HEAT TRACES AND EXISTENCE OF SCATTERING RESONANCES FOR BOUNDED POTENTIALS

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1 HEAT TRACES AND EXISTENCE OF SCATTERING RESONANCES FOR BOUNDED POTENTIALS HART SMITH AND MACIEJ ZWORSKI Abstract. We show that any real valued bounded potential with compact support, V L c (R n ; R, n odd, has some scattering resonances. For n 3 that was known before only for sufficiently smooth potentials. The proof is based on the following inverse result: V L c (R n ; R, t n tr(e t( V e t C ([, V C c (R n ; R. 1. Introduction and statements of results Let V L c (R n ; R be a bounded, compactly supported, real valued potential and let n 3 be odd. We consider the Schrödinger operator, P V = + V (x, (1.1 and ask the question whether P V always (for V has infinitely many scattering resonances. Scattering resonances are defined defined as poles of the meromorphic continuation of the resolvent R V (λ := ( + V λ 1, n odd, (1. from Im λ > to λ C. The multiplicity of a resonance at λ is defined as m V (λ = rank R V (ζdζ, (1.3 where the integral is over a circle around λ enclosing no other singularities of R V than (possibly λ see [DyZw, 3.]. At λ = we put m V ( = 1 rank R V (ζdζ + rank R V (ζζdζ. (1.4 If m V ( = m N then P V has an eigenvalues of multiplicity m at. If m V ( = m+ 1, m N then in addition, P V has a zero resonance see [DyZw, 3.3], [Je9] and [JeKa79]. These poles have many interesting interpretations and in particular appear in expansions of solutions to the wave equation see and references given there. For n even the situation is more complicated as the meromorphic continuation has a logarithmic 1 λ

2 HART SMITH AND MACIEJ ZWORSKI branch singularity at λ = see [ChHi1] and references given there. Here we prove that Theorem 1. Suppose that V L c (R n ; R and that n is odd. Then the meromorphic continuation of the resolvent (1., R V (λ : L c(r n L loc(r n, λ C, has at least one pole. If V L c (R n ; R H n 3 (R n then R V has infinitely many poles. For V Cc (R n ; R existence of infinitely many resonances was proved by Melrose [Me95] for n = 3 and by Sá Barreto Zworski [SaZw96] for all odd n. Soon afterwards quantitative statements about the counting function, N(r, of resonances in { λ r} were obtained by Christiansen [Ch99] and Sá Barreto [Sa1]: lim sup r N(r r For potentials generic in Cc (R n ; F or L c (R n ; F, F = R or C, Christiansen and Hislop [Ch5],[ChHi5] proved a stronger statement lim sup r log N(r log r >. = n. (1.5 This means that the upper bound N(r Cr n from [Zw89] is optimal for generic complex or real valued potentials. The only case of asymptotics r n for non-radial potentials was provided by Dinh and Vu [DiVu13] who proved that a large class of L potentials supported in B(, 1 has resonances satisfying a Weyl law. Tanya Christiansen pointed out that our argument provides the following inverse result: Theorem. Suppose that V j L c (R n ; R, j = 1, and n is odd. If, in the notation of (1.3 and (1.4, then for any m N, m V1 (λ = m V (λ, λ C, (1.6 V 1 H m (R n V H m (R n. This is interesting because of the dearth of results on resonance inverse problems. It is known that resonances alone may not determine the potential uniquely see Korotyav [Kor], [Zw1] and also Autin [Au11], [Ch8] where references to more general isopolar problems can be found. In the positive direction Datchev Hezari [DaHe1] showed that in the semiclassical setting certain radial potentials are determined by the asymptotic behaviour of resonances. That paper contains further references to inverse problems for resonances.

3 HEAT TRACES AND EXISTENCE OF RESONANCES 3 To prove Theorem 1 we proceed by contradiction, as in [BaSa95], [Me95] and [SaZw96], and assume that there are no resonances. By a direct argument (Proposition.1 this implies that the scattering phase is a polynomial. This in turn implies (Proposition. that the heat trace has an asymptotic expansion. The main result of this note, Theorem 3 below, shows that this implies that V Cc, and since it is real valued we obtain a contradiction by [Me95] and [SaZw96]. (We provide a direct argument of the contradiction in. See.4 for why our arguments do not yield a contradiction for a finite number of resonances if n 5 and V L c (R n, R. Although we expect (1.5, or possible even N(r > r n /C when r 1, to be true for all non-zero real valued potentials, Christiansen gave classes of examples of non-zero V Cc (R n ; C which have no resonances. Our argument outlined above depends on the following, which is the principal new result of this paper. Theorem 3. Suppose that P V is given by (1.1, and V L c (R n ; R, where n 1 may be even or odd. If t n tr ( e tp V e tp C ([, (1.7 then V C c (R n ; R. Theorem 3 is a direct consequence of a more precise result presented in Theorem 4 in 3. The study of heat expansions has a very long tradition going back to Kac, Berger and McKean Singer see [CdV1],[Gil4],[HiPo3] for more recent accounts and references. Theorem 3, although not surprising, seems to be new. However, closely related inverse results are well known. They concern recovering Sobolev norms from the coefficients of expansion of smooth potentials, and using those a priori bounds to prove compactness of sets of isospectral potentials see Brüning [Br84] and Donnelly [Do4], and for the origins of that approach, McKean van Moerbeke [McMo75]. The paper is organized as follows. In we review the scattering theory needed for the proofs of Theorems 1 and. For detailed arguments we refer to the original papers and to the on-line notes [DyZw]. The section on the heat trace 3 is by contrast completely self-contained. Some aspects of the approach in 3 appear to be new, in particular the use of Gagliardo Nirenberg Moser inequalities in a bootstrap regularity scheme. Acknowledgements. We would like to thank Gunther Uhlmann for a helpful discussion, in particular for reminding us of the references [Br84] and [McMo75], and Tanya Christiansen for helpful comments on the first version of this note, and especially for suggesting Theorem. This material is based upon work supported by the National Science Foundation under Grants DMS (HS and DMS-11417(MZ.

