Virtual Eigenvalues of the High Order Schrödinger operator II

Transcription

1 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-090 Wien, Austria Virtual Eigenvalues of the High Order Schrödinger operator II Jonathan Arazy Leonid Zelenko Vienna, Preprint ESI 706 (2005) October 3, 2005 Supported by the Austrian Federal Ministry of Education, Science and Culture Available via

2 Virtual eigenvalues of the high order Schrödinger operator II Jonathan Arazy and Leonid Zelenko Abstract We consider the Schrödinger operator H γ = ( ) l + γv (x) acting in the space L 2(IR d ), where 2l d, V (x) 0, V (x) is continuous and is not identically zero, and lim x V (x) = 0. We study the asymptotic behavior as γ 0 of the non-bottom negative eigenvalues of H γ, which are born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(h 0) of the unperturbed operator H 0 = ( ) l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type. Mathematics Subject Classification 2000, Primary: 47F05, Secondary: 47E05, 35Pxx Keywords. Schrödinger operator, virtual eigenvalues, coupling constant, asymptotic behavior of virtual eigenvalues, Puiseux-Newton diagram, Lieb-Thirring estimates Introduction In the present paper, which is a continuation of [Ar-Z], we consider the elliptic differential operator of order 2l (l IN) H γ = ( ) l + γv (x) (.) acting in the space L 2 (IR d ). Here V (x) is the multiplication operator in L 2 (IR d ) by the real-valued, continuous, not identically zero, non-negative function V (x), defined on IR d and tending to zero sufficiently fast as x. We denote this operator briefly by V. We assume that the coupling constant γ is real. As it is conventional in the literature, we call the operator H γ the Schrödinger operator of order 2l and we call the function V (x) the potential. We consider the so-called virtual eigenvalues of the operator H γ. Recall that these are the negative eigenvalues which are born at the moment γ = 0 from the endpoint λ = 0 of the gap (, 0) of the spectrum σ(h 0 ) of the unperturbed operator H 0 = ( ) l, while γ varies from 0 to a small negative value γ 0 (see Definitions 3.4 and 3.3 of [Ar-Z]). Both authors were partially supported from the Israel Science Foundation (ISF), grant number 585/00, and from the German-Israeli Foundation (GIF), grant number I /200. The second author was partially supported also by the KAMEA Project for Scientific Absorption in Israel.

3 In [Ar-Z] we studied the asymptotic behavior of the bottom virtual eigenvalue of the operator H γ as γ 0. To this end we have developed in Section 3 of [Ar-Z] a supplement to the Birman-Schwinger theory ([Bi3], [Sc], [Re-Si], [S]), in which we studied the process of the birth of eigenvalues in a gap of the spectrum of the unperturbed operator for a small coupling constant. This is a generalization (to the case of relatively compact perturbations) of the theory developed in our earlier paper [Ar-Z] for the case of finite-rank perturbations. In Section 4 of [Ar-Z] we have extracted a finite-rank portion Φ(λ) from the Birman-Schwinger operator X V (λ) = V 2 ( ( ) l λi ) V 2 (λ < 0), such that the norm of the remainder X V (λ) Φ(λ) is uniformly bounded with respect to λ in ( δ, 0) for some δ > 0. In Section 5 of [Ar-Z], in order to obtain the asymptotic expansion of the bottom virtual eigenvalue of the operator H γ in the case of IR d with d odd, we have used a simple version of Schrödinger method ([Bau], Ch. 3, n o 3..2) for a power expansion of the maximal eigenvalue µ 0 (λ) of the operator Φ(λ), The use of this method was possible thanks the fact that the quantity t 2l d µ 0 ( t 2l ) is born at the moment t = 0 from a simple eigenvalue of the operator Φ 0 = lim t 0 t 2l d Φ( t 2l ). (.2) The leading terms of the desired asymptotic expansion of the bottom virtual eigenvalue have been obtained via an inversion of the asymptotic expansion of the maximal eigenvalue of the operator Φ(λ). In the case of an even dimension d we have used a modification of the method mentioned above. The goal of the present paper is to obtain asymptotic estimates for non-bottom virtual eigenvalues of the operator H γ as γ 0. It turns out that in most cases the leading coefficients of these estimates are algebraically computable in the sense that they are algebraic functions of power moments of the potential V (x). In some particular cases these coefficients can be calculated explicitly (see Theorem 3.4 and Examples 3., 4.). Notice that in [N-W] (Lemma 5.) the only result concerning the asymptotic of the non-bottom virtual eigenvalues is that their rate of decay as γ 0 is bigger than the rate of decay of the bottom virtual eigenvalue. On the basis of the results mentioned above we obtain some new asymptotic formulas of Lieb-Thirring type. The paper is divided into five sections and an appendix. After this introduction (Section ), we give in Section 2 the list of notation used in the paper. In Section 3 we obtain asymptotic estimates for the non-bottom virtual eigenvalues of the operator H γ as γ 0 in the case of an odd dimension d and 2l > d (Theorem 3.3). The leading terms of these asymptotic estimates are of power type, because the finiterank portion Φ( t 2l ) of the Birman-Schwinger operator X V ( t 2l ) is a meromorphic operator function in the case of an odd dimension d (see Lemma 4.3 and Proposition 4.4 of [Ar-Z]). We propose an algorithm for evaluation of leading coefficients of these asymptotic estimates. Notice that we cannot use in this case the simple version of Schrödinger method for a power expansion of unbounded branches µ j ( t 2l ) of nonmaximal eigenvalues of the operator Φ( t 2l ), because the corresponding quantities t 2l d µ j ( t 2l ) are born at the moment t = 0 from a multiple eigenvalue µ = 0 of the operator Φ 0 defined by (.2). Therefore we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions 2

4 explained in the Appendix of the paper (see [Bau], A.7 and [Va-Tr], Ch. I, 2). In the one-dimensional case (d = ) we derive from Theorem 3.3 explicit formulas for the leading coefficients in the asymptotic representation of the non-bottom virtual eigenvalues (Theorem 3.4). In Section 4 we obtain asymptotic estimates for non-bottom virtual eigenvalues of the operator H γ as γ 0 in the case of an even dimension d (Theorem 4.2). In this case the finite-rank portion Φ( t 2l ) of the Birman-Schwinger operator X V ( t 2l ) is no longer a meromorphic operator function because it contains summands with ln( t ) in its expansion near the point t = 0 (see Lemma 4.6 and Proposition 4.8 of [Ar-Z]). We try to overcome this difficulty with the help of some tricks. Notice that B. Simon has shown that for d = 2 and l = the unique virtual eigenvalue of the operator H γ has an exponential rate of decay for γ 0 (see [S], [S] and Remark 5.7 of [Ar-Z]). It turns out that also in the general case of an even d and 2l d the operator H γ always has virtual eigenvalues with an exponential rate of decay for γ 0, but all the rest of the virtual eigenvalues (if they exist) have a power rate of decay. We succeeded in obtaining asymptotic formulas with algebraically computable leading coefficients only for virtual eigenvalues with a power rate. For virtual eigenvalues with an exponential rate we have obtained in general only two-sided exponential estimates (see assertion (iv) of Theorem 4.2). But in some particular cases we get an asymptotic estimate of logarithm of such eigenvalues with algebraically computable leading coefficients (see assertion (i) of Theorem 5.6 of [Ar-Z] and assertion (ii) of Theorem 4.3). In Section 5 we obtain asymptotic formulas of Lieb-Thirring type, mentioned above, on the basis of the results of Sections 3 and 4 (Theorem 5.3). Notice that in [N-W] estimates (ordinary or asymptotic with respect to γ) of the sum of the form λ ν (γ) κ (γ < 0, κ > 0), ν have been carried out, where λ ν (γ) are negative eigenvalues of the operator H γ. In contrast to this, we consider groups of virtual eigenvalues {λ ν (γ)} ν Nj with the same power rate of decay as γ 0 and obtain asymptotic estimates for the sums of the form ν N j λ ν (γ) κj (.3) with suitable powers κ j > 0. These asymptotic estimates enable us to get an explicit information about asymptotic behavior of generalized means of these groups of virtual eigenvalues (see Remark 5.4). Notice that in general we cannot write an asymptotic formula for each individual virtual eigenvalue with an explicitly calculated leading coefficient, because to this end we must solve an algebraic equation of a high order (see assertion (iv) of Theorem 3.3). In some cases for the group of exponentially decaying virtual eigenvalues we obtain an asymptotic estimate of the sum, which is a logarithmic analog of the sum (.3) (see assertion (iii) of Theorem 5.3). The Appendix is devoted to the above mentioned Puiseux-Newton diagram for a class of polynomial matrix functions. We add the label A to the number of propositions and formulas in the Appendix. 2 Notation In this section we give a list of notation used in the present paper. 3

