ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-19 Wien, Austria The Negative Discrete Spectrum of a Class of Two{Dimentional Schrodinger Operators with Magnetic Fields A. Laptev O. Safronov Vienna, Preprint ESI 66 (1998) September, 1998 Supported by Federal Ministry of Science and Transport, Austria Available via http://www.esi.ac.at
THE NEGATIVE DISCRETE SPECTRUM OF A CLASS OF TWO-DIMENTIONAL SCHR ODINGER OPERATORS WITH MAGNETIC FIELDS by A. Laptev and O. Safronov Abstract. We obtain an asymptotic formula for the number of negative eigenvalues of a class of two-dimensional Schrodinger operators with small magnetic elds. This number increases as a coupling constant of the magnetic eld tends to zero. 1.Introduction and Notations. Let us consider a selfadjoint Schrodinger operator in L (R ) formally written as (1.1) H = (ir + A)? V; > : Here the electric potential V = V (x), x R, is a nonnegative function and A = A(x) = (A 1 (x); A (x)) is a vector potential. The precise denition of the operator H can be given via the corresponding quadratic form (1.) h[u] = ZR jiru + A uj? V juj dx; u H 1 (R ): Under certain conditions on V and A the form (1.) is closed on the Sobolev space H 1 (R ) and generates the selfadjoint operator H in L (R ). We shall study a class of operators whose discrete negative spectrum is innite when = and nite when >. The number of the negative eigenvalues of the operator H, > is denoted by N() = N(H ) = N(; h) depending on what is more convenient. In this paper we obtain an asymptotic formula for the value N(), as! and therefore give a quantative description of how a magnetic eld \lifts up" the spectrum of the operator H. This setting becomes possible because the spectral point zero is the resonance state for a two-dimensional Schrodinger operator and is thus sensitive to perturbations either by electric or vector potentials. In order to formulate the main result we consider a gauge transformation which annihilates the radial component of the vector potential A (see [T, Section 8.4.]) and which sometimes is called by the transversal (or Poincare) gauge. The latter corresponds to a unitary transformation of the original operator and does not change its spectrum. Most of our discussions will be described in polar coordinates 1991 Mathematics Subject Classication. 35P15, 35P. The rst author was supported by the Swedish Natural Sciences Research Council, Grant M-AA/MA 9364-3. 1 Typeset by AMS-TEX
BY A. LAPTEV AND O. SAFRONOV (r; ) [; 1) T at x R, where T = fx R : jxj = 1g. We denote by e r = x=r the radial unit vector and by e the unit vector which completes e r up to the oriented orthonormal basis at x 6=. Proposition 3.1 shows that if A is smooth enough, then one can gauge away the radial component of the vector potential A or nd real functions and a, such that (1.3) A + r = a(x)e : Then (1.) transfers into a more convenient form for which we use the same notation (1.4) h[u] = ju rj + jir?1 u + auj? V juj ) r ddr: In the next section we formulate our main result (Theorem.1) concerning the asymptotics of the value N() as! and give its proof in Section 4. Theorem.1 deals with the special behaviour of the vector and electric potentials near innity and covers the important applications including Aharonov-Casher magnetic elds. If = then the innite number of the negative eigenvalues is prescribed by the behaviour of V at 1. In particular, our result shows that a compactly supported magnetic eld with a small coupling constant (the case = in Theorem.1) makes the negative spectrum nite (an immediate inuence at 1) and gives the rate of increase of the number of negative eigenvalues as the coupling constant turns to zero. We show in Section 5 that similar eects hold true if a has some prescribed asymptotic properties at x R and nally we apply our results to vector potentials corresponding to the Aharonov-Bohm eect in Section 6. The result of Theorem.1 can be easily generalized to the case when Condition () (see Section ) is satised only asymptotically as r! 1. In fact a similar result holds true for a much wider class of magnetic Schrodinger operators without a prescribed asymptotic behavior at innity, but involving some cumbersome quantative assumptions on the mean values of V and a over T. Finally we decided to consider here a simple case covering, however, the most important applications. Denote by < ; > the real scalar product of vectors in R. Let B(p; q) = t p?1 (1? t) q?1 dt be the Beta function. An open ball with its center at x R and radius r > is denoted by B(x; r); R + := (; 1), N and Zare sets of positive and all integers respectively. For the scalar product in L (T) we use (; ) Ṫ The class of functions u L p (R ), 1 p 1, whose distributional derivatives belong to L p (R ), is denoted by W 1;p (R ). The corresponding local class of functions is W 1;p loc (R ). Let us also agree that the number of negative eigenvalues corresponding to a quadratic form will sometimes be denoted by its reference number, say N(; h) = N(; (1:)). Let G be a function of and G()! 1, as!. In a similar way we shall denote the functionals (1.5) G (h) = G ((1:)) = lim sup N(; h)=g()! and G (h) = G ((1:)) = lim inf N(; h)=g():!