4 4 HART SMITH AND MACIEJ ZWORSKI. Review of scattering theory Here we recall various facts in scattering theory and show how Theorem 1 follows from Theorem The scattering matrix. The continued resolvent, R V (λ, given in (1. does not have any poles on R \ {} that is a well known consequence of the Rellich uniqueness theorem see [DyZw, 3.6]. This implies that, for λ R \ {} and ω S n 1, there exist (unique solutions to (P V λ w(x, λ, ω =, w(x, λ, ω = e iλ x,ω + u(x, λ, ω, u(x, λ, ω = x n 1 e iλ x ( b(λ, x/ x, ω + O( x 1, x. (.1 The radiation pattern b(λ, θ, ω, is the observed field in a scattering experiment. The scattering matrix, S V (λ, can be defined using b(λ, θ, ω. This definition is not the most intuitive, and we refer to [DyZw, 3.7] for motivation. Here we define S V (λ : L (S n 1 L (S n 1 as S V (λf(θ = f(θ + a(λ, θ, ωf(ωdω, S n 1 a(λ, θ, ω := (π n 1 e π 4 (n 1i λ n 1 We also have the following useful representations of a(λ, θ, ω: b(λ, θ, ω. (. a(λ, θ, ω = a n λ n R n e iλ x,θ V (xw(x, λ, ωdx = a n λ n R n e iλ x,ω θ V (x(1 e iλ x,ω R V (λ(e λ,ω V (xdx, (.3 where a n = (π n+1 /i. The scattering matrix is unitary for λ real, and from (.3 we see that it continues meromorphically to all of C. Hence we have Another symmetry comes from changing λ to λ: S V (λ 1 = S V ( λ, λ C. (.4 S V (λ 1 = JS V ( λj, Jf(θ := f( θ. (.5 The operator S V (λ I is of trace class, and hence det S V (λ is well defined. The following result, see [DyZw, Theorem 3.4] or [Zw97], is important for the investigation of scattering resonances:

5 HEAT TRACES AND EXISTENCE OF RESONANCES 5 Proposition.1. Suppose that V L c (R n ; R, where n is odd. Then det S V (λ is a meromorphic function of order n. More precisely, ( K det S V (λ = ( 1 m iµ k + λ P ( λ iµ k λ P (λ, (.6 where µ k, µ 1 < µ µ K are the eigenvalues of P V, included according to multiplicity, P (λ is entire and non-zero for Im λ, and P (λ C ε e Cεrn+ε, for any ε >. (.7 The power m in (.6 is the multiplicity of the zero resonance, m = or 1 for n = 1, 3 and m = for for n 5; see [DyZw, 3.3] and [JeKa79]. We make the following observation based on the second representation in (.3: λ is a pole of det S V = λ is a pole of S V = λ is a pole of R V. (.8 A more precise statement is possible (see [DyZw, Theorem 3.4] but we do not need it for Theorem 1. To show existence of poles of R V we only need to show existence of poles of det S V... A trace formula. The tool connecting the scattering matrix to the heat trace is the Birman Kreĭn trace formula. In 3 we will recall the argument showing that e tp V e tp is of trace class. Proposition.. Suppose that V L c (R n ; R. Then, in the notation of Proposition.1, tr(e tp V e tp = 1 πi tr ( S V (λ 1 λ S V (λ e tλ dλ + K e tµ k + 1m. (.9 If V Cc, this is proved for n = 3 in [CdV81], and for n 5 in [Gu81] and references given there. The proofs for V L c can be found in [DyZw, 3.8, 4.6]. Since det S V (λ = 1 for λ R (which follows from (.4, the unitarity of the scattering matrix we can define the winding number of the scattering phase: σ(λ := 1 πi log det S V (λ, σ (λ = 1 πi tr ( S V (λ 1 λ S V (λ, λ R. In the case of V Cc (R n, R, n odd, σ(λ admits a full asymptotic expansion for λ, with only odd powers of λ except for the constant term. When n = 3, ( θ(+ θ( = K + 1m, θ(λ := σ(λ + λ 1 V (xdx, R

