. (a)compact operators (see [3]). Let T be a compact operator. We use the notation s n (T ); n 2 N, for its singular values (s?values) and denote by n(s; T ) = #fs n > sg; s > 0 their distribution function. Recall that s n (T ) are dened as eigenvalues of the selfadoint operator (T T ) 1=2, so that Moreover, the Weyl inequality holds: n(s 2 ; T T ) = n(s; T ): (2.16) n(s 1 + s 2 ; T 1 + T 2 ) n(s 1 ; T 1 ) + n(s 2 ; T 2 ): (2.17) We denote by S p ; p 1; the Neumann-Schatten classes of compact operators with the norm T p = n s n (T ) p 1 p : It is easy to see that n(s; T ) s?p T p p ; 8s > 0: (2.18) (b) Birman-Schwinger principle. Let N () = #f () <?g; > 0 be the number of negative eigenvalues of the operator P. The quantities (2.10) can be represented as follows: M () =? 1 0 N () dn () = = sup N () ; >0 1 0?1 N ()d; 9 >= >; (2.19)

9 8 A.V. SOBOLEV which reduces the problem to the study of the function N (). To estimate it we use the following classical argument. For a function Y dened on D(A ) denote Then K (; Y ) = Y (A + )?1 Y : (2.20) N () n? 1; K (; Y ) ; Y = (V? + ) 1 2? ; 8 2 (0; ]; (2.21) for any > 0. This result is a version of the Birman-Schwinger principle. (c) Diamagnetic inequality. Let H a be the Schodinger operator with a magnetic eld. Then for any > 0; { 0 one has the following point-wise estimate: R(?; H a ) { u(x) R(?; H 0 ) { u(x); 8u 2 L 2 (R 2 ); a.a. x 2 R 2 : This inequality yields (see [2] and references therein) Proposition 2.5. Let be multiplication by a measurable function and { > 0. Then for any > 0 and for any positive integer n R(?; H a ) { R(?; H 0 ) { R(?; H a ) { 2n R(?; H 0) { 2n : (2.22) The rst part of this proposition with { = 1=2 implies Corollary 2.6. Let be as in Proposition 2.5. Then the inequality u 2 H 0 [u; u] + Mu 2 ; 8u 2 C 1 0 (R 2 ); with some positive and M, implies that u 2 H a [u; u] + Mu 2 ; 8u 2 C 1 0 (R2 ): An estimate for the r.h.s. of (2.22) can be proved with the help of a simple Cwicel estimate for the operators of the form a(x)b(?i@) (see, e.g. [15]). The latter provides for any 2 L p (R 2 ); p 2; and > 0 the bound R(?; H 0 ) { p C?{+ 1 p L p; 8{ > p?1 : (2.23) (d) Multiplicative inequality. In the proof of Theorem 2.3 we shall use the following inequality (see [13] and also [5]): Lemma 2.7. Let u 2 C 1 0 The constant C q does not depend on u. (R 2 ). Then for any q 2 [2; 1) one has u L q C q 1? ; = 2q?1 : This Lemma is nothing but a convenient version of the embedding theorem in R 2 with the \critical" exponent (see [5] for details).

10 LIEB-THIRRING ESTIMATES 9 3. Proof of Theorem Partition of unity. The rst step in the proof is to construct a partition of unity associated with the function `(x) introduced in the beginning of Sect. 2.: Lemma 3.1. Let `(x) be a continuous function satisfying (2.). Then there exists a set x 2 R 2 ; 2 N such that the open diss D = D(x ) (see (2.6) for denition) form a covering of R 2 with "the nite intersection property" (i.e. each dis intersects no more than N = N(%) < 1 other diss). Moreover, there exists a set of non-negative functions 2 C 1(D 0 ); 2 N, such that and uniformly in. (x) 2 = 1; m (x) C m`(x)?m ; 8m; (3.2) Note that more common denition of the partition of unity requires P = 1 instead of (3.1). Nevertheless, for us the square will be convenient. Proof of this Lemma is analogous to that of Theorem from [9] and we do not reproduce it here. We single out an important consequence of the nite intersection property for diss D. Denote M = f 2 N : D \ D 6= ;g: (3.3) and dene by induction the sets M (r+1) Then = fn 2 N : D n \ D 6= ;; 8 2 M (r) g; M (0) = M ; r = 0; 1; : : : : (3.) card M (r)? N(%) + 1 N(%) r ; r = 0; 1; : : : (3.5) with the number N(%) dened in Lemma Proof of Theorem 2.3. It suces to verify that for any " < 1 there exists a constant C = C(V; B; ") such that V 1=2 u 2 "A [u; u] + Cu 2 ; 8u 2 C 1 0 (R 2 ): (3.6) Step 1. We claim that for any W 2 L p (R 2 ); p > 1; and u 2 C 1 0 bound where (R2 ) one has the W 1=2 u 2 CW L ph! (u); 8! > 0; (3.7) H! (u) =!H a [u; u] +!?1 u 2 ; = 1? p?1 : Indeed, by Holder inequality and Lemma 2.7, for q = 2p(p?1)?1 > 2 and = 2q?1 we have W 1 2 u2 W L pu 2 L q CW L 2(1?) :