THE NEGATIVE DISCRETE SPECTRUM OF SCHR ODINGER OPERATORS 3. Assumptions and Results. It is now convenient to formulate our main assumptions in terms of the function a introduced in (1.3) (see also Proposition 3.1). Introduce the mean values a and V of the functions a and V over T (.1) a = 1 Z a(r; ) d and 1 V = V (r; ) d: We assume now that the following four conditions are satised: There exist constants C 1 ; C >, a function L 1 (T) and r > 1 such that (1) C 1 r (log r) V (r) C r ; r r (log r) ; < < ; () () a(r; ) = r r r(log r) = ; < ; Suppose also that there is a constant C >, such that for any r 1 ; r > r "(3)" (1) ja? (r 1 ) V (r 1 )? a? (r ) V (r )j Cj log(r 1 r?1 )j(min(log r 1; log r ) + 1)?1?+ : In order to characterize the behaviour of V; a for r < r we will suppose that (1) a; V L 1 (R ): Conditions (1) and (3) follow from the corresponding conditions in Theorem XIII.8 [RS] applied to the eective potential given in (4.9). If Conditions (1)- (4) are satised, then the quadratic form (1.4) is semi-bounded, closed on H 1 (R ) and therefore denes the selfadjoint operator H. Note that for the mean value of we have = r(log r) = a; r > r : Therem.1. Let the functions V and a satisfy the conditions (1)-(4), 6= and < <. Then the number of the negative eigenvalues N(H ) of the operator (1.1) satises the asymptotic formula (.) lim N(H )=G() = 1;! where G() = 1 Z 1= V dx (jxj)? a R + jxj : Remark.1. If a bounded potential V (x) cjxj? log? jxj as jxj! 1 and >, c >, then the Schrodinger operator?? V in L (R d ), d 3 has nite number of negative eigenvalues (see [BS] and [L]). However, if d = and < <, then this fact is no longer true and the negative spectrum of the operator?? V is innite due to the lack of the Hardy inequality (see [BL], [Sol]). Theorem 1.1 shows that a nontrivial magnetic eld \lifts" the negative spectrum up and gives quantative characteristics of this lifting in terms of asymptotics with respect to a small coupling constant at the vector potential.