6 6 HART SMITH AND MACIEJ ZWORSKI and for n 5, θ(+ θ( = K, θ(λ := σ(λ c k (V λ n k, where c k (V are the coefficients in the expansion of σ(λ. For proofs see [CdV81], [Gu81], [DyZw, 3.7], and for less regular potentials but fewer expansion terms [Je9]..3. Proof of Theorem 1. If V has no resonances then Proposition.1 shows that det S V (λ = P ( λ P (λ, where P (λ is an entire function with no zeros and of order n. This implies that P (λ = e G(λ where G(λ is a polynomial of degree at most n; see for instance [Ti64, 8.4]. Defining the odd polynomial g(λ = (G( λ G(λ/(πi, we obtain n 1 det S V (λ = e πig(λ, σ (λ = g (λ. The unitarity of S V (λ for λ real shows that g(λ has real coefficients, g(λ = a λ n + a 1 λ n + + a n 1 λ. Hence, where a n := a n Γ(n/ j + 1. σ (λ e tλ dλ = t n a jt j, (.1 We now insert (.1 into the trace formula (.9 to see that t n/ tr(e tp V e tp has a full asymptotic expansion j= c jt j as t +. That means that the assumption of Theorem 3 is satisfied, and hence V Cc (R n ; R. But the result of [SaZw96] (see also [DyZw, 3.7] then contradicts our assumption that V has no resonances: every nonzero potential in Cc (R n, R has to have infinitely many resonances. Christiansen s argument [Ch99] that there must be at least one resonance for nonzero V Cc (R n ; R is simple and elegant, and we reproduce it here. As above, absence of resonances would imply that σ (λ = a λ n 1 + a 1λ n 3 + a n. Comparison with the heat expansion shows that a = c n V. That immediately provides a contradiction in the case of n = 3. When n 5 we use the representation (.3: σ (λ = tr S V (λ λ S V (λ = λ a(λ, θ, θdθ + S n 1 S n 1 n 1 j= S n 1 a(λ, ω, θ λ a(λ, ω, θdωdθ. Under the assumption that R V is holomorphic, that is no poles, (.3 then shows that σ (λ = O(λ n 3 as λ. But this contradicts a, since that would imply a lower order of vanishing at λ =.

7 HEAT TRACES AND EXISTENCE OF RESONANCES 7 We now use Theorem 4 to show that if V L c (R n, R H n 3 (R n, then R V has infinitely many poles. This is again seen by contradiction, by assuming that det S V (λ has only finitely many resonances. In that case, let µ 1 < µ µ K <, µ k >, denote the negative eigenvalues of P V, and let iρ j, ρ j <, j = 1,..., J 1, λ j λ j, j = 1,..., J the remaining finite set of resonances. Proposition.1 gives Hence for λ R, That implies that K det S V (λ = ( 1 m g(λ e σ (λ g (λ = 1 π 1 π λ + iµ k λ iµ k K J µ j J 1 λ + µ j λ + iρ j λ iρ j 1 π J 1 J λ λ j λ λ j λ λ j λ + λ j. ρ j λ + ρ j ( Im λj λ λ j + Im λ j λ + λ j, (.11 (σ (λ g (λ dλ = 1 K + 1 J 1 + J. (.1 where K K is the number of negative eigenvalues. We compare this with Proposition. and the expansion in Theorem 4: if V L c (R n, R H n 3 (R n, then (3.1 shows that n 1 tr(e tp V e tp = c kt n +k + O(t 1. In particular, tr(e tp V e tp n 1 c kt n +k, t +. (.13 Since the terms on the right hand side of (.11 make bounded contributions, comparison with (.9 shows that n 1 Using (.9 and (.1 we obtain tr(e tp V e tp n 1 c k t n +k = 1 πi g (λe tλ dt. c kt n +k = tr(e tp V e tp 1 πi = 1 πi (σ (λ g (λe λt dλ + g (λe tλ dλ K e µ k t + 1m.