11 10 A.V. SOBOLEV By the Young inequality, the r.h.s. does not exceed CW L p!@u 2 +!?1 u 2 ; 8! > 0: In view of Corollary 2.6 this leads to the estimate (3.7). Step 2. Let D be the diss forming the covering of R 2 described above and let be the set of functions constructed in Lemma 3.1. Set b = b(x ); ` = `(x ). Denote by the characteristic function of D. By (3.1) the l.h.s. of (3.6) equals h V u ; u i; u = u: Applying (3.7) with W = V and u = u, we see that V 1=2 u 2 C V L ph! (u ); 8! > 0: Adding and subtracting the term!hbu ; u i, one obtains H! (u ) =!A [u ; u ] +!?1 u 2!hBu ; u i: Estimate the rst summand. Due to (2.) and (3.2), one has A [u ; u ] 2 Q u 2 + 2[Q ; ]u 2 2 Q u 2 + C`?2 u 2 : Moreover, hbu ; u i hb u ; u i. Therefore, in view of (2.9) and (2.7), H! (u ) 2! Q u 2 + C!?1 u 2 +!b u : 2 (3.8) The last term is bounded by Let us pic C!?1? 1 +! 1 b u 2 :! = (1 + b )? ; 1; and plug it in (3.8), taing into account that 1? = 1=p: V 2 1 u 2 C V L p 2(1 + b )? Q u 2 +?1 (1 + b ) 1? u 2 C V L p(1 + b ) p 1 2 Q u 2 +?1 u : 2 With the notation L = v-sup x2r 2 D(x) this estimate transforms into the bound V (y) p? 1 p 1 + b(y) dy V 1 2 u 2 CL Q u 2 + C()L u 2 : Let us sum up the contributions from dierent diss D : V 1 2 u 2 CLA [u; u] + CL u 2 : Since the number of mutual intersections of D is bounded, the last term does not exceed C 0 Lu 2. Letting = (CL)?1 ", we arrive at (3.6).

12 LIEB-THIRRING ESTIMATES 11. Proof of Theorem Properties of the operator A. In the sequel we shall retrieve spectral properties of the operators A by comparing it with the operator H = H a + W (x); W 2 L 1 loc (R2 ); W 0. In view of (2.2) H = A B + W: Our basic tool will be the resolvent identity relating R(z; H) and R(z; A ). Let 2 C 1 0 (R2 ). Then R(z; A ) = R(z; H) + R(z; H) (; W B)R(z; A ); (; f) = Q + Q + f: ) (.1) To derive (.1) we use Lemma 2.2. In the proof of Theorem 2. we use (.1) with the functions and dierent operators H ; = H a + B + b ; b = b(x ). Since H ;? A = B + b, one has 2 R(z; A ) = R(z; H ; ) + R(z; H ; ) ; R(z; A ); ; = ( ; B + b ): Note that H ; H a + b, which implies that ) (.2) R(?; H ; ) { R??( + b ); H a { ; 8 > 0; { 2 [0; 1]: (.3) In order to control the second term in (.2) we need the following elementary properties of the resolvent R(z; A ): Lemma.1. Let a 2 L 2 loc (R2 ) and = 0; 1. Then Q R(; A ) 1 2?1 2 ; 8 > 0: (.) Suppose that Assumption 2.1 is fullled. Let 2 C 1 0 Then (R 2 ) and f = v-sup x2supp f(x); f 2 L 1 loc (R2 ): Q R(; A ) ?1? 2 L1B + 2 L1 + 2 L1 : (.5) Proof. For = 0 (.) is obvious. For = 1 the bound (.) follows from the inequality Q u 2 (A + ) 1 2 u 2 ; 8u 2 C 1 0 (R2 ): (.6) Let us prove (.5). For any u 2 C 1 0 (R2 ) the relations (2.3) and (2.) ensure that Q u 2 = ha u; ui = ha u; ui 2hB u; ui Q u L1B u 2 It remains to use (.6). 2 Q 2 + 2[Q ; ]u L1B u L1Q u L1 + 2 L1B u 2 : If B obeys the conditions of Theorem 2., then Lemma.1 leads to