4 BY A. LAPTEV AND O. SAFRONOV Remark.. The asymptotic formula (.) depends only on the mean values of the potentials V and a over T and, for example, holds true even when the supports of these functions do not intersect. Remark.3. The value a appearing in Theorem.1 is invariant with respect to gauge transformations. Indeed, for any and r > r by the Stocks formula we nd (.3) a = 1 r?1 r a(r; ) d = 1 Z r?1 (@ x1 A? @ x A 1 ) dx: B(;r) Remark.4. If =, then the magnetic eld B = curl A = for r > r. Thus Theorem.1 establishes the following phenomena: A small perturbation by a compactly supported magnetic eld can make the innite negative spectrum of a Schrodinger operator nite. The number of its negative eigenvalues satises the asymptotic formula (.), as the coupling constant!. In some special cases the formula (.) can be simplied. Corollary.. Let Conditions () and (4) be satised. Assume also that Then V (r; ) = () r (log r) ; r r ; < < : lim { N(H ) = c ; ({+1)=?{ ;! { where c ; =?1 (? )?1 B ; 3 ; { =?? : 3. Preliminary Results. We shall show here that under some assumptions of smoothness of the vector potential A, its radial component can be gauged away. Proposition 3.1. Let A : R! R be a vector potential in R and assume that its radial component A rad =< A; e r > satisfy the following conditions r?1 A rad L 1 loc(r ) and 1 @A rad r @ L 1 loc(r ): Then there exists a real function W 1;1 loc (R ), such that A(x) + r (x) = a(x)e ; a.e. where (3.1) a(x) = a(r; ) =< A; e >? r?1 @ @ Z r A rad (; ) d: Proof. Let (r; ) =? R r A rad(; ) d (for almost all this function is dened for for all r > ). Then W 1;1 and loc Z r = r e r + r?1 @ r e =? < A; e r > e r? r?1 A rad (; ) d e ; @
and thus THE NEGATIVE DISCRETE SPECTRUM OF SCHR ODINGER OPERATORS 5 The proof is completed. < A + r ; e r >= ; < A + r ; e >= a a.e. Remark 3.1. Proposition 3.1 can be easily generalized into a corresponding statement for vector potentials in R d for any d. We shall see later that the asymptotics does not change if we change the potentials a and V on a compact set. Therefore we assume that the function a is smooth for jxj < r. When studying the quadratic form (1.4) we use spectral properties of the selfadjoint dierential operator K ;r dened by (3.) (K ;r ')() = i @ @ '() + r a(r; )'(); < ; ' H 1 (T) and depending on r and as parameters. The spectrum of this operator is discrete and its eigenvalues f k g kz and the complete orthonormal system of eigenfunctions f' k g kz are given by the expressions: (3.3) k = k (; r) = k + r a(r); k Z; and (3.4) ' k = ' k (; r; ) = 1 p e?i( k(;r)? r R a(r; ) d ) ; k Z: In particular, k and ' k are dierentiable with respect to r and we also always have (3.5) (' k ; (' k ) r) T = ; r > r ; k Z: Condition () also gives us that for r > r (3.6)!(r) = 1 j(' ) rj d = 1 = 1? log?= r r Z (r(a? a)) r d Z We introduce now the following decomposition (3.7) u = z + g; u L (R ); where d ( ()? ) d d = o(r? (log r)? ); as r! 1: (3.8) z = z(; r; ) = (u; ' ) T ' (; r; ) = ' (; r; ) u(r; )' (; r; ) d; r-a.e.;
6 BY A. LAPTEV AND O. SAFRONOV and (3.9) g = u? z: Obviously (g; ' ) T = g(; r; )' (; r; ) d = for almost all r. Therefore the equations (3.7)-(3.9) induce the corresponding orthogonal decomposition of the Hilbert space L (R ). They also give a direct (not necessary orthogonal) decomposition of the space H 1 (R ). Using the property ' W 1;1 (R ) we nd that v(r)' (r; ) H 1 (R ) if and only if v H 1 (R + ). This fact will be important for us when we study the restriction of the form h on the class of functions (3.8). The decomposition (3.7) is not invariant with respect to the action of the operator H. We show, however, that there is an \asymptotic invariance" as!, which allows us to compute the asymptotic formulae (.) and (.3). 