8 8 HART SMITH AND MACIEJ ZWORSKI Taking the limit as t + we obtain (σ (λ g (λ dλ + K + 1 m = K 1 K + 1 m + 1 J 1 + J >. But this contradicts ( Why not infinitely many? A frustrating aspect of the argument in.3 is that for V L c (R n, R, n 5, it only shows existence of one resonance. The reason for that is the strong assumption in Theorem 3. If we allowed, for example, a unique (non-zero resonance λ = iρ (it has to be purely imaginary, as the symmetry λ λ would otherwise imply that there are two then the factorization argument above would imply det S V (λ = e πig(λ iρ + λ iρ λ, σ (λ = g(λ 1 ρ. π λ + ρ We now note that 1 e sr π 1 + r dr 1 es + s 1 b j s j, s +. (.14 To see (.14, let I(s := (1/π e s(1+r /(1 + r dr. Then the right hand side of (.14 is e s I(s, while s I(s = (1/π e s(1+r dr = αe s /s 1, α = 1/ π. Hence I(s = I( + α s e s 1 s s 1 j= b js j. Multiplying by e s gives (.14. Inserting (.14 into the trace formula (.9, and noting that if ρ > we have an eigenvalue, gives tr(e tp V e tp = t n/ a j t + 1 eρ t, and we cannot use Theorem 3 to conclude that V is smooth. The same problem arises if we assume that we have two (or more resonances, λ, λ. The following simple example does not fit into our hypotheses, but it suggests a possible complication. Consider n = 1 and V = δ. Then there is only one resonance, at λ = i, and the heat trace has an expansion with both integers and half-integers..5. Proof of Theorem. We again use the Birman Kreĭn formula (.9 to see that, under the assumption that the eigenvalue and zero resonance contributions cancel, tr ( e tp V 1 e tp V = 1 πi j= λ det ( S V (λ 1 S V1 (λ e tλ dλ. The assumption (1.6 and [DyZw, Theorem 3.4] show that det S Vj (λ, j = 1,, have the same poles and zeros (with the agreement of multiplicities. Arguing as in.3 we then see that det ( S V (λ 1 S V1 (λ = e πih(λ, h(λ = b λ n + b 1 λ n + b n 1 λ.

9 Then as in (.1 we see that HEAT TRACES AND EXISTENCE OF RESONANCES 9 tr ( e tp V 1 e tp V = t n/ b jt j, b n = n 1 j= b n Γ(n/ j + 1. That means that (3.1 holds for V 1 if and only if it holds for V, and the Theorem follows from Theorem Heat trace expansions For P V given by (1.1 with V L c (R n ; C, it is well known that e tp V e tp is trace class for t >, and if V C c it is known that tr ( e tp V e tp admits a full asymptotic expansion see for instance [vdb91] and references given there. Theorem 3 is a consequence of a converse result which gives a sharp relation between existence of a finite expansion for the trace, and a given finite order of Sobolev regularity for V, assuming that V is real-valued. Theorem 4. Suppose that V L c (R n, R, and that for some m N one can write tr ( e tp V e tp = (4πt n (c 1 t + c t + + c m+1 t m+1 + r m+ (tt m+, (3.1 where r m+ (t C for < t 1. Then V H m (R n. Conversely, if V H m (R n then (3.1 holds with such an r m+ (t, and lim t + r m+ (t = c m+ exists. The proof of Therem 4 begins by using iteration to expand the heat kernel for P V = + V. The formula is e tp V e tp = W k (t, where W k (t = <s 1 < <s k <t e (t s kp V e (s k s k 1 P V V e (s s 1 P V e s 1P ds 1 ds k. Convergence of the expansion in the L operator norm follows from W k (t L L V k L tk /k!, which holds since for all s j and t the integrand is L bounded by V k L, and the volume of integration is t k /k!. We also have a bound on the trace class norm: W k (t L 1 C k k n t k n /k!, (3. where n is the dimension. For this we use that the trace class is an ideal, so it suffices to show that one pair of successive terms in the product has L 1 bound less than C k n t n.

10 1 HART SMITH AND MACIEJ ZWORSKI We then observe that at least one of t s k, s j+1 s j or s 1 is greater than t/k, and for that term we use the trace bound e sp χ L 1 C s n/, (3.3 where χ Cc is chosen to be 1 on the support of V. To prove (3.3 we choose χ 1 Cc equal to 1 on supp χ. Then the explicit Schwartz kernel, K 1 (x, y of (1 χ 1 e sp χ satisfies α K 1 C α,n s N (1 + x + y N, for any α and N. Hence (1 χ 1 e sp χ L 1 = O(s. On the other hand, if K (x, y is the Schwartz kernel of e sp/ χ 1 then K (x, y dxdy Cs n/ which provides an estimate O(s n/4 on the Hilbert Schmidt norm. These two bounds give (3.3: e sp χ L 1 C χ 1 e sp χ 1 L 1 + (1 χ 1 e sp χ L 1 C χ 1 e sp / L + C Ns N C s n/. Using (3., we see that e tp V e tp is of trace class for t >. The trace can be brought into the sum, and we write ( tr e tp V e tp = tr ( W k (t. It is well known, and we include the proof, that tr ( W 1 (t = (4πt n t V (y dy, which shows that c 1 = V, and the expansion (3.1 is equivalent to tr ( W k (t ( = (4πt n c t + + c m+1 t m+1 + r m+ (t. k= Theorem 4 will then follow as a result of the following two propositions that concern the asymptotics of the individual terms tr ( W k (t. Proposition 3.1. If V L c (R n, R H m (R n, then one can write tr ( W (t = (4πt n (c, t + + c,+m t +m + ε(tt +m, (3.4 with lim t + ε(t = and c,+j = a j D j V L for j m, for constants a j. Conversely, assuming V L c (R n, R H m 1 (R n, if one can write tr ( W (t = (4πt n (c, t + + c,1+m t 1+m + r,+m (tt +m, (3.5 where r,+m (t C for < t 1, then V H m (R n, and hence (3.4 holds. Proposition 3.. If V L c (R n, R H m (R n, then for k 3 one can write tr ( W k (t = (4πt n (c k,k t k + + c k,k+m 1 t k+m 1 + r k,k+m (tt k+m, (3.6