5 Z is the ring of all integers; IN is the set of all natural numbers, 2,...; Z + = IN {0}; IR is the field of all real numbers; IR + = [0, ); CI is the field of all complex numbers; R(z), I(z) are the real and the imaginary parts of a number z CI. #S is the number of elements of a finite set S; If X Y then we shall occasionally write X for Y \ X if Y is understood. O(x) is generic notation for a neighborhood of a point x. If M is a metric space, then dist(x, y) and dist(x, Y ) are the distance between points x, y M and the distance between a point x M and a set Y M. span(m) is the closure of the linear span of a subset of a Hilbert space H. CI d = d j= CI; IRd = d j= IR; Zd + = d j= Z +. x y = d j= x jy j is the canonical inner product of vectors x = (x, x 2,..., x d ) and y = (y, y 2,..., y d ) belonging to IR d ; x = x x is the Euclidean norm in IR d ; k = d j= k j is the l -norm of a multi-index k = (k, k 2,..., k d ) Z d +. x k = d j= xkj j, where x = (x,..., x d ) IR d and k = (k,..., k d ) Z d +; k! = d j= k j!, where k = (k,..., k d ) Z d +. f G is the restriction of a mapping f : A A 2 on a subset G A. If A is a closed linear operator acting in a Hilbert space H, then: R(A) is the resolvent set of A, that is the set of all λ CI such that A λi is continuously invertible; R λ (A) (λ R(A)) is the resolvent of A, that is R λ (A) = (A λi) ; σ(a) = CI \ R(A) is the spectrum of A. P G is the orthogonal projection on a closed subspace G of a Hilbert space H. S 2 is the Hilbert-Schmidt class of operators acting in a Hilbert space H; T 2 is the Hilbert-Schmidt norm of T S 2. On the set Z d + we define the ordering relation in the following manner: for k,n Z d + we say that k n, if either k < n, or k = n and in the last case the sequence k = (k, k 2,..., k d ) is lexically less than the sequence n = (n, n 2,..., n d ). According to this definition, we can write: We also denote: Z d + = {k ν } ν=0 and 0 = k 0 k k n.... (2.) K j = {k Z d + k = j} (j Z + ); 4

6 S j = #K j ; It is evident that K 0 = {0}, hence S 0 = ; G j = {k Z d + k j} (j Z + ); G j = #G j. Assume that j Z + and C K j. We denote: ( j ) C = ν=0 K ν C, if j > 0; If j = 0 and C = {0}, we put C = {0}; If j = 0 and C =, we put C =. The entries of the matrices which will be considered below are indexed by the elements of a linearly ordered set A with an ordering relation. A t is the transposed matrix for a matrix A; A is the matrix whose entries are complex conjugate to the corresponding entries of the matrix A; A = A t is the conjugate matrix to a matrix A; colon(c n,c n2,...,c nl ) (n n 2,..., n l ) is a matrix with the columns c n,c n2,...,c nl ; A k,n (k,n A) is the entry of a matrix A lying in the row with the index k and in the column with the index n; A H,G (H, G A) is the submatrix of the matrix A, in which the rows have indices from the set H and the columns have indices from the set G; if H = G, we write A H ; A H,G (H, G A) is the minor of the matrix A, in which the rows have indices from the set H and the columns have indices from the set G, that is A H,G = det(a H,G ); if H = G, we write A H. Let H and G be finite subsets of A, H G and H = G \ H. We denote: H,H = ( ) ν, where ν (= ν H,H ) is the number of pairs (k,n) such that n H, k H and k n. 3 Asymptotic estimates for non-bottom virtual eigenvalues in case of an odd dimension In this and in the next sections we shall obtain asymptotic estimates for non-bottom virtual eigenvalues of the operator H γ as γ 0. We propose an algorithm for evaluation of leading coefficients of these asymptotic estimates. We use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions explained in Appendix. In this section we shall consider the case of IR d with d odd. 3. o Assume that 2l > d. Let us recall that in the case of an odd d the finiterank portion Φ(λ) has been extracted from the Birman-Schwinger operator X V (λ) in Lemma 4.3 and Proposition 4.4 of [Ar-Z], that is it has the form: Φ(λ) = k,n: k+n 2m λ k+n 2m 2l ξ k+n (, h n )h k, (3.) 5

7 where and the quantities ξ k are defined by ξ k = (2π) d m = l d + 2 IR d s k ds s 2l + (3.2) ( k < 2l d), (3.3) and they have the following explicit form: d ξ k = (2π) d π l sin ( j= ( Γ ( ) m j + 2 π d l 2 + m )) Γ (. (3.4) d 2 + m ) Furthermore, recall that h k (x) = (ix)k (V (x)) 2 ( k 2m). (3.5) k! We shall assume that the potential V (x) satisfies the conditions and the condition V ( ) C(IR d ), (3.6) V (x) 0 x IR d, (3.7) lim V (x) = 0. (3.8) x IR d x 2(2l d) V (x)dx <, (3.9) which ensures the membership of the functions h k (x) ( k 2m) to the class L 2 (IR d ). Our immediate goal is to get a convenient matrix representation for the operator Φ(λ). The entries of all the matrices, considered below, are indexed by elements of the linearly ordered set Z d + with the ordering relation defined in Notation (see (2.)). Let us carry out the orthogonalization of the sequence h k (x) ( k 2m) by Schmidt, that is consider in L 2 (IR d ) the orthonormal sequence constructed in the following manner: {f j } G2m j=0 (G 2m = #{k Z d + k 2m}), f 0 = h 0 h 0, (3.0) where As is known, f j = h k j P Lj h kj h kj P Lj h kj (j =, 2,..., G 2m ), (3.) L j = span ( {h 0, h k,..., h kj } ). (3.2) j P Lj h kj = λ j,ν h kν, (3.3) ν=0 6