13 12 A.V. SOBOLEV Lemma.2. Let the vector-potential a and the eld B be as in Theorem 2.. Then for any > 0 and 2 N one has ; R(?; A ) 1 2 C? b 1 2 +? 1 2 b ; b = b(x ): where the constant does not depend on or the functions B; b. Proof. By (2.7) and denition (.1) ; R(?; A ) 1 2 Q ( ) R(?; A ) ( ) Q R(?; A ) Cb R(?; A ) 1 2 : Denote ` = `(x ). In view of Lemma.1 and (3.2) the r.h.s. does not exceed C? 2 1?`?2 2 + b 1 `?1 + `?1 + C 0`?1 + Cb? 2 1? C`?1 + C 0? 2 1?`?1 2 `?1 + b 1 + Cb? C b 2 +? 2 1 b : To obtain the last inequality we used (2.9). 2. Spectral estimates for the operator K (; Y ). Here we obtain an estimate for the counting function n(s; K ) where K = K (; Y ) is the operator dened in (2.20). For the sae of brevity below we sometimes omit " " from the notation and write K; A; Q instead of K ; A ; Q. For some functions Y; b and an open set R 2 dene the integrals Our aim is I 1 (Y ; ) = Y (x) dx; I 2 (Y; b; ) = Y (x) 2 b(x)dx: (.7) Theorem.3. Let Assumption 2.1 be fullled and the eld B obey the conditions (2.7){(2.9). Let Y be a function such that Y 2 D[A]. If I 1 (Y ; R 2 ) < 1; I 2 (Y; b; R 2 ) < 1, then the operator K(; Y ) is compact for any > 0 and n? s; K(; Y ) C(s)?1 I 1 (Y ; R 2 ) + I 2 (Y; b; R 2 ) ; 8s > 0: (.8) To study properties of the operator K(; Y ), we rewrite it as follows: K(; Y ) = Y (A + )?1 Y = Y (A + )?1 (A + )(A + )?1 Y = Y (A + )?1 Q Q(A+ )?1 Y + Y (A + )?1 (A + )?1 Y : p Let S denote either the operator Q or the multiplication operator. Dene T = T (; Y; S) = Y (A + )?1 S :

14 LIEB-THIRRING ESTIMATES 13 It is clear that K(; Y ) = T (; Y; Q)T (; Y; Q) + T (; Y; p )T (; Y; p ) ; which ensures, in view of (2.17) and (2.16), that n +? 2s 2 ; K(; Y ) n? s; T (; Y; Q) + n? s; T (; Y; p ) : (.9) Thus it is sucient obtain (.8) for the operator T. Using the partition of unity constructed in Lemma 3.1, one can represent T as T = F ; F = F (; Y; S) = 2 T (; Y; S): According to (.2), each F breas up into three operators: F = F (1) + F (2) + F (3) ; (.10) F (1) = Y R(?; H )S ; F (2) =? Y R(?; H )[S ; ]; F (3) = Y R(?; H ) R(?; A)S : In the next lemma the constants in all the bounds do not depend on. They may depend only on the constants in (2.8), (2.9),(3.2) and the parameter % from (2.5). As before, we use the notation b = b(x ); ` = `(x ). Lemma.. Let the magnetic potential a and the eld B be as in Theorem 2. and Y 2 L loc (R2 ). Let I 1 ; I 2 be dened in (.7). Then F (1) + F (2) F (3) 2 C?1 I 1 (Y ; D ); (.11) 2 C?1 I 2 (Y; b; D ): (.12) Proof. Observe rst of all that the inequalities (.3), (2.22) and (2.23) lead to the bounds Y R(?; H ) { Y R??( + b ); H 0 { C( + b )?{+ 1 I1 (Y ; D ) 1 ; 8{ > 1=; (.13) Consequently, Y R(?; H ) { 2 Y R??( + b ); H 0 { 2 C( + b )?{+ 1 2 I2 (Y; 1; D ) 1 2 ; 8{ > 1=2: (.1) F (1) Y R(?; H ) 1 2 R(?; H ) 1 2 S C? 1 I1 (Y ; D ) 1 :