11 HEAT TRACES AND EXISTENCE OF RESONANCES 11 where, for a constant C depending on k and m, for j m, c k,k+j C V k L V H j, sup r k,k+m (t C V k L V H. m <t<1 The fact that V L c (R n, R H m (R n implies existence of the asymptotic expansion (3.1 of order m + is an easy consequence of the above propositions. By the bound W k (t L 1 C k k n t k n /k! we have that tr W k (t C t m+3 n, < t 1. (3.7 k=m+3 On the other hand, Propositions 3.1 and 3. show that W k (t = (4πt n m+ tr (c 1 t + c t + + c m+1 t m+1 + c m+ t m+ + ε(tt m+, where for j we have c j = j k= c k,j. The other direction of Theorem 4, that existence of an asymptotic expansion implies regularity, is carried out by induction. Assume m 1 and V L c H m 1 (R n, which trivially holds if m = 1 since L c L (R n. Assume (3.1 holds. By (3.7 this implies W k (t = (4πt n m+ tr k= where r m+ (t C. (c 1 t + c t + + c m+1 t m+1 + r m+ (tt m+, By Proposition 3., since V L c H m 1 (R n the same relation holds, with different coefficients that can be bounded from L and H j norm bounds for V with j m 1, for tr m+ k=3 W k(t. Hence the relation (3.5 holds, and we conclude V H m (R n Calculating tr ( W 1 (t. We calculate the trace of W 1 (t by integrating over the diagonal tr ( W 1 (t t = (4π n (t s n s n e x y 4(t s V (y e y x 4s ds dx dy. R n The integral dx is carried out leading to R n e R n x y 4 t n (t ss dx = (4π t n n n (t s s tr ( W 1 (t = (4πt n t V (y dy. (From now on the integrals without integration limits will denote integrals over R n.

12 1 HART SMITH AND MACIEJ ZWORSKI 3.. Calculating tr ( W (t. Again we integrate over the diagonal to write tr ( W (t as (4π 3n (t s n (s r n r n e x y 4(t s y z 4(s r z x 4r V (y V (z dr ds dx dy dz. <r<s<t We let u = t s and x = ( ( r r+u y + u r+u z and carry out the integral over x by writing r+u<t <r,u x y + u z x r which expresses tr ( W (t as ( (4π n (t u r n (u + r n e y z 4 = r + u ru x x + 1 r + u y z (3.8 1 t u r + 1 u+r V (y V (z dr du dy dz. Let r = tv u, so dr du = t dv du, the integrand is then independent of u, the new limits are < u < tv, < v < 1, and we get t (4πt n 1 (1 v n v n +1 e y z 1 4t v(1 v V (y V (z dv dy dz. Since V is real we can use the Plancherel theorem to write this as 1 t (4πt n v ((π n e t(1 vv ξ V (ξ dξ dv. By symmetry under v 1 v we can also write this as 1 1 t (4πt n ((π n e t(1 vv ξ V (ξ dξ dv. The term in parentheses is continuous in t, and at t = equals V L, so tr ( W (t = 1 ( t (4πt n V L + ε(t, lim ε(t =. (3.9 t + This settles the case m = of Theorem 4 which, since L c H (R n = L (R n, is nontrivial only for the existence of the expansion (3.1 for m =. It also shows that we can recover V L from lim t + r (t. Remark. If we were to assume V is Hölder-α, then to get van den Berg s bounds [vdb91] we would write V (yv (z = 1 (V (y + V (z ( V (y V (z and writing V (y V (z y z α would lead to a gain of t α for the last term on the right; the other two terms would lead to the desired leading term, so we would get ε(t t α.