8 where λ j,ν = det(γ(j, ν)) det(γ(j)), (3.4) Γ(j) is the Gram matrix of the sequence h 0, h k,..., h kj, that is and Γ(j, ν) is a matrix of the form: Γ(j) r,s = (h kr, h ks ) (r, s {0,,..., j }) (3.5) Γ(j, ν) r,s = { (hkr, h ks ), if s ν, (h kr, h kj ), if s = ν (r, s, ν {0,,..., j }). (3.6) Recall that k 0 = 0 (see Notation, (2.)). Thus, we have the following orthogonalization formulas: f 0 = ω 0,0 h 0 f = ω,0 h 0 + ω, h k (3.7) f G = ω G,0 h 0 + ω G, h k + + ω G,G h kg (G = G 2m ), where and ω j,j = ω 0,0 = h 0, (3.8) h kj j r=0 λ j,rh kr (j {, 2,..., G 2m }) (3.9) λ j,ν ω j,ν = h kj j r=0 λ j,rh kr (j {, 2,..., G 2m }, ν {0,,..., j ). (3.20) Recall that the numbers λ j,ν are defined by (3.4), (3.5) and (3.6). We shall need the following Lemma 3.. Let F be the subspace of L 2 (IR d ) of the form ( ) F = span {h kj } G2m j=0 and {f j } G2m j=0 be the orthonormal sequence obtained from {h kj } G2m j=0 via the orthogonalization process (3.7). Let W be a linear operator acting in F and defined by the conditions: f j = Wh kj (j = 0,,..., G 2m ). (3.2) Then the matrix representation of the operator W in the basis {f j } G2m j=0 has the form: W ν,ρ = { ων,ρ, for ρ ν, 0, for ρ < ν, (ν, ρ {0,,..., G 2m }), (3.22) where ω ν,ρ are coefficients used in the orthogonalization formulas (3.7). 7

9 Proof. Equalities (3.7) imply that where the matrix h 0 = α 0,0 f 0 h k = α,0 f 0 + α, f (3.23) h kg = α G,0 f 0 + α G, f + + α G,G f G (G = G 2m ), A ν,ρ = { αν,ρ, for ν ρ, 0, for ν < ρ (ν, ρ {0,,..., G 2m }) is inverse to the matrix B ν,ρ = { ων,ρ, for ν ρ, 0, for ν < ρ, (ν, ρ {0,,..., G 2m }) which is the transposed matrix for the matrix, defined by (3.22). On the other hand, equalities (3.23) mean that the matrix A t is the representation of the operator W in the basis {f j } G2m j=0. These circumstances mean that the matrix, defined by (3.22), is the representation of the operator W in this basis. The lemma is proven. We now turn to the main statement of this subsection. Proposition 3.2. Assume that 2l > d. Then for any fixed λ < 0 the set of all nonzero eigenvalues µ = µ(λ) of the operator Φ(λ), defined by (3.), coincides with the set of all non-zero roots of the equation where L(λ) is the matrix of the form: L(λ) k,n = det( L(λ) µt ) = 0, { λ k+n 2m 2l ξ k+n for k + n 2m 0 for k + n > 2m (k,n G 2m), (3.24) the integer m is defined by (3.2), the numbers ξ k are defined by (3.4), the matrix T has the form T = WW (3.25) and W is the matrix of the form (3.22), in which ω ν,ρ are coefficients used in the orthogonalization formulas (3.7) and calculated by (3.8), (3.9), (3.20), (3.4), (3.5) and (3.6). Proof. We see from (3.) that the subspace ( ) F = span {h kj } G2m j=0 of the space L 2 (IR d ) is an invariant subspace of the operator Φ(λ) and, furthermore, σ(φ(λ)) = σ(φ F (λ)) {0}, 8

10 where Φ F (λ) = Φ(λ) F. Then it is enough to prove the assertion of the proposition for the operator Φ F (λ). Let f j L (j = 0,,..., G 2m ) be functions obtained from h kj by the procedure of the orthogonalization (3.7). Consider a linear operator W acting in the space F and defined by the conditions (3.2). Since the sequence {h kj } G2m j=0 is linearly independent, the operator W realizes a linear topological automorphism of the finitedimensional space F. We see from (3.) that Φ F (λ) = W L(λ)(W ), (3.26) where L(λ) is a linear operator acting in L, whose matrix representation in the orthonormal basis {f j } G2m j=0 has the form (3.24). Representation (3.26) together with Lemma 3. imply the assertion of the proposition. 3.2 o We now turn to the main result of this section. We assume that the virtual eigenvalues of the operator H γ at λ = 0 are indexed by the elements of the linearly ordered set Z d + (see Notation, (2.)) such that for γ < 0 λ 0 (γ) λ k (γ) λ kj (γ).... Theorem 3.3. Assume that d is odd, 2l > d, the potential V (x) is not identically zero and it satisfies conditions (3.6)-(3.8) and condition (3.9). Denote m = l d+ 2. Then: (i) The operator H γ, defined by (.), has r = #({k Z d + k m} = ( ) m+d d (3.27) virtual eigenvalues {λ k (γ)} k m at the endpoint λ = 0 of the gap (, 0) of σ(h 0 ) ; (ii) For the bottom virtual eigenvalue λ 0 (γ) the asymptotic expansion, described in Theorem 5.3 of [Ar-Z], is valid; (iii) For the rest of virtual eigenvalues λ k (γ) (k 0) the following asymptotic representation is valid for γ 0: ( ) 2l λ k (γ) = c k γ 2m+ 2 k + O( γ 2m+ 2 k ) (0 < k m); (3.28) (iv) The constants c k in (3.28) are calculated by the following formula for k = j {, 2,..., m}: 2m+ 2 k c k = ek, (3.29) in which the numbers {e k } k =j are positive and form the set of roots of the following algebraic equation: S j ( e) Sj Ξ Kj T Kj + ( e) Sj ν C,D K j: #C=#D=ν ν= 2l Ξ C, D T C, D C, C D, D = 0. (3.30) This fact have been established in [W] (Corollary 6.), but we prove it in the process of the proof of this theorem. 9

11 Here the matrix Ξ has the form Ξ k,n = { ξk+n, if k + n 2m, 0, if k + n > 2m ( k 2m, n 2m) (3.3) and ξ k is expressed by (3.4). Recall that the matrix T is defined by (3.25), (3.22), (3.8), (3.9), (3.20), (3.4), (3.5), (3.6) and (3.5). Proof. For λ < 0 consider the Birman-Schwinger operator X V (λ), Recall that it is defined in the following manner: X V (λ) = V 2 Rλ (H 0 )V 2. (3.32) By Proposition 4. of [Ar-Z], this operator is compact. By Proposition 4.4 of [Ar-Z], the representation is valid: X V (λ) = Φ(λ) + T(λ), (3.33) where, in view of formulas (4.7) and (4.7) from [Ar-Z] for the kernel Φ(x,y, λ) of the integral operator Φ(λ), the latter is a self-adjoint bounded operator of the rank G 2m having the form (3.). Furthermore, T(λ) is the integral operator belonging to the Hilbert-Schmidt class. Taking into account that T(λ) T(λ) 2, we obtain from estimate (4.8) of [Ar-Z]: Denote and T > 0, λ < 0 : T(λ) T. (3.34) t = λ 2l (3.35) Φ(t) = Φ( t 2l ). (3.36) By Proposition 3.2, for any fixed t the set of non-zero eigenvalues of the operator Φ(t) coincides with the set of all non-zero roots of the equation det( L( t 2l ) µt ) = 0, where the matrix L(λ) is defined by (3.24), that is it has the form: in which L(t) k,n = L( t 2l ) = L(t), t2m+ { ξk+n t k+n, if k + n 2m, 0, if k + n > 2m ( k 2m, n 2m). Observe that, in view of (3.25), the matrix T is positive-definite. Furthermore, the matrix Ξ Gm is positive-definite too, because in view of (3.3) and formula (3.3) for ξ k, it is a Gram matrix of a linearly independent system. The above circumstances and Proposition A.2 with N = 2m, p = m imply that the identically non-zero branches µ(t) of eigenvalues of the operator Φ(t) can be indexed by the elements of the linearly ordered set Z d + such that t O(0) (0, ) : µ 0 (t) µ k (t) µ k2 (t) µ kn+ (t) > 0 > µ kn+ (t)... (3.37) 0