4. Proof of Theorem.1. We shall now discuss two side estimates for the quadratic form (1.4) employing the above mentioned asymptotic invariance of this form restricted on the functions (3.8) and (3.9). Upper spectral estimate. In order to obtain an upper estimate for the negative spectrum of the operator (1.1) we begin by estimating the form (1.4) from below (4.1) h[u] = h[z + g] = j(z + g) r )j ) + ji r?1 (z + g) + a(z + g)j? V (r; )jz + gj r ddr: Let < " < 1. Applying the Cauchy inequality we obtain (4.) h[u] h 1 [z] + h [g]; where (4.3) h 1 [z] = and (4.4) h [g] = Since (g; ' ) T = we nd (1? ")jz rj + ji r?1 z? a zj? (1 + ")V jzj r ddr (1?"?1 )jg rj +ji r?1 g? a gj?(1+"?1 )V )jgj r ddr: ji r?1 g? a gj d r? 1(; r) jgj d; where 1 is dened in (3.3) (k = 1). It is now easy to see that there exists >, such that the operator generated by the form (4.4) has a nite number of negative
THE NEGATIVE DISCRETE SPECTRUM OF SCHR ODINGER OPERATORS 7 eigenvalues uniformly with respect to <. Indeed, from (3.3) and Condition (3) we observe that if < 1= and r > r, then for any < we have 1 (; r) > 1=. Since the potential V is a bounded function satisfying V (r; ) = o(r? ), r! 1, we conclude that [(r)?? (1 + "?1 )V ]? is a bounded function whose support is a compact set and thus the negative spectrum of the form h is nite. Let us study the form h 1 [z]. By substituting z(r; ) = v(r)' (r; ) and using (3.5) we obtain (4.5) h 1 [z] = (1? ")j(v' ) rj + r? (; r)jvj j' j? (1 + ")V jvj j' j r ddr (4.6) = (1? ")jv j +? (1? ")! + a? (1 + ") V ) jvj r dr; where! is dened in (3.6). Note that in order to obtain a lower estimate for the form (4.6), we can omit the term with!, since this function is positive. Let us x an arbitrary r 1 >. Using rather standard variational arguments we can reduce the study of the upper estimate for the value N(; (4:6)) to that of the upper estimate for N(; (4:7)), where (4.7) r 1 (1? ")jv j +? a? (1 + ") V ) jvj r dr; v C 1 (r 1; 1): so the term with the function! is omitted and we take the integration over the interval (r 1 ; 1). Without loss of generality we can assume that a(r; ) = () r?1 (log r)?= for any r > r 1 i.e. a is substituted by its behavior valid for r > r. In order to prove this fact it is sucient to note that the number of the negative spectrum of the form Z r1 ((1? ")jv j? (1 + ") V jvj ) r dr is nite and independent of. Thus by the splitting principle, the asymptotics for N(; (4:6)) does not change if we change the function a on a compact set and integrate in (4.6) from r 1 to 1. Returning to the study of the quadratic form (4.7) we introduce the substitution (4.8) y(t) = (log r)?=4 v(r); r = exp(((? )t=)? ): We also put (4.9) (t) = V (e ((?)t=)? )=a (e ((?)t=)? ) ; and choose r 1 = exp(((? )=)? ):
8 BY A. LAPTEV AND O. SAFRONOV Then we nd that N(; (4:7)) = N(; (4:1)), where (4.1) (1? ")jy j + (1? ")ct? +?? (1 + ")(t) jyj dt; 1 y C 1 (1; 1); with some constant c > which can be precisely calculated. Applying Theorem XIII.8 [RS] to the operator generated by the form (4.1), we arrive at (4.11) G (h 1 ) G ((4:1)) 1 = lim sup (G())?1! p 1? " 1 1 = lim sup (G())?1! p 1? "? ((1 + ")? ) + (1? ")ct? 1= dt +? (1 + ") V? a 1= + dr: Let us show that " appearing in the right hand side of (4.11) can be substituted by. Indeed, using the inequality p (a + b) +? p a + p b + and Condition (1) we have lim sup! (G())?1 Z?(1 + ") V? a 1= + dr? 1 p " lim sup (G())?1 V! Z(1+") 1= dr C p "; V a where C = C( V ; a). Since " is arbitrary this implies G (h 1 ) 1 which is equivalent to the inequality lim sup(g())?1 N(H ) 1:! Lower spectral estimate. The proof of the estimate of the value N(H ) from below coincides with the corresponding proof of the upper estimate. We only need to write the opposite inequalities, change " by?" and use the functional G instead of G. MOreover, we should not ignore the term containing ~' but use the estimate (3.6). This gives us lim sup(g())?1 N(H ) 1:! The theorem is proved. 5. Local singularities. Let us now consider the problem generated by the form (1.4), where a and V have compact supports and satisfy the following local conditions at x = : There exist constants C 1 ; C >, a function L 1 (T) and a number r < 1 such that (1*) (*) C 1 r (log r) V (r) C r (log r) ; r r ; < < ; a(r; ) = () r(log r) = r r ; < ;