13 HEAT TRACES AND EXISTENCE OF RESONANCES Proof of Proposition 3.1. First consider the case m = 1, and suppose that we have an expansion tr ( W (t = (4πt n ( c t + O(t 3, t 1. From (3.9 we must have c = 1 V L. This leads to the estimate 1 ( 1 e t(1 vv ξ V (ξ dξ dv C, < t 1. t The integrand is positive, so by Fatou s lemma we get ( 1 (1 vv dv ξ V (ξ dξ C, implying that V H 1 (R n. Conversely, if V H 1 (R n L c (R n, R we would get such an expansion by dominated convergence. To consider higher values of m, write e s = m 1 j= ( 1 j j! s j + r m (s ( 1m m! s m, (3.1 where r m (s is a smooth function, and by the Lagrange form for the remainder, r m (s 1 if s, r m ( = 1, s r m ( = 1 m + 1. (3.11 Now suppose that V H m (R n for some m 1. Then we can expand 1 ( m 1 ( e t(1 vv ξ V (ξ 1 dξ dv = j! + ( 1m m! ( 1 j= 1 ( (1 v j v j dv ξ j V (ξ dξ t j r m ( t(1 vv ξ (1 v m v m ξ m V (ξ dξ dv t m. The coefficient of t m is continuous in t, and converges to a m D m V L as t, where a m. Thus, if we can write tr ( W (t ( m = (4πt n c j t j + O ( t m+1, t 1, j= then c j = a j D j V L for j m, and in addition we have uniform bounds for < t 1 1 ( ( 1 rm t(1 vv ξ (1 v m v m ξ m V (ξ dξ dv C. t Then by Fatou s lemma and (3.11 we get ( 1 1 ( (1 v m+1 v m+1 dv ξ (m+1 V (ξ dξ C, m + 1

14 14 HART SMITH AND MACIEJ ZWORSKI so necessarily V H m+1 (R n, completing the proof of Proposition Trace of W k (t for k 3. To estimate products of derivatives, we will use the following particular case of the Gagliardo Nirenberg Moser inequalities. Lemma 3.3. Suppose {α j } k are multi-indices, with α j m, and j α j = m. If u L (R n H m (R n, then for a constant C depending only on n and m, k ( α j ( k k ( k u L j 1 C u j L D m u j L. Proof. We use the following bound [Tay11, (3.17 in 13.3]. Assuming u L H m, α j u j m α L j C u j 1 α j m L α j Dm m u j L. The result follows by Hölder s inequality after taking the product over j. We now write tr ( W k (t for t > as an integral <s 1 < <s k <t e x y k 4(t s k y k y k 1 4(s k s k 1 y 1 x 4s 1 V (y k V (y 1 dy (4π n (k+1 (t s k n (s s 1 n (s 1 n 1 dy k ds 1 ds k dx. After integrating over x, and letting s j = tr j, then letting Σ R k denote the set {r R k : < r 1 < < r k < 1}, we obtain t k Σ (R n k e y k y k 1 4t(r k r k 1 y y 1 4t(r r 1 y 1 y k 4t(1+r 1 r k V (y k V (y 1 (4πt n k (r k r k 1 n (r r 1 n (1 + r 1 r k n dy dr. To analyse this, we introduce variables u 1 = y 1, and u j = y j y 1 for j k. Then du 1 du k = dy 1 dy k, so the formula for tr ( W k (t becomes t k G (4πt n r,t (u V (u 1 + u k V (u 1 + u V (u 1 du dr, (3.1 Σ (R n k where G r,t (u is the Gaussian function of u = (u,..., u k (R n k 1 ( e 1 uk + u k u k 1 + u 3 u + u 4t 1+r 1 r k r k r k 1 r 3 r r r 1 G r,t (u,..., u k =. (4πt n (k 1 (1 + r 1 r k n (r k r k 1 n (r r 1 n Applying successively the following equality, which is a special case of (3.8, u j+1 u j r j+1 r j + u j r j r 1 = r j+1 r 1 (r j+1 r j (r j r 1 u j r j r 1 r j+1 r 1 u j r j+1 r 1 u j+1

15 HEAT TRACES AND EXISTENCE OF RESONANCES 15 we can write the quadratic term in the exponent of G r,t as u k k 1 (1 + r 1 r k (r k r 1 + j= (r j+1 r 1 (r j+1 r j (r j r 1 In particular we see that, for all t > and r Σ, (R n k 1 G r,t (u du = 1. u j r j r 1 r j+1 r 1 u j+1 (3.13 For t > consider the k-linear form B t (V 1,..., V k = G r,t (u V k (u 1 + u k V (u 1 + u V 1 (u 1 du dr. Σ (R n k By Hölder s inequality applied to the integral over u 1, we have k B t (V 1,..., V k V j L k (R n, and thus B t is uniformly continuous on bounded sets in L k (R n k. The quadratic form (3.13 is bounded below by c u, for c > independent of r Σ. An approximation to the identity argument then shows that B t is continuous over t [,, for fixed elements of L k (R n k, where we set B (V 1,..., V k = 1 V k (u 1 V 1 (u 1 du 1. k! R n Consequently, we can write tr ( W k (t = (4πt n t k B t (V, B t (V C ( [,, B (V = 1 k! V (y k dy. Here we set B t (V = B t (V,..., V, which, by the above, is for each t a continuous function of V L k (R n. We start by demonstrating an m-th order expansion of B t (V when V Cc (R n, R, after which we will show it applies to V L c (R n, R H m (R n by taking limits. For j k we write V (u j + u 1 = (π n and plug this into (3.1 to express B t (V = (π n(k 1 Σ e iη j (u 1 +u j V (ηj (R n k 1 e tqr(η V (ηk V (η V (η + + η k dη dη k dr. where Q r (η is the quadratic form inverse to (3.13, and where V ( ζ = V (ζ since V is real valued.