12 for some neighborhood O(0) and these branches have the form: e k φ t 2m+ 2 k k (t), if k m, µ k (t) = tψ k (t), if m < k 2m, (3.38) where the functions φ k (t) and ψ k (t) are analytic in O(0) and φ k (0) =. Furthermore, the numbers e k ( k m) are positive and they are calculated according to the rule indicated in assertion (iv) of the theorem. We see from (3.38) that lim µ k (t) = t 0 { + for k m, 0 for k > m. (3.39) Let {µ + k j (λ)} m+ j=0 be the positive characteristic branches of the operator H 0 with respect to V on the gap (, 0) of σ(h 0 ) (see Definition 3.2 of [Ar-Z]), indexed by elements of the linearly ordered set Z d + and arranged in the non-increasing ordering: µ + 0 (λ) µ+ k (λ) µ + k m+ (λ) > 0. (3.40) Recall that for any fixed λ < 0 the sequence {µ + k j (λ)} m+ j=0 coincides with the set of all positive eigenvalues of the Birman-Schwinger ) operator X V (λ). Observe that, in view of (3.35), (3.36) and (3.37), { µ kj ( λ) 2l } n+ j=0 is the sequence of all positive branches of eigenvalues of the operator Φ(λ). Here λ varies in ( σ, 0) for some σ > 0. In view of representation (3.33), estimate (3.34) and orderings (3.37), (3.40), we get by Lemma 3.4 of [Ar-Z] for λ ( σ, 0): ) µ + k j (λ) µ kj ( λ) 2l T, ( ) if µ kj λ) 2l T > 0 (3.4) and ) µ kj ( λ) 2l µ + k j (λ) T, if µ + k j (λ) T > 0. (3.42) Let {µ + k j (λ)} l(0) j=0 be the main characteristic branches of H 0 with respect to V near the endpoint λ = 0 of the gap (, 0) of σ(h 0 ) (see Definition 3.9 of [Ar-Z]). Here l(0) is the corresponding asymptotic multiplicity M(0, H 0, V ) of the endpoint λ = 0. This means that { limµ + + for 0 j l(0), λ 0 k j (λ) = < + for j > l(0). Then properties (3.39), (3.4) and (3.42) imply that l(0) = r, where r is defined by (3.27). So, assertion (i) of the theorem is proven. Moreover, in view of (3.38), the asymptotic representation is valid for λ 0: µ + k (λ) = e k λ 2m+ 2 k 2l ( + O ( )) λ 2l ( k m). By Proposition 3.6 of [Ar-Z], the latter circumstances mean that there exist exactly r branches λ k (γ) ( k m) of virtual eigenvalues of the operator H γ at λ = 0 and they have the form: λ k (γ) = ( ( ) µ + ) k (γ < 0), γ

13 hence asymptotic representation (3.28) is valid, in which the coefficients c k are calculated according to the rule indicated in assertion (iv) of the theorem. The theorem is proven. 3.3 o In the one-dimensional case (d = ) Theorem 3.3 yields explicit formulas for the leading coefficients in the asymptotic representation of the non-bottom virtual eigenvalues. Before formulating the corresponding theorem, let us specify the notation introduced above for the one-dimensional case. We have for d = : m = l d + 2 = l r = #{k Z + k m} = l; for j Z + K j = {k Z + k = j} = {j}, (3.43) hence for j IN and C K j : S j = #K j = ; (3.44) C = { {0,,..., j}, if C = {j}, {0,,..., j }, if C =. (3.45) For d = the sequence of functions (3.5) has the form: h j (x) = (ix)j (V (x)) 2 (j {0,,..., 2l 2}). (3.46) j! In the case d = the matrix Ξ, defined by (3.3), has the form: Ξ ν,ρ = { ξν+ρ, if ν + ρ 2l 2, 0, if ν + ρ > 2l 2 (ν, ρ {0,,..., 2l 2}), (3.47) where ξ j = s j ds s 2l +. (3.48) The orthogonalization formulas (3.7) acquire the form in the case d = : f 0 = ω 0,0 h 0 f = ω,0 h 0 + ω, h (3.49) f G = ω G,0 h 0 + ω G, h + + ω G,G h G (G = 2l 2). We now turn to the theorem promised in the beginning of this subsection. Theorem 3.4. Let H γ be the operator, defined in L 2 (IR) by the operation: ( d2 ) l + γv (x), dx 2 2

14 where the potential V (x) is not identically zero and satisfies conditions (3.6)-(3.8) (with d = ) and the condition x 2(2l ) V (x) dx <. Then: (i) The operator H γ has l virtual eigenvalues {λ j (γ)} l j=0 (γ < 0) at the endpoint λ = 0 of the gap (, 0) of σ(h 0 ); (ii) For the bottom virtual eigenvalue λ 0 (γ) the asymptotic expansion, described in Theorem 5.3 of [Ar-Z], is valid with d = and m = l ; (iii) For the rest of virtual eigenvalues λ j (γ) (j 0) the following asymptotic representation is valid for γ 0: ( ) 2l λ j (γ) = c j γ 2l 2j + O( γ 2l 2j ) (j =, 2,..., l ); (3.50) (iv) The constants c j in (3.50) are positive and they are calculated by the following formula for j {, 2,..., l }: where 2l 2l 2j c j = ej, (3.5) e j = Ξ {0,,...,j} Ξ {0,,...,j } (dist(h j, L j )) 2, (3.52) L j = span ({h 0, h,..., h j }) (3.53) and the matrix Ξ is defined by (3.47) and (3.48). Recall that the integral (3.48) is calculated by the formula (3.4) with d =. Proof. By virtue of Theorem 3.3, it remains only to prove formula (3.52). In view of (3.43), (3.44) and (3.45), equation (3.30) (see assertion (iv) of Theorem 3.3) takes the following form in the case d = : Recall that ( e)ξ {0,,...,j } T {0,,...,j } + Ξ {0,,...,j} T {0,,...,j} = 0. (3.54) where W is a triangular matrix of the form: T = WW, (3.55) W ν,ρ = { ων,ρ, for ρ ν, 0, for ρ < ν, (ν, ρ {0,,..., 2l 2}). Here ω ν,ρ are coefficients used in the orthogonalization formulas (3.49). The root of the equation (3.54) is following: e j = Ξ {0,,...,j} Ξ {0,,...,j } T {j+,j+2,...,2l 2} T {j,j+,...,2l 2}. (3.56) 3

15 Since W is an upper triangular matrix, then (3.55) implies that T {j,j+,...,2l 2} = W {j,j+,...,2l 2} ( W {j,j+,...,2l 2}), hence T {j,j+,...,2l 2} = W {j,j+,...,2l 2} W {j,j+,...,2l 2}. (3.57) On the other hand, since W {j,j+,...,2l 2} is a triangular matrix, then W {j,j+,...,2l 2} = W {j,j+,...,2l 2} = ω j,jω j+,j+ ω 2l 2,2l 2. The latter equality, (3.56) and (3.57) imply that On the other hand, in view of (3.9) and (3.3), e j = Ξ {0,,...,j} Ξ {0,,...,j } ωj,j 2. (3.58) ω j,j = h j P Lj h j = dist(h j, L j ), (3.59) where the subspace L j is defined by (3.53). From (3.58) and (3.59) we get the desired formula (3.52). The theorem is proven. Remark 3.5. In formula (3.52) the quantity dist(h j, L j ) can be expressed explicitly in the following manner: j dist(h j, L j ) = h j λ j,ν h ν, ν=0 where and Γ(j, ν) r,s = λ j,ν = det(γ(j, ν)) det(γ(j)), Γ(j) r,s = (h r, h s ) (r, s {0,,..., j }) { (hr, h s ), if s ν, (h r, h j ), if s = ν (r, s, ν {0,,..., j }). 3.4 o In this subsection we shall consider an example of the operator H γ on the basis of application of Theorem 5.3 from [Ar-Z] and of Theorem 3.3. In this example the coefficients of the asymptotic representation can be found in an explicit form. First we shall prove a lemma, which will be used also in the next sections. Observe that the set K = {k Z d + k = } consists of d vectors: where k j has the form: K = {k,k 2,...,k d }, (k j ) ν = {, ν = j, 0, ν j. 4

