Perturbations of Gibbs Semigroups and the Non-Selfadjoint Harmonic Oscillator

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Perturbations of Gibbs semigroups and the non-selfadjoint harmonic oscillator

Citation for published version: Boulton, L 2020, 'Perturbations of Gibbs semigroups and the non-selfadjoint harmonic oscillator', Journal of Functional Analysis, vol. 278, no. 7, 108415. https://doi.org/10.1016/j.jfa.2019.108415

Digital Object Identifier (DOI): 10.1016/j.jfa.2019.108415

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Published In: Journal of Functional Analysis

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Download date: 01. Oct. 2021 arXiv:1806.06374v2 [math.SP] 10 Jul 2018 atUiest,Eibrh H44S ntdKndm E-ma Kingdom. United 4AS, EH14 Edinburgh, University, Watt M MSC2010: AMS Schrödinger operators. selfadjoint Keywords: ∗ eateto ahmtc n awl nttt o Mathem for Institute Maxwell and Mathematics of Department etrain fGbssmgop n the and semigroups Gibbs of Perturbations h eeao faGbssmgope semigroup Gibbs a of generator the h yo-hlisepnini utbeShte-o Neu con Schatten-von the suitable of in terms expansion in Dyson-Phillips conditions the these determine We semigroup. h o-efdon amncoclao,on oscillator, harmonic non-selfadjoint the emn omi hscs for case this in norm Neumann h eeao faGbssmgop ntefis afo hspa this of half first the In semigroup. on conditions Gibbs sufficient a give of generator the nerbeptnilgoiglike growing potential integrable sals htteDsnPilp xaso ovre in converges expansion Dyson-Phillips the that establish T rmti edtriehg nryaypoisfrteeigen the for of asymptotics norm energy resolvent high the determine we this From o all for buddwith -bounded ntescn afo h ae econsider we paper the of half second the In Let o-efdon amncoscillator harmonic non-selfadjoint > t T etraino ib eirus yo-hlisepnin non- expansion, Dyson-Phillips semigroups, Gibbs of Perturbation etegnrtro a of generator the be aGbssmgop.Let semigroup). Gibbs (a 0 70,8Q2 81Q15. 81Q12, 47D06, T budeult eo ngeneral In zero. to equal -bound H ϑ + ynl Boulton Lyonell n uy2018 July 2nd V A . othat so Abstract C r 0 ag nuhadso that show and enough large -semigroup | x 1 | α − ( T H tifiiyfr0 for infinity at ϑ + + A V A L ) eaohrcoe operator, closed another be τ 2 T e stegnrtro Gibbs a of generator the is ( ∗ − for R = t T and ) H | hc so rc class trace of is which arg ϑ T il: = + τ A [email protected] − ≤ | ≤ A tclSine,Heriot- Sciences, atical r = e annorms. mann − ih o be not might Schatten-von < α π 2 V i ϑ egneof vergence ausand values | − ∂ locally a , H x 2 ϑ +e ϑ .We 2. e we per + 6 | = i ϑ V x π 2 2 is , . 1 Introduction

The non-selfadjoint harmonic oscillator

iϑ 2 iϑ 2 π π H = e− ∂ +e x <ϑ< , ϑ − x − 2 2 2 2 \2 acting on L (R) with domain D(Hϑ) = H (R) H (R) studied by Exner in [12] and Davies in [7], has become one of the∩ reference models in the theory of pseudospectra and non-selfadjoint phenomena. Cf. [19, 26, 20, 4], [9, Chapter 14] and [23, p.105]. This operator is J-selfadjoint with respect to the conjugation Ju(x)= u(x) and Hϑ∗ = H ϑ, so it is selfadjoint only when ϑ = 0. − Hϑτ As it is also m-sectorial, Hϑ is the generator of a C0-semigroup e− for all π arg τ 2 ϑ . In fact, the classical Mehler’s formula extends to ϑ = 0 and |non-real| ≤τ in− a| maximal| angular semi-module which is much larger than6 this sector, rendering a trace class (Gibbs) semigroup. See [26, 1] and Theorem 4 below. In this paper we consider perturbations of Hϑ by locally integrable complex potentials V such that