THE NEGATIVE DISCRETE SPECTRUM OF SCHR ODINGER OPERATORS 9 Suppose also that there is a constant C >, such that for any < r 1 ; r < r (3*) ja? (r 1 ) V (r1 )? a? (r ) V (r )j Cj log(r 1 r?1 )j(min(log r 1; log r ) + 1)?1?+ : (4*) the functions a and V are compactly supported and a; V L 1 (R n B(; r )): If Conditions (1*)-(4*) are satised, then the quadratic form (1.4) is semibounded on C 1 (R n fg), therefore its closure generates the selfadjoint operator (1.1) for which we use the same notation H. Nothing new is needed in order to obtain the following result: Therem 5.1. Let the functions V and a satisfy Conditions (1*)-(4*), 6= and < < then the asymptotic formula (.) is fullled. Obviously we can also consider a combined problem where the functions a and V are singular both at innity and at zero (or at any other (several) point(s)). In order to prove the corresponding results one needs to use the splitting arguments together with Theorems.1 and 5.1. 6. Aharonov-Bohm-type magnetic elds. Finally we would like to discuss a special case, where (6.1) a(r; ) = () ; L 1 (T): r The corresponding magnetic eld is zero everywhere except x =. In this case the exponential change of variables (4.8) applied to the whole R + leads to a study of the number of the negative eigenvalues of a Schrodinger operator on R and we obtain: Theorem 6.1. Let a be dened by (6.1). Assume that there is < < such that r (1 + j log rj )V (r; ) L 1 (R ) and either lim inf r!1 r (log r) V (r; ) > ; or lim sup r j log rj V (r; ) > : r! Suppose also that there is a constant C >, such that for any r 1 ; r R + jr 1 V (r 1 )? r V (r )j Cj log(r 1 r?1 )j(min(log r 1; log r ) + 1)?1? : If now 6=, then lim! N(H )=G() = 1;
1 BY A. LAPTEV AND O. SAFRONOV where G() = 1 ZR 1= V (jxj)? dx jxj + jxj : Acknowledgments. The authors are grateful to T.Weidl for useful discussions. The rst author would like to express his gratitude to T. Homan-Ostenhof for inviting him to the International Erwin Schrodinger Institute in Vienna in June 1998, where he was able to discuss the results of this paper with the participants of the conference on magnetic Schrodinger operators. References [BL] M.Sh.Birman and A.Laptev, The negative discrete spectrum of a two-dimensional Schrodinger operator, Comm. Pure Appl. Math. XLIX (1996), 967{997. [BS] M.Sh.Birman and M.Z.Solomyak, Estimates for the number of negative eigenvalues of the Schrodinger operator and its generalizations., Advances in Soviet Math. 7 (1991), 1{55. [L] A. Laptev, Asymptotics of the negative discrete spectrum of a class of Schrodinger operators with large coupling constant., Proc. of AMS 119 (1993), 481{488. [RS] M.Reed and B.Simon, Methods of mordern mathematical physics. 4, vol. 4, Academic Press, New York, San Francisco, London, 1978. [Sol] M.Solomyak, Piecewise-polynomial approximation of functions from H l ((; 1) d ), l = d, and applications to the spectral theory of Schrodinger operator, Israel J. Math. 86 (1994), 53{76. [T] B. Thaller, The Dirac equation., Texis and Monographs in Physics., Springer-Verlag, 199. Department of Mathematics, Royal Institute of Technology, S-1 44 Stockholm, Sweden E-mail address: laptev@math.kth.se Department of Mathematics, Royal Institute of Technology, S-1 44 Stockholm, Sweden E-mail address: safronov@math.kth.se