16 Lemma 3.6. For any k K the following equality is valid: T {k} = T {0} ( h k 2 (h k, h 0 ) 2 ) h 0 2. (3.60) Proof. Assume that k = k j (j {, 2,..., d}). Making use of the formula for an inverse matrix, we have: ( T {k} = T {0,kj} = T {0} T {0} ). (3.6) k j,k j On the other hand, since T = WW and the matrix W is upper triangular, then Let us recall that the entries of the matrix T {0} = W {0} ( W {0} ). (3.62) W t ν,ρ = { ωρ,ν, for ρ ν, 0, for ρ > ν, (ν, ρ {0,,..., G 2m }) are used in the orthogonalization formulas (3.7). Then the equalities hold: h 0 = α 0,0 f 0 h k = α,0 f 0 + α, f (3.63) h kg = α G,0 f 0 + α G, f + + α G,G f G (G = G 2m ), where the matrix A ν,ρ = { αν,ρ, for ν ρ, 0, for ν < ρ (ν, ρ {0,,..., G 2m }) is inverse to the matrix W t. It is evident that ( ( A {0} = W {0} ) ) t. Hence we get from (3.62) that (T {0} ) = A {0} ( A {0} ) t. (3.64) Then, taking into account the facts that {f j } G2m j=0 is an orthonormal system and α 0,0 = h 0, we get from (3.63): (A {0} ( A {0} ) t ) k j,k j = j α j,ν 2 = ν= h kj 2 α j,0 2 = h kj 2 (h kj, f 0 ) 2 = h kj 2 (h k j, h 0 ) 2 h 0 2. The latter equality and equalities (3.6), (3.64) imply the desired equality (3.60). The lemma is proven. 5

17 We now turn to an example of the operator H γ, promised in the beginning of this subsection. Example 3.. Consider the case l = d = 3 and assume that the potential V (x) satisfies the conditions of Theorem 5.3 from [Ar-Z] and of Theorem 3.3 with l = d = 3. In this case m = l d+ = and r = ( ) m+d 2 m = 4. The latter means that the operator H γ has four virtual eigenvalues at the endpoint λ = 0 of the gap (, 0) of σ(h 0 ). We shall write their asymptotic representation making use of the theorems mentioned above. First of all, observe that in our case G 2 = #{k Z 3 + k 2} = 0 and the sets have the form: and where K j = {k Z 3 + k = j} (j {0,, 2}) K 0 = {k 0 } = {0}, K = {k,k 2,k 3 } K 2 = {k 4,k 5,k 6,k 7,k 8,k 9 }, k = {0, 0, }, k 2 = {0,, 0}, k 3 = {, 0, 0}, k 4 = {0, 0, 2}, k 5 = {0,, }, k 6 = {0, 2, 0}, k 7 = {, 0, }, k 8 = {,, 0}, k 9 = {2, 0, 0}. Making use of Theorem 5.3 from [Ar-Z], we can write the following asymptotic expansion for the bottom virtual eigenvalue of the operator H γ for γ 0: ( ) 3 λ 0 (γ) = γ 2 δ 0 + δ γ O(γ), (3.65) where δ 0 and δ are coefficients of the polynomial p(ǫ) = ǫ(δ 0 + δ ǫ), which is calculated by the following procedure: t 0 (ǫ) = 0, t (ǫ) = ǫ (θ 0 (t 0 (ǫ))) 0, p(ǫ) = ǫ (θ (t (ǫ))). Here hence t (ǫ) = ǫν and θ 0 (t) = ν 2 3 0, θ (t) = (ν 0 + ν t) 2 3, p(ǫ) = ǫν ( + 2 ) 3 ν ν 3 0 ǫ. These circumstances mean that the asymptotic expansion (3.65) acquires the following explicit form: λ 0 (γ) = γ 2 (ν ν ν 3 0 γ O(γ) ) 3. 6

18 Recall that ν 0 = ξ 0 h 0 2 = ξ 0 IR 3 V (s) ds, ν = (Φ X 0, X 0 ), X 0 = h 0 V (x) h 0 = V (s) ds IR 3 and Φ = ξ k+n (, h n )h k. k,n: k+n =2 Also recall that the numbers ξ k are calculated by formula (3.4) and the functions h k are defined by (3.5). We now turn to an asymptotic representation of the rest of virtual eigenvalues of the operator H γ. Let Ξ be the matrix with the size 0 0, defined by (3.3), (3.4) and T be the matrix with the same size, defined by (3.25), (3.22), (3.8), (3.9), (3.20), (3.4), (3.5) and (3.6). Observe that, in view of properties (i) and (ii) of the numbers ξ k given in Proposition 4.5 of [Ar-Z], { 0, if j ν, Ξ kj,k ν = (j, ν {, 2, 3}). ξ (2,0,0), if j = ν Hence we get: Ξ C, D = { 0, if C D, ξ 0 ( ξ(2,0,0) ) #C, if C = D (C, D K ). (3.66) By Theorem 3.3, for the non-bottom virtual eigenvalues λ k (γ) ( k = ) of the operator H γ the following asymptotic representation holds as γ 0: where λ k (γ) = c k γ 6 ( + O(γ)), c k = (e k ) 6 and, in view of (3.30) and (3.66), the quantities e k, e k2, e k3 are the roots of the following cubic equation: ( ) e 3 T {k,...,k 9} + e 2 ξ (2,0,0) T{k2,k 3,...,k 9} + T {k,k 3,...,k 9} + T {k,k 2,k 4,...,k 9} e ( ) 2 ( ) ξ (2,0,0) T{k3,...,k 9} + T {k2,k 4,...,k 9} + T {k,k 4,...,k 9} + ( ξ (2,0,0) ) 3 T{k4,...,k 9} = 0. Observe that, by Lemma 3.6, T {k2,k 3,...,k 9} = T {k,k 2,...,k 9} ( h k 2 (h k, h 0 ) 2 ) h 0 2, T {k,k 3,...,k 9} = T {k,k 2,...,k 9} ( h k2 2 (h k 2, h 0 ) 2 ) h 0 2 and T {k,k 2,k 4,...,k 9} = T {k,k 2,...,k 9} ( h k3 2 (h k 3, h 0 ) 2 h 0 2 ). 7