V (x) a x α + b x R (1) | |≤ | | ∀ ∈ for some 0 α< 2, a> 0 and b R. As V is H -bounded with relative bound ≤ ∈ ϑ 0, the non-selfadjoint Schrödinger operators Hϑ +V are also J-selfadjoint in the same domain D(Hϑ + V )=D(Hϑ). In Section 4 we show that Hϑ + V is the generator of a Gibbs semigroup (Hϑ+V )τ π e− for all arg τ 2 ϑ when ϑ = 0. According to the work of Angelescu et al [2]| and of| Zagrebnov ≤ − | | [27], see also6 [5], a class perturbation1 of an m-sectorial operator whose real part is the generator of aP Gibbs semigroup is also the generator of a Gibbs semigroup. But what is remarkable and not obvious for Hϑ + V from this, is the fact that the trace class property extends all the way to the edges of the maximal sector. We obtain the latter, by showing that the Dyson-Phillips expansion of the perturbed semigroup converges in an r Schatten-von Neumann norm for sufficiently large r (it does not converge for r 2 for α too close to 2). ≤As the framework turns out to be general and may be applicable in other contexts, we begin by developing an abstract perturbation theory of generators of Gibbs semigroups in Section 2. The results in the papers [2] and [27] mentioned above, rely on an inequality due to Ginibre and Gruber, [15], which cannot be easily extended to the non-sectorial setting. Therefore we take here a completely different route, that of the Dyson-Phillips expansion. This allows generators which are not necessarily m-sectorial, but the perturbations ought to be more than just of class . They must satisfy an analogous condition of integrability, but with respectP to a Schatten-von Neumann norm. Details in Lemma 1 and Definition 1 below. 1See [6, p70].

2 The spectrum of Hϑ is Spec(Hϑ) = 2n +1 n∞=0 , with corresponding normalised eigenfunctions [8] { }

iϑ iϑ Ψn(x)=e 4 Φn e 2 x where Φ ∞ are the normalised eigenfunctions of H . In Section 4 (Corol- { n}n=0 0 lary 3) we show that the eigenvalues of Hϑ + V have a real part growing at least like n and a distance from the rays arg(z) = ϑ growing at least like n1/2 as n . We know [4, 20] that the distance| | from| | these rays to points z → ∞ 1/3 on the boundary of the pseudospectra of Hϑ increases exactly like (z) for z . Therefore, despite of the fact that V might be unbounded,ℜ the eigen- → ∞ values of the non-selfadjoint Schrödinger operator Hϑ + V eventually lie way in the interior of the pseudospectra of Hϑ. This is surprising, if we recall that the ε-pseudospectrum is the union of the spectra of all bounded perturbations of Hϑ with norm less than (or equal) ε. The study of eigenvalue asymptotics of non-selfadjoint Schrödinger operators has attracted interest from different communities in recent years, see [14] and references therein. Since condition on the decay of the Schatten-von Neumann norm of the resolvent at infinity are related to conditions of integrability of the semigroup at small times via the Laplace transform, our approach is closer to the framework of relative Schatten-von Neumann class perturbation developed in [10]. Our final statement, Corollary 4, gives an indication about the shape of the pseudospectra of Hϑ + V . We show that

iϑ 1 lim (Hϑ + V e± ρ β)− =0 ρ →∞ k − − k for all β R. Therefore the distance from the real axis to points z on the ∈ boundary of the pseudospectra of Hϑ + V is o( (z)) as z . This com- plements findings in [8, 28, 21, 11, 19], about theℜ resolvent→ norm ∞ growth for non-selfadjoint Schrödinger operators with potentials large at infinity. Our results refine in various ways those published in [3, Chapter 3] many years ago. I have to thank A. Doiku and P. Siegl with whom I sustained a number of useful discussions. Also D. Krejˇciˇr´ıkand J. Viola for their valuable comments.

2 Gibbs semigroups and their perturbations

Let be a Hilbert space. Below the operator T acting on is said to be the H T t H generator of a C -semigroup e− for t > 0, if T is so in the usual sense [17, 0 − Chapter X and 10.6]. We are only concerned about C0-semigroups of compact operators and begin§ by brieﬂy recalling various well-known facts. T t T t Let e− be compact for all t > 0. Then e− is continuous in the uniform operator topology for all t> 0, the resolvent of T is compact and

T t λt Spec(e− )= e− : λ Spec(T ) 0 . (2) { ∈ }∪{ }

3 Let T t at ϕ(T ) = inf a R : M > 0, e− Me t> 0 . { ∈ ∃ k k≤ ∀ } For any t0 > 0 [25, pro.1.2.2],

T t0 T t log rad(e− ) log e− ϕ(T )= = lim k k. t t t 0 →∞ Here rad(W ) is the spectral radius of W . Combining this with (2) yields

ϕ(T )= inf (λ): λ Spec(T ) . − {ℜ ∈ } T t That is, the spectral bound and the uniform growth bound of e− coincide, due to compactness. This property will play an important role below. The following statement is not a direct consequence of classical results such as the Hille-Yosida theorem. It will serve our purposes later on, hence we include a self-contained proof. Many more precise asymptotic properties of similar nature are known, cf. [25] and [9, Ch. 8].