16 16 HART SMITH AND MACIEJ ZWORSKI We expand exp ( tq r (η as in (3.1. The first m 1 terms give contributions to B t (V of the form m 1 (π n(k 1 ( 1 j t j Q(η j V (ηk j! V (η V (η + + η k dη dη k, j= where Q(η is the quadratic form obtained by integrating Q r (η over r. observation we need is that we can write Q(η j = C αk,...,α 1 η α k k η α (η + + η k α 1 where k i=1 α i = j, and α i j for every i. Thus, the coefficient of t j is such a linear combination of terms of the form (π n(k 1 ( α k V (ηk ( α V (η ( α 1 V (η + + η k dη dη k, The key This integral is equal to ( α k V (y ( α V (y ( α 1 V (y dy, which by Lemma 3.3 is bounded by C V k L Dj V L. This establishes the bounds of Proposition 3. on the coefficients c k,j+k, provided V C c (R n. The m-th order remainder is a constant times 1 t m (1 s m 1 Σ (R n k 1 e stqr(η Q r (η m V (ηk V (η V (η + + η k dη dr ds, which by a similar argument can be written as an integral over r and s of various polynomials in r, s times t m e stqr(η ( α kv (ηk ( α V (η ( α 1 V (η + + η k dη dη k, with α i m, and i α i = m. We now show that, uniformly over r Σ, and t >, 1 k e tqr(η v k (η k v (η v 1 (η + + η k dη v j L p j, (3.14 (π n(k 1 whenever p j and j p 1 j = 1. Here we note that the proof of Lemma 3.3 bounds the right hand side, with p j = m/ α j and v j = α j V, by V k L V Hm. The bounds on r k,k+m (t in Proposition 3. will follow for V Cc (R n. The left hand side of (3.14 equals G r,t (y x,..., y k x v k (y k v (y v 1 (x dx dy dy k. The kernel G r,t is positive and has total integral 1, so for proving the bound we may assume each v j is nonnegative. By interpolation, we may restrict to the case that two

17 HEAT TRACES AND EXISTENCE OF RESONANCES 17 of the p j s are equal to, and the rest equal. There are then two distinct cases to consider: p 1 = p =, or p = p 3 =. In the first case, we dominate the integral by v k L v 3 L K(y x v (y v 1 (x dy dx (3.15 where K(z = G r,t (z, y 3,..., y k dy 3 dy k. Since K = 1, by Young s inequality the integral in (3.15 is bounded by v L v 1 L. In case p = p 3 =, we bound the integral by v k L v 4 L v 1 L K(y, y 3 v 3 (y 3 x v (y x dy dy 3 dx, (3.16 where now K(y, y 3 = G r,t (y, y 3, y 4,..., y k dy 4 dy k. Thus K(η, η 3 = e tqr(η,η 3,,...,. Writing v and v 3 in terms of their Fourier transforms, and integrating out y and y 3, expresses the integral in (3.16 as (π n e ix(η +η 3 e tqr( η, η 3,,..., v 3 (η 3 v (η dη dη 3 dx = (π n e tqr( η,η,,..., v 3 ( η v (η dη, which is bounded by v 3 L v L by the Schwarz inequality, as Q r. It remains to show the expansion holds for general V L (R n, R H m (R n. We set φ ε V = V ε Cc (R n, where φ ε = ε n φ(ε 1 is a family of smooth compactly supported mollifiers. Recall that tr ( W k (t = (4πt n/ t k B t (V. Since for each t, B t (V is continuous in V in the L k (R n topology, then B t (V = lim ε B t (V ε. Furthermore, since V ε L V L, V ε H m V H m, we have the following bounds, uniform for t > and ε >, r k,k+m (t, V ε C V k L V H m. It thus remains to show that lim ε c k,k+j (V ε = c k,k+j (V if j m 1, for appropriately defined c k,k+j (V satisfying the bounds of Proposition 3.. Recall that c k,k+j (V ε can be written as a linear combination of terms of the form ( α k V ε (y ( α 1 V ε (y dy, (3.17 where α i j for all i, and k i=1 α i = j. We define c k,k+j (V by the same formula, which by Lemma 3.3 is well defined, and absolutely dominated by V k L Dj V L.