19 4 Asymptotic estimates for non-bottom virtual eigenvalues in case of an even dimension We see from Lemma 4.6 and Proposition 4.8 of [Ar-Z] that in the case of IR d with d even the finite-rank portion Φ( t 2l ) of the Birman-Schwinger operator X V ( t 2l ) is no longer a meromorphic operator function, because it contains summands with ln ( t) in its expansion near the point t = 0. In this section we try to overcome this difficulty with the help of some tricks. 4. o Before formulating the main theorem of this section, we prove the following Lemma 4.. Let L(t) be an operator function defined on an interval (a, b] and taking values in the set of linear self-adjoint operators acting in a Hilbert space H of a finite dimension N. Assume that there exists a subspace N H, whose codimension is equal to n, such that lim sup t a sup (L(t)v, v) <. (4.) v N, v = For each t (a, b] consider the eigenvalues of the operator L(t) arranged in the nonincreasing ordering: µ (t) µ 2 (t) µ N (t). Then at most n branches of the above eigenvalues have the property: limµ k (t) = +. (4.2) t a Proof. Assume, on the contrary, that at least n + branches have property (4.2). Let µ (t) µ 2 (t) µ n+ (t) e (t), e 2 (t),..., e n+ (t) be the eigenvectors corresponding to these eigenvalues. Consider the family of subspaces: E n+ (t) = span ( {e j (t)} n+ j= ). In view of (4.2) with k =, 2,..., n +, lim t a Since codim(n) = n, we have By condition (4.), inf v E n+(t), v = lim sup t a F(t) = E n+ (t) N = {0}. sup v F(t), v = On the other hand, in view of (4.3), we obtain the relation which contradicts (4.4). lim t a inf v F(t), v = (L(t)v, v) = +. (4.3) (L(t)v, v) <. (4.4) (L(t)v, v) = +, 8

20 We now turn to the main result of this section. As above, we assume that the virtual eigenvalues of the operator H γ at λ = 0 are indexed by the elements of the linearly ordered set Z d + (see Notation, (2.)) such that for γ < 0 λ 0 (γ) λ k (γ) λ kj (γ).... Theorem 4.2. Assume that d is even, 2l d, the potential V (x) is not identically zero and it satisfies conditions (3.6)-(3.8) and the condition IR d x 2(2l d) (ln( + x )) 2 V (x)dx <. Denote m = l d 2. Then: (i) The operator H γ, defined by (.), has r = #({k Z d + k m} = ( ) m+d d (4.5) virtual eigenvalues {λ k (γ)} k m at the endpoint λ = 0 of the gap (, 0) of σ(h 0 ) 2 ; (ii) For the bottom virtual eigenvalue λ 0 (γ) the asymptotic expansion, described in Theorem 5.6 of [Ar-Z], is valid. For the rest of virtual eigenvalues λ k (γ) (k 0) the following asymptotic properties are valid: (iii) If k = j {, 2,..., m }, the asymptotic estimate holds for γ 0: ( ( )) l λ k (γ) = c k γ m k + O γ 2(m k ) ln γ, (4.6) where the constants c k are calculated by the following formula: l m k c k = ek. (4.7) Here the numbers {e k } k =j are positive and form the set of roots of the algebraic equation (3.30), in which the matrix T is defined by (3.25), (3.22), (3.8), (3.9), (3.20), (3.4), (3.5), (3.6) and (3.5), and the matrix Ξ is defined by (3.3) and (3.4); (iv) If k = m, then for a small enough γ < 0 ( ) ( ) f f+ exp λ k (γ) exp, (4.8) γ γ where f + and f are positive numbers, which do not depend on γ. Proof. For λ < 0 consider the finite-rank portion Φ(λ), extracted from the Birman- Schwinger operator X V (λ) in Section 4 of [Ar-Z]. This is an integral operator with the kernel Φ(x,y, λ) defined by (4.48) of [Ar-Z], where F(x,y, λ) is defined by (4.22) of [Ar-Z] (see Lemma 4.6 and Proposition 4.8 of [Ar-Z]). For k 2m consider 2 As in the case of an odd d, we prove this fact in the process of the proof of this theorem, despite it have been established in [W] (Corollary 6.). 9

21 also the functions h k (x) of the form (3.5). Thus, for m > 0 the operator Φ(λ) can be written in the following manner: Φ(λ) = k,n: k+n 2m +ln ( λ ) λ 2m k+n 2l k,n: k+n =2m ξ k+n (, h n )h k + η k+n (, h n )h k. (4.9) Recall that the quantity ξ k is defined by formula (3.3) and η k is defined by the formula: η k = (2π) d s k ds ( s 2l ( k < 4l d), (4.0) + ) 2 or in an explicit form: η k = (2π) d IR d d «2 + m l π Qd lsin( π l ( d + m )) j= Γ(mj+ 2) 2 Γ( d 2 + m ), if k < 2l d, Q d j= Γ(mj+ 2) Γ( d 2 + m )l, if k = 2l d. (4.) Consider the Birman-Schwinger operator X V (λ) defined by (3.32). Let µ + k ν (λ) be the positive characteristic branches of the operator H 0 with respect to the operator V on the gap (, 0) of σ(h 0 ), that is they are positive eigenvalues of the operator X V (λ), arranged in the non-increasing ordering for any fixed λ < 0 (see Definition 3.2 of [Ar-Z]). Let µ + k ν (Φ(λ)) be the branches of positive eigenvalues of the operator Φ(λ), arranged in the same manner. We shall write briefly: µ + ν (λ) = µ + k ν (λ), µ + ν (Φ(λ)) = µ + k ν (Φ(λ)). By Proposition 4.8 of [Ar-Z], for some λ 0 < 0 and for any λ (λ 0, 0) X V (λ) Φ(λ) T, (4.2) where T > 0 does not depend on λ. Thus, for a small enough λ < 0 the estimates are valid: µ + ν (λ) µ+ ν (Φ(λ)) T, if µ + ν (Φ(λ)) T > 0 (4.3) and Consider the subspace µ + ν (Φ(λ)) µ + ν (λ) T, if µ + ν (λ) T > 0. (4.4) F = span ( {h k } k 2m ) of the space L 2 (IR d ). In view (4.9), F is an invariant subspace of the operator Φ(λ). Consider the operator Φ F (λ) = Φ(λ) F. It is evident that the set of all non-zero eigenvalues of the operator Φ(λ) coincides with the set of non-zero eigenvalues of the operator Φ F (λ). Let W be a linear operator acting in F and defined by the conditions: f j = Wh kj (j = 0,,..., G 2m ), 20

22 where the sequence {f j } G2m j=0 is obtained from the sequence {h kj } G2m j=0 by the orthogonalization process (3.0), (3.), (3.2). By Lemma 3., the operator W has the matrix representation (3.22) in the basis {f j } G2m j=0 of F, where ω ν,ρ are coefficients used in the orthogonalization formulas (3.7). Like in the proof of Proposition 3.2, we get from (4.9) that Φ F (λ) = W K(λ)(W ), (4.5) where K(λ) is a linear operator in F, whose matrix representation in the orthonormal basis {f j } G2m j=0 has the following form for m > 0: K(λ) k,n = ξ k+n λ 2m k+n 2l for k + n 2m, ( ) ln λ η k+n for k + n = 2m, 0 for k + n > 2m Consider the following subspace of F: where E = W (Ẽ), Ẽ = span ( {f j } m< kj 2m). ( k, n 2m). (4.6) It is evident that E has codimension r in F, where r is defined by (4.5). Furthermore, from the matrix representation (4.6) of the operator K(λ) and formula (4.5) we see that (Φ F (λ)y, y) = ( K(λ)(W ) y, (W ) y ) = 0 for any y E. Then, by Lemma 4., the self-adjoint operator Φ F (λ) has at most r branches of eigenvalues having the property limµ + ν (Φ(λ)) = +. λ 0 The latter circumstance, estimate (4.4) and the fact that µ + ν (λ) are increasing functions force the asymptotic multiplicity (Definition 3.9 of [Ar-Z]) to satisfy M(0, H 0, V ) r. Our immediate goal is to prove that M(0, H 0, V ) r and to get a lower estimate for the main characteristic branches {µ + ν (λ)} ν=0 r of H 0 with respect to V near the endpoint λ = 0 of the gap (, 0) of σ(h 0 ) (see Definition 3.9 of [Ar-Z]). To this end consider another form of the matrix K(λ), connected with formula (4.2) from [Ar-Z] for the function F(x, y, λ) (which takes part in the representation (4.48) from [Ar-Z] for the kernel Φ(x, y, λ) of the integral operator Φ(λ)): K(λ) k,n = 2lη k+n λ 2m k+n 2l (2m k+n ) for k + n 2m, ( ) ln λ η k+n for k + n = 2m, 0 for k + n > 2m, where k, n 2m. Also consider the r-dimensional subspace F = span ( ) {f j } kj m (4.7) 2