T t Theorem 1. Let T be the generator of a C0-semigroup such that e− is compact for all t> 0. Then for all r< ϕ(T ) ﬁxed, − 1 lim (T r iy)− =0. y →±∞ k − − k Proof. Without loss of generality we can assume that ϕ(T ) = 0. Taking inverse Laplace transform [6, thm. 2.8], we get

1 ∞ zt T t (T z)− f = e e− f dt (3) − Z0 for all f and z such that (z) < 0. Let z = r + iy, let M > 1 be such that T t ∈ H ℜ e− M for all t> 0 and let k k≤ log M s> > 0. − r Then s zs 1 z(t s) T t ∞ z(t s) T t e− (T z)− f = e − e− f dt + e − e− f dt − 0 s Z s Z z(t s) T t ∞ zt T (t+s) = e − e− f dt + e e− f dt 0 0 Z s Z z(t s) T t T s 1 = e − e− f dt +e− (T z)− f − Z0 for all f . Applying the triangle inequality and then placing all the resolvent norms to∈ the H left hand side, yields

rs s 1 e− iyt rt T t (T r iy)− e e e− dt . k − − k≤ e rs M − Z0 −

4 Here the constant outside the norm is positive and does not depend upon y. T t Since e− is continuous in the uniform operator topology, it is also locally integrable (Riemann and Bochner) with respect to the associated norm. Let

s iyt rt T t Fˆ(y)= e− e e− dt. Z0 Then Fˆ(y) is the Fourier transform of the operator-valued function

rt T t t e e− ξ (t) 7→ [0,s] which lies in L1(R; ( )). Here s is fix. Thus, a version of the Riemann- Lebesgue lemma forB BochnerH spaces (the proof is identical to the classical result [13, sec. 7.2] as the integrand above is a limit of step functions) ensures that Fˆ(y) 0 as y . k k → → ±∞ Let q 1. We denote by the q Schatten-von Neumann operator ideal and ≥ Cq by q the corresponding norm. As usual, here is the compact operators. Belowk·k we write for operators. C∞ k·k∞ ≡k·k T (s+t) T s T t By virtue of the semigroup property, e− = e− e− , combined with T t the fact that q is non-increasing as q increases, it follows that e− t>0 q k·k T t { } ⊂C for some q < if and only if e− t>0 1. A C0-semigroup with this property is often∞ called a Gibbs semigroup,{ } ⊂ [24, C 2, 27]. We will adhere to this terminology. T t T t If e− is a Gibbs semigroup, then t e− is continuous in the trace norm for all t > 0. If the generator is unbounded,7→ the C -semigroup is always k·k1 0 discontinuous at t = 0 in and hence in all of the other norms q. We now determine a class of perturbationsk·k∞ of the generator which preservek·k the finite trace. A closed operator A is some times said to be a class perturbation of the generator T iff P

1 T t T t D(A) e− ( ) and Ae− dt< , ⊃ H k k∞ ∞ t>0 0 [ Z see [6, p70]. If A is a class perturbation of T , then D(A) D(T ), A is T - bounded with relative boundP equal to zero and the closed operator⊃ (T + A) is the generator of a C0-semigroup on . The perturbed C0-semigroup is given in terms of the unperturbed one via aH Dyson-Phillips expansion

(T +A)t ∞ k e− = ( 1) W (t) (4) − k Xk=0 which is absolutely convergent in for all 0

5 and T t W0(t)=e− t (5) T s Wk(t)= Wk 1(t s)Ae− ds. − − Zs=0 The integrals are convergent in for all t> 0. From this it follows that the variation of parameter formula k·k∞

t (T +A)t T t (T +A)(t s) T s e− =e− + e− − Ae− ds (6) Z0 holds true, where the integral converges in for all 0