18 18 HART SMITH AND MACIEJ ZWORSKI To see that (3.17 converges, as ε, to the same expression with V ε replaced by V, we note that, by the proof of Lemma 3.3, α i V L m α i, so for α i >, lim ε α i V ε α i V L m α i =. Thus, the product over the terms in (3.17 with α i converges in L m j to the same product with V ε replace by V. Since m > 1, the integral in (3.17 converges as ε j by the fact that V ε V in L p for all p <. References [Au11] A. Autin, Isoresonant complex-valued potentials and symmetries. Canad. J. Math. 63(11, [BaSa95] R. Bañuelos and A. Sá Barreto, On the heat trace of Schrödinger operators, Communications in Partial Differential Equations, (1995, [vdb91] M. van den Berg, On the trace of the difference of Schrödinger heat semigroups, Proc. Roy. Soc. Edinburgh Sect. A 119 (1991, no. 1-, [Br84] J. Brüning, On the compactness of isospectral potentials, Communications in Partial Differential Equations, 9(1984, [Ch99] T. Christiansen, Some lower bounds on the number of resonances in Euclidean scattering, Math. Res. Letters 6(1999, [Ch5] T. Christiansen, Several complex variables and the distribution of resonances in potential scattering. Comm. Math. Phys. 59(5, [Ch6] T. Christiansen, Schrödinger operators with complex-valued potentials and no resonances. Duke Math. J. 133(6, [Ch8] T. Christiansen, Isophasal, isopolar, and isospectral Schrödinger operators and elementary complex analysis. Amer. J. Math. 13(8, [ChHi5] T. Christiansen and P. Hislop, The resonance counting function for Schrödinger operators with generic potentials, Math. Res. Lett. 1(5, [ChHi1] T. Christiansen and P. Hislop, Maximal order of growth for the resonance counting functions for generic potentials in even dimensions. Indiana Univ. Math. J. 59(1, [CdV81] Y. Colin de Verdière, Une formule de traces pour l opérateurs de Schrödinger dans R 3, Ann. Sci. Éc. Norm. Sup. 4éme série, 14(1981, [CdV1] Y. Colin de Verdière, Semiclassical trace formulas and heat expansions, Anal. PDE 5(1, [DaHe1] K. Datchev and H. Hezari, Resonant uniqueness of radial semiclassical Schrdinger operators, Applied Mathematics Research Express, 1(1, [DiVu13] T.-C. Dinh and D.-V. Vu, Asymptotic number of scattering resonances for generic Schr dingier operators, Comm. Math. Phys. 36(14, [Do4] H. Donnelly, Compactness of isospectral potentials, Trans. A.M.S. 357(5, [DyZw] S. Dyatlov and M. Zworski, Mathematical theory of scattering resonances, book in preparation; [Gil4] P. B. Gilkey, Asymptotic formulae in spectral geometry, CRC, Boca Raton, FL, 4. [Gu81] L. Guillopé, Asymptotique de la phase de diffusion pour l opérateur de Schrödinger avec potentiel, C. R. Acad. Sci. Paris Sér. I Math. 93(1981,

19 HEAT TRACES AND EXISTENCE OF RESONANCES 19 [HiPo3] M. Hitrik and I. Polterovich, Regularized traces and Taylor expansions for the heat semigroup, J. London Math. Soc. 68(3, [Je9] A. Jensen, High energy asymptotics for the total scattering phase in potential scattering theory. Functional-analytic methods for partial differential equations (Tokyo, 1989, , Lecture Notes in Math., 145, Springer, Berlin, 199. [JeKa79] A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J. 46(1979, [Kor] E. Korotyaev, Inverse resonance scattering on the real line, Inverse Problems 1(5, [McMo75] H. McKean and P. van Moerbeke, The Spectrum of Hill s Equation, Invent. Math. 3(1975, [Me95] R.B. Melrose, Geometric Scattering Theory, Cambridge University Press [Sa1] A. Sá Barreto, Remarks on the distribution of resonances in odd dimensional Euclidean scattering, Asymptot. Anal. 7(1, [SaZw96] A. Sá Barreto and M. Zworski, Existence of resonances in potential scattering. Comm. Pure Appl. Math. 49(1996, [Tay11] M.E. Taylor, Partial Differential Equations III: Nonlinear Equations, Applied Mathematical Sciences, 118, nd edition, Springer 11. [Ti64] E.C. Titchmarsh, The theory of functions, nd edition, Oxford University Press, [Zw89] M. Zworski, Sharp polynomial bounds on the number of scattering poles, Duke Math. J. 59(1989, [Zw97] M. Zworski, Poisson formula for resonances, Séminaire E.D.P. ( , Exposé no XIII, 1 p. [Zw1] M. Zworski, A remark on isopolar potentials, SIAM J. Math. Anal., 3(1, [Zw1] M. Zworski, Semiclassical analysis, Graduate Studies in Mathematics 138, AMS, 1. address: hart@math.washington.edu Department of Mathematics, University of Washington, Seattle, WA 98195, USA address: zworski@math.berkeley.edu Department of Mathematics, University of California, Berkeley, CA 947, USA