23 of the space F. Let us put λ = t 2l and consider for t > 0 the operator function M(t) = P F K( t2l ) F, having the following matrix representation in the basis {f j } kj m: M(t) k,n = 2l t 2m k+n (2m k+n ) η k+n for k + n 2m, 2l ln ( t) ηk+n for k + n = 2m Let us calculate the derivative of the latter matrix: ( ) d dt M(t) 2l = t η 2m+ k+n k+n ( k, n m). k,n ( k, n m). We see from the latter formula and formula (4.0) for η k that the matrix function N(t) = t 2m+ d M(t) (4.8) dt satisfies the same conditions as the matrix function L(t) in Proposition A.2 with N = p = m. Hence by this proposition the minimal eigenvalue of the matrix N(t) has the form for a small enough t > 0: where ẽ > 0. Thus, in view of (4.8), Then for t (0, δ) Hence, since λ = t 2l, µ min (N(t)) = ẽt 2m ( + O(t)), δ > 0, C > 0, t (0, δ) : d dt M(t) C t I. M(t) M(δ) = δ t ( ddt ) ( ) δ M(t)( dt C ln I. t λ 0 < 0, C > 0, λ (λ 0, 0) : P F K(λ) F C ln Consider the following r-dimensional subspace of the space F: G = W ( F). Then, in view of (4.5) and (4.9), for any f G and λ (λ 0, 0): (Φ(λ)f, f) = (Φ F (λ)f, f) = ( K(λ)(W ) f, (W ) f ) ( ) ( ) C ln (W ) f 2 C λ W ln f 2. 2 λ ( ) I. (4.9) λ Hence, in view of (4.2), λ 0 < 0, C 2 > 0, λ (λ 0, 0) : ( ) (X V (λ)f, f) inf C f G, f 0 f 2 2 ln. λ 22

24 Thus, by Proposition 3.7 of [Ar-Z], M(0, H 0, V ) r. Since the inverse estimate have been proved above, we have: M(0, H 0, V ) = r. So, we have proved assertion (i) of the theorem. Moreover, Proposition 3.7 of [Ar-Z] yields the following estimate for the main characteristic branches of H 0 near λ = 0: λ 0 < 0, C 2 > 0, λ (λ 0, 0) ν {0,,..., r } : ( ) µ + ν (λ) C 2 ln. (4.20) λ We now turn to the asymptotic estimation of the first r main characteristic branches µ + k (λ) of the operator H 0 near λ = 0, where r = #{k Z d + k m }, (4.2) and to the upper estimation of the rest of the main characteristic branches. Observe that, in view of property (ii) of ξ k given in Proposition 4.5 of [Ar-Z], ξ k = 0 for k = 2m. Then we can write the matrix K(λ), defined by (4.6), in the form: where and K (λ) k,n = ξ k+n λ 2m k+n 2l K(λ) = K (λ) + ln ( λ for k + n 2(m ), 0 for k + n > 2(m ) 0 for k + n 2m, R k,n = η k+n for k + n = 2m, 0 for k + n > 2m ) R, (4.22) ( k, n 2m) (4.23) ( k, n 2m). Making use of (4.5), consider also the corresponding representation of the operator Φ F (λ): ( ) Φ F (λ) = Φ (λ) + ln R, (4.24) λ where and Φ (λ) = W K (λ)(w ) R = W R(W ). Hence for any fixed λ < 0 the set of eigenvalues of the operator Φ (λ) coincides with the set of roots of the equation det(k! (λ) µt ) = 0. Recall that the matrix T is defined by (3.25), hence it is positive-definite. Observe that, in view of (4.23) and formula (3.3) for ξ k, the matrix t 2m K ( t 2l ) satisfies the same conditions as the matrix L(t) in Proposition A.2 with N = 2m and p = m. Thus, by Proposition A.2, the branches of eigenvalues µ k (Φ (λ)) of the operator Φ (λ) can be chosen such that µ k (Φ (λ)) = ) e k φ λ m k k ( λ 2l, if k m l ψ k ( λ 2l ), if m k 2m, (4.25) 23

25 where the functions φ k (t), ψ k (t) are analytic in a neighborhood O(0), φ k (0) = and e k > 0 ( k m ). Furthermore, for any fixed j {, 2,..., m } the numbers {e k } k =j form the set of roots of the algebraic equation (3.30). Applying the same arguments as in the proof of Theorem 3.3 and making use of (4.3), (4.4), (4.24) and of Lemma 3.4 from [Ar-Z], we get the following estimates for the main characteristic branches µ + k (λ) of H 0 at λ = 0: λ 0 < 0, M > 0, λ (λ 0, 0) : ( ) ( ) µ + k (Φ (λ)) M ln µ + k λ (λ) µ+ k (Φ (λ)) + M ln λ ( k m). Thus, in view of (4.25), we have the asymptotic estimates for λ 0: µ + k (λ) = e ( ( )) ( ( )) k + O λ λ m k 2l + O ln k : k m l λ and the estimates for a small enough λ < 0: ( ) µ + k (λ) M ln λ k : k = m, where M > 0 does not depend on λ. The latter estimates, estimate (4.20) and Proposition 3.6 of [Ar-Z] imply assertions (iii) and (iv) of the theorem. The theorem is proven. 4.2 o In the particular case m = l d = it is possible to strengthen assertion 2 (iv) of Theorem 4.2, replacing the two-sided exponential estimate (4.8) of non-bottom virtual eigenvalues by an asymptotic representation of the logarithm of these eigenvalues with algebraically computable leading coefficients. Furthermore, in this case it is possible to get a simple asymptotic expansion for the bottom virtual eigenvalue on the basis of Theorem 5.6 of [Ar-Z]. The following theorem says about this: Theorem 4.3. Assume that, in addition to the conditions of Theorem 4.2, m =, that is d = 2l 2. Then for r = d+ virtual eigenvalues {λ k (γ)} k of the operator H γ at λ = 0 the following asymptotic properties are valid: (i) For the bottom virtual eigenvalue λ 0 (γ) the following asymptotic expansion is valid if γ 0: ( ) l λ 0 (γ) = γ (ν l 0 + lν ν 0 γ ln + O(γ)), (4.26) γ where ν 0 = ξ 0 h 0 2 = ξ 0 IR d V (s) ds, (4.27) ν = (Ψ X 0, X 0 ), (4.28) X 0 = h 0 V (x) h 0 = V (s) ds, IR d (4.29) Ψ = k,n: k+n =2 η k+n (, h n )h k. (4.30) 24