∞ ∞ T = (in3 + n) e e and A A = bn e e | nih n| ≡ b | nih n| n=1 n=1 X X in their maximal domains. Then, T + Ab is: the generator of a Gibbs semigroup for b > 1, only the generator of a unitary group for b = 1 and not even a − − 1 generator of a C0-semigroup for b < 1. This, despite of the fact that AbT − − T t 1 is trace class for all b R. Note that Abe− t− as t 0, so Ab is not ∈ k k∞ ∼ → a class perturbation of T . Also that Ab destroys the m-sectoriality of T for b< 1.P In− the next lemma, T is not assumed to be m-sectorial and (T ) might not be the generator of a compact semigroup. The proof follows closelyℜ the line of arguments in [6, theorems 3.1-3.5], replacing the operator norm with the norm of q. In this proof we could have used directly the variation of parameters formulaC (as we do later on), but we prefer to highlight the range of absolute convergence of the Dyson-Phillips expansion in . k·kq T t Lemma 1. Let T be the generator of a C0-semigroup such that e− 1 for all t> 0. Let A be a closed operator such that ∈ C

1 T t T t D(A) e− ( ) and Ae− dt< (7) ⊃ H k kq ∞ t>0 0 [ Z for some q < . Then T + A with domain D(T ) is also the generator of a ∞ (T +A)t C -semigroup such that e− for all t> 0. Moreover 0 ∈C1 (T +A)t T t + e− = O( e− ) t 0 . (8) k kq k kq → Proof. The hypotheses ensure that A is a class perturbation of T . Hence T +A P (T +A)t with domain D(T ) is the generator of a C0-semigroup. Moreover e− is given by (4) convergent in for t> 0 small enough. By considering T ϕ(T ) k·k∞ −

6 instead of T for the general case, we can assume without loss of generality that T t e− M for all t> 0. k k∞ ≤ Let us show ﬁrst that in (5), Wk(t) q for all k N and t> 0. We begin with k = 1. Since ∈C ∈

n 1 T s T (s+n 1) Ae− q ds = Ae− − q ds n 1 k k 0 k k Z − Z 1 T s M Ae− ds< n N. ≤ k kq ∞ ∀ ∈ Z0 Then t T s Ae− ds< t> 0. k kq ∞ ∀ Zs=0 Fix t> 0. Then

t t T (t s) T s T s e− − Ae− ds M Ae− ds< . k kq ≤ k kq ∞ Zs=0 Zs=0 T s The q-valued function s e− is continuous in q for all s> 0, because it is C 7→ k·k T (t s) T s continuous in the trace norm. Then the -valued function s e− − Ae− Cq 7→ is also continuous with respect to the norm q. Indeed, for ﬁx b > 0 such t t k·k that b< 2 and for s> 2 , we have

T t T s T b T (t b) T (s b) Ae− Ae− q Ae− e− − e− − q 0 s t. k − k ≤k k∞k − k → →

Therefore the integrand in the expression for W1(t) is Riemann integrable in q in all segments of the form [α, 1] for α> 0. Note that this integral is improperC in the norm q in the segment (0, 1] but the right hand side of (7) ensures that this improperk·k integral is convergent. Hence W (t) and 1 ∈Cq t T s W (t) M Ae− ds. k 1 kq ≤ k kq Zs=0 Now consider k = 2. Let

e Tx x> 0 F (x)= − (0 otherwise.

7 Then

t s T (t s) T (s u) T u e− − Ae− − Ae− q du ds s=0 u=0k k Z Z t s M AF (s u)AF (u) q du ds ≤ s=0 u=0 k − k Z t Z t = M AF (s u)AF (u) q du ds s=0 u=0 k − k Z t Z t M AF (s u) AF (u) q ds du ≤ u=0 s=0 k − k∞k k Z t Z t M AF (u) q AF (s u) qds du ≤ u=0 k k s=0 k − k Z t Z t u − = M AF (u) q AF (x) qdx du u=0 k k x= u k k Z Z − t t M AF (u) du AF (x) dx ≤ k kq k kq Zu=0 Zx=0 t 2 T s = M Ae− ds < . k kq ∞ Zs=0 T (t s) T (s u) T u Hence, by continuity, the q-valued function (s,u) e− − Ae− − Ae− is integrable (Riemann withC the improper integral7→ once again convergent) with respect to the norm q in the region 0 0, k =3, 4,... k k kq ≤ k kq ∀ Zs=0 (T +A)t In order to show that e− , it is then enough to prove the conver- ∈Cq gence in the norm of q of the series at the right hand side of (4) for t> 0 small enough. Choose a> C0 such that

a T s Ae− ds< 1. k kq Zs=0

Then for all t (0,a] the series k∞=1 Wk(t) q < . This guarantees the ∈ k k(T +A)∞t convergence of the right hand side of (4) and e− q for 0 a is a consequenceP of the semigroup∈C property.≤ Hence (T +A)t e− is also a Gibbs semigroup.