15 1 A.V. SOBOLEV If S = p, then F (2) = 0. If S = Q, then in view of (2.) and (3.2), and hence (.13) yields that [S ; ] C`?1 ; F (2) C Y R(?; H ) `?1 C( + b )? 3 `?1 By (2.9) b?1=2 `?1 C. This proves (.11) for F (1) and F (2). Proof of (.12). By Lemma.2 I1 (Y ; D ) 1 : R(?; A)S R(?; A) 1 2 R(?; A) 1 2 S C? b 1 2 +? 1 2 b : Consequently, using (.1) we get F (3) 2 C? b 1 2 +? 1 2 b Y R(?; H ) 2 C? b 1 2 +? 1 2 b ( + b )? 1 2 I2 (Y; 1; D ) 1 2 C? 1 2 b 1 2 I2 (Y; 1; D ) 1 2 ; which provides (.12) by virtue of (2.8). In accordance with (.10) the operator T can be presented as the sum T = T (1) + T (2) + T (3) ; T (l) = F (l) ; l = 1; 2; 3: Next lemma puts together contributions from dierent F 0 s. Lemma.5. Under the conditions of Theorem.3 one has T (1) ; T (2) 2 S ; T (3) 2 S 2 and T (1) + T (2) C?1 I 1 (Y ; R 2 ); (.15) T (3) 2 2 C?1 I 2 (Y; b; R 2 ): (.16) Proof. We prove (.15) rst. Let M ; M (r) be the sets dened in (3.3), (3.). For F (l)? F (l) =? F (l) F (l) = 0; =2 M ; l = 1; 2; one has T (l) =? T (l) T (l)? T (l) T (l) 2M m2m n2m m 1 F (l) F (l) F (l) m F n (l) 2M (2) F (l) :

16 LIEB-THIRRING ESTIMATES 15 For the number card M (2) Furthermore, by (3.5) is bounded uniformly in (see (3.5)), one can estimate 2M (2) F (l) C 2M (2) F (l) : T (l) C card M (2) F (l) C0 F (l) : In combination with (.11) this leads to T (l) 1 C?1 I 1 (Y ; D ): In view of (3.5) with r = 0 this implies (.15). Exploiting the same argument, one proves that T (3) 2 2 =? T (3) T (3) 1 This provides (.16) due to (3.5). F (3) 2 F (3) 2M 2 C?1 Proof of Theorem.3. According to (2.17), (2.18) and (.15), I 2 (Y; b; D ): Now (.8) follows from (.9). n(s; T (1) + T (2) ) Cs??1 I 1 (Y ; R 2 ); n(s; T (3) ) Cs?2?1 I 2 (Y; b; R 2 ): 3. Proof of Theorem 2.. According to the Birman-Schwinger principle (2.21) with = =2,? 1=2 N () n(1; K ); K = K =2; (V + =2)? : Consequently, plugging the bound (.8) into (2.19), one obtains that M () C = C 1 0?2 (V (x) + ) 2? + (V (x) + )? b(x) V? (x) +1 dx 1 0 t?2 (1? t) 2 dt + C dxd V? (x) b(x)dx 1 0 t?2 (1? t)dt: For > 1 the r.h.s. is nite and therefore (2.11) follows. To prove (2.12) we use (2.21) with = 0, which provides the desired result due to (.8) and (2.19).