26 Recall that the quantities ξ k and η k are defined by (3.4) and (4.), respectively, and the functions h k (x) are defined by (3.5); (ii) If k =, then for γ 0 the following asymptotic representation is valid: ln ( ) λ k (γ) = ( + O(γ)), (4.3) e k γ where the numbers e k ( k = ) are positive and they form the set of roots of the following algebraic equation: ( e) d T {0} + d ( e) d ν ( ) ν η (2,0,...,0) ν= C K : #C=ν T C = 0. (4.32) Recall that the matrix T is defined by (3.25), (3.22), (3.8), (3.9), (3.20), (3.4), (3.5), (3.6) and (3.5). Proof. Let us prove assertion (i). By Theorem 5.6 of [Ar-Z], the bottom virtual eigenvalue λ 0 (γ) of the operator H γ has the following asymptotic expansion for γ 0: where the function η(ǫ) has the form: λ 0 (γ) = γ l (η( γ ) + O(γ)) l, η(ǫ) = δ 0 + δ ǫ ln and it is calculated by the following procedure: ( ) ǫ t 0 (ǫ) = 0, t (ǫ) = ǫ (θ 0 (t 0 (ǫ)) 0, η(ǫ) = ν 0 + lν (t (ǫ)) ln ( ). ǫ Observe that in our case θ 0 (ǫ) = ν 0, hence t (ǫ) = ǫν 0. These circumstances imply the desired asymptotic expansion (4.26). Let us prove assertion (ii). As in the proof of Theorem 4.2, consider the operator Φ(λ), defined by (4.9), the operator Φ F (λ) = Φ(λ) F, where and the operator F = span ( {h k } k 2 ), K(λ) = WΦ F (λ)w. Recall that W is a linear operator acting in F and defined by the conditions: f j = Wh kj (j = 0,,..., G 2m ), where the sequence {f j } G2m j=0 is obtained from the sequence {h kj } G2m j=0 by the orthogonalization process (3.0), (3.), (3.2). Taking into account Remark 4.7 of [Ar-Z], we have: η k = 0 for k =. (4.33) 25

27 Then in our case m = the matrix representation (4.7) of the operator K(λ) in the basis {f j } G2m j=0 acquires the form: l η λ 0 for k = n = 0, l 0 K(λ) k,n = ( ) for k + n =, ln ( k, n 2). λ η k+n for k + n = 2, 0 for k + n > 2 We put t = t(λ) = λ l l ( ) ln. (4.34) λ We see that the function t(λ) is continuous and decreasing for a small enough λ < 0, and t(0) = lim λ 0 t(λ) = 0. Then there exists the continuous inverse function λ = λ(t) defined for a small enough t 0 and λ(0) = 0. Let us put K(t) = l λ(t) l K(λ(t)). Then, taking into account (4.33), we can write K(t) in the form: K(t) k,n = { ηk+n t k+n for k + n 2, 0 for k + n > 2 ( k, n 2). (4.35) Thus, taking into account (4.5) and the fact that T = WW, we get the following form of the eigenvalues of the operator Φ F (λ): µ k (Φ F (λ)) = µ k(t(λ))l, (4.36) λ l where µ k (t) are the roots of the equation ( ) det K(t) µt = 0. (4.37) In view of formula (4.0) for η k, the matrix K(t), defined by (4.35), satisfies the same conditions as the matrix L(t) in Proposition A.2 with p =. Furthermore, observe that the matrix T is positive-definite. Thus, by Proposition A.2, equation (4.37) has r = d + branches of roots with the asymptotic representation µ k (t) = e k t 2 k ( + O(t)) ( k ), (4.38) where e k > 0 and the quantities e k with k = form the set of roots of the equation: ( e) d η 0 T {0} + d ( e) d ν ν= C,D K : #C=#D=ν and Π is a matrix of the form: { ηk+n for k + n 2, Π k,n = 0 for k + n > 2 Π C, D T C, D C, C D, D = 0 (4.39) ( k, n 2). 26

28 Observe that, by Remark 4.7 of [Ar-Z], { 0, if k n, Π k,n = η (2,0,0), if k = n (k,n K ). Hence we get: Π C, D = { 0, if C D, η 0 ( η(2,0,0) ) #C, if C = D (C, D K ). (4.40) Thus, equation (4.39) acquires the form (4.32). In view of (4.36), (4.34) and (4.38), we have the following asymptotic estimate for λ 0: ( ) ( ( ( ) )) µ k (Φ F (λ)) = e k ln + O λ l ln. λ λ In the same manner as in the proof of Theorem 4.2, we obtain the desired estimate (4.3), making use of the latter estimate, estimates (4.3), (4.4) for the main characteristic branches µ + k (λ) and Proposition 3.6 of [Ar-Z]. The theorem is proven. Example 4.. We now turn to a particular case of the situation, considered in Theorem 4.3. Namely, put l = d = 2. In this case m =, where K 0 = {0}, K = {k,k 2 }, K 2 = {k 3,k 4,k 5 }, k = {0, }, k 2 = {, 0}, k 3 = {0, 2}, k 4 = {, }, k 5 = {2, 0}. By Theorem 4.3, for three virtual eigenvalues of the operator H γ we have the following asymptotic formulas if γ 0: (i) For the bottom virtual eigenvalue we have: ( ) 2 λ 0 (γ) = γ (ν ν ν 0 γ ln + O(γ)), γ where the numbers ν 0 and ν are defined by (4.27)-(4.30); (ii) For j {, 2} we have: ( ) ln = ( + O(γ)), λ kj (γ) e kj γ where e k, e k2 are the roots of the quadratic equation: ( e) 2 ( ) T {k,...,k 5} + ( e)η (2,0) T{k2,k 3,k 4,k 5} + T {k,k 3,k 4,k 5} + ( η (2,0) ) 2 T{k3,k 4,k 5} = 0. Recall that η 2,0 = s 2 ds ds 2 ((s 2 + s2 2 )2 + ) 2 27

29 and η 2,0 is calculated by formula (4.) with k = (2, 0). Also observe that by Lemma 3.6: ( T {k2,k 3,k 4,k 5} = T {k,k 2,...,k 5} hk 2 (h k, h 0 ) 2 h 0 2) = T {k,k 2,...,k 5} ( ( ) 2 x 2 x V (x, x 2 ) dx dx 2 V (x, x 2 ) dx dx 2 V (x, x 2 ) dx dx 2 and ( T {k,k 3,k 4,k 5} = T {k,k 2,...,k 5} hk2 2 (h k2, h 0 ) 2 h 0 2) = T {k,k 2,...,k 5} ( ( ) 2 x 2 x 2V (x, x 2 ) dx dx 2 2V (x, x 2 ) dx dx 2 V (x., x 2 ) dx dx 2 Recall that the matrix T is defined by (3.25), (3.22), (3.8), (3.9), (3.20), (3.4), (3.5), (3.6) and (3.5). In connection with Theorems 4.2 and 4.3 the following problem appears: Problem 4.. Does the asymptotic estimate of the form ( ) ln = f k λ k (γ) γ ( + O(γ)) (f k > 0) as γ 0 hold for k = m = l d also in the case where 2l d > 2 and d is even? If the 2 answer is positive, find an algorithm for the computation of the constants f k. 5 Asymptotic formulas of Lieb-Thirring type There are papers devoted to estimates (ordinary and asymptotic with respect to the coupling constant γ) of the sum of the form λ j (γ) κ (γ < 0, κ > 0), j where λ j (γ) are negative eigenvalues of the operator H γ arranged in the non-decreasing ordering ([L-Thr], [N-W]). As above, we assume that these eigenvalues are indexed by the elements of the linearly ordered set Z d + (see Notation, (2.)), that is for γ < 0 λ 0 (γ) λ k (γ) λ kj (γ).... It turns out that the asymptotic formulas for λ k (γ), obtained in Sections 3 and 4, enable us to get asymptotic formulas with respect to a small γ < 0 for the sums of the form λ k (γ) κj k: k =j with suitable powers κ j, where { {,..., m}, if d is odd, j {,..., m }, if d is even. 28