17 16 A.V. SOBOLEV Appendix. On the operators Q Let Q be the operators dened in (1.3). As was mentioned in Sect. 2, Q are closable on C 1 0 (R2 ), since Q Q. In this appendix we nd conditions on the magnetic eld, which guarantee Q = Q. Suppose that there exists a function B 2 C 1 (R 3 ) such that B(x) B(x); B(x) 1; rb(x) CB(x); 8x 2 R 3 : (A1) (A2) Note that unlie (2.7), the inequality (A1) is required for both positive and negative parts of the magnetic eld. Theorem A1. Let a 2 C 1 (R 2 ) and the eld B obeys the conditions (A1), (A2). Then Q = Q. To prove Theorem A1 it suces to establish essential self-adointness of the symmetric operator 0 Q? = (A3) Q + 0 on D = C 1 0 (R2 ) C 1 0 (R2 ). Observe that = H, where H is the Pauli operator (1.1): H = A+ 0 : 0 A? It is clear that on D one has H = 2. Theorem A1 will result from the following abstract commutator lemma (see [3], [16]): Lemma A2. Let M be an essentially self-adoint positive-denite operator. Suppose that a symmetric operator S satises the conditions D(S) = D(M) and Sf CMf; 8f 2 D(M); (Sf; Mf)? (Mf; Sf) C 0 (Mf; f); 8f 2 D(M): (A) (A5) Then S is essentially self-adoint. Proof. We shall use Lemma A2 with M = H + BI; S = ; D(S) = D(M) = D: Since B 1 and H 0, the operator M is positive-denite. In view of (2.2) and (A1), B? B 0 M = H a I+ ; = 0 B 0: + B Therefore M is essentially self-adoint on D (see [1]). Since B 1, for any f 2 D one has f 2 = h 2 f; fi MfBf Mf 2 ;

18 LIEB-THIRRING ESTIMATES 17 which guarantees (A). Now, L[f] = hf; ( 2 + BI)fi? h( 2 + BI)f; fi =?h[; BI]f; fi: It follows from the denition of that so that Therefore, by (A2) [; BI] =?i 1 B? i@ 2 1 B + i@ 2 B 0?rBI i[; BI] rbi: ; L[f] hrbif; fi ChBIf; fi ChMf; fi: Consequently, (A5) is fullled. Now Lemma A2 yields essential self-adointness of the operator on D, which in its turn provides the equality Q = Q. Acnowledgements Part of the paper was written during the author's visit to the International Erwin Schrodinger Institute for Mathematical Physics in Vienna in December 199. The author is grateful to L. Erd}os for fruitful discussions on the subect and critical comments on the paper. References 1. Y. Aharonov, A. Casher, Ground state of spin-1/2 charged particle in a two-dimensional magnetic eld, Phys. Rev. A19, 261{ J. Avron, I. Herbst, B. Simon, Schrodinger operators with magnetic elds, I. General interactions, Due Math. J. 5 (1978), 87{ M.S. Birman, M.. Solomya, Spectral theory of self-adoint operators in Hilbert space, Reidel, H.L. Cycon, R.G. Frose, W. Kirsch, B. Simon, Schrodinger Operators, Texts and Monographs in Physics, Springer, Berlin Heidelberg, D.E. Edmunds, A. A. Ilyin, On some multiplicative inequalities and approximation numbers, Quart. J. Math. Oxford (2)5 (199), 159{ L. Erd}os, Ground state density of the two-dimensional Pauli operator in the strong magnetic eld, Lett. Math. Phys. 29 (1993), 219{20. 7., Magnetic Lieb-Thirring inequalities, Commun. Math. Phys (1995). 8., Magnetic Lieb-Thirring inequalities and estimates on stochastic oscillatory integrals, Ph.D. Thesis, Princeton University, L. Hormander, The analysis of linear partial dierential operators, I, Springer, Berlin, E. H. Lieb, J.P. Solove, J. Yngvason, The ground states of large quantum dots in magnetic elds, Princeton University Preprint, , Quantum dots, Proceedings of the conference on partial dierential equations and mathematical physics, Birmingham, Alabama, 199, International Press, In press. 12., Asymptotics of heavy atoms in high magnetic elds: II. Semiclassical regions, Comm. Math. Phys. 161 (199), 77{ O.A. Ladyzensaa, V.A. Solonniov, N.N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of mathematical monographs, V. 23, AMS, Providence, M. Reed, B. Simon, Methods of modern mathematical physics, II, Academic Press, New Yor, , Methods of modern mathematical physics, III, Academic Press, New Yor, , Methods of modern mathematical physics, IV, Academic Press, New Yor, MAPS, University of Sussex, Falmer, Brighton, BN1 9QH, UK