8 Finally, note that for 0

(T +A)t T t ∞ e− e− + W (t) k kq ≤k kq k k kq Xk=1 a k ∞ T t T s e− q + M Ae− qds . ≤k k 0 k k kX=1 Z As the series at the right hand side converges independent of t, then there exists M˜ > 0 independent of t, such that

(T +A)t T t e− e− + M˜ 0

If T and A satisfy the hypothesis of Lemma 1, then the improper integral in the variation of parameters formula (6) converges in q. Indeed the map (T +A)(t s) T s k·k s e− − Ae− is continuous and 7→ k·kq (T +A)(t s) T s (T +A)(t s) T s e− − Ae− e− − Ae− q ≤ q ∞ where

(T +A)(t s) T (t s) e− − = O e− − = O(1) s 0 and s t. → → ∞ ∞

Remark 1. Both the results of [2] and those of [27] concerning perturbations of m-sectorial generators, are consequence of an inequality originally found by Ginibre and Gruber [15] extended from the selfadjoint setting. Details apparently missing in [27] were completed in [5]. In the latter, this extension was formulated for m-sectorial operators. Unfortunately we do not have an analogue inequality at hand under the more general hypothesis above. Lemma 1 induces the following terminology which will simplify the discussions below.

Deﬁnition 1. Let 1 q . The closed operator A is said to be a class q ≤ ≤ ∞ T t PC perturbation of the generator T of a C0-semigroup e− t>0 1, if (7) are satisﬁed. { } ⊂ C

If A is a class q perturbation of T , it is also a class p perturbation of T for all p > q. PC PC If two closed operators A and A are class perturbations of the gener- 1 2 PCq ator of a Gibbs semigroup, it is not necessarily the case that the sum A1 + A2 (on a suitable domain) is closable. For this reason, the class described in Deﬁ- nition 1 is not additive. By following the ideas of [17, 13.3-13.5], it is possible to extend this deﬁnition to perturbations that are not necessarily§ closable, then obtain an additive class and an equivalence relation for generators. The details of this require developing extra notation that will not serve our focused purpose

9 in the next section when considering T = Hθ. Therefore we do not address this for the time being. Now an example. Let T = T ∗ be the selfadjoint operator with compact resolvent given by ∞ T = n e e , | nih n| n=1 X where e ∞ is an orthonormal basis of . Then { n}n=1 H

T t ∞ nt e− = e− e e . | nih n| n=1 X Hence t T t e− e− = k k1 1 e t − − T t and e− is a Gibbs semigroup. For α 1, let ≤ ∞ A = T α = nα e e α | nih n| n=1 X in its maximal domain. Then

α tn T t maxn N n e− q = A − ∈ ∞ αe q = αq tqn 1/q k k (( ∞ n e− ) 1 q< n=1 ≤ ∞ T t P α + For q = , we have Aαe− t− as t 0 . Then Aα is a class (class )∞ perturbationk of T fork∞ all∼α< 1. For →q < , PC∞ P ∞ T t q tq Aαe− q = Li( αq)(e− ) k k − where Lis(z) is the polylogarithm function. Since

1 s lim(1 z) − Lis(z) = Γ(1 s) s< 1 z 1 → − − ∀ [16, 9.557], for all qα > 1 − +1 T t qα + A e− t− q t 0 . k α kq ∼ → 1 Then, Aα is a class q perturbation of T if and only if q > 1 α (assuming q 1 as in the deﬁnitionPC above). This shows that, the smaller− the q, the “multiplicative≥ smaller” the perturbation of a generator of a Gibbs semigroup should be, in order to be included in the class q. It also shows that, although they are nested, these classes are not equal inPC general. Note that for α = 0, A0 = I is not a class q perturbation of T for q = 1, but it is so for all q> 1. PC 1 We can relate this example to the harmonic oscillator by taking T = 2 (H0 + 1). Let us now determine that the Deﬁnition 1 is symmetric. The next lemma follows the template of [17, Lemma 13.5.1].

10 T t Lemma 2. Let T be the generator of e− 1 for all t> 0. If A1 and A2 are two closed operators such that they are both∈C class perturbations of T , then PCq 1 (T +A1)t (T +A1)t D(A ) e− ( ) and A e− dt< 2 ⊃ H k 2 kq ∞ t>0 0 [ Z 2 Proof. Since Aj are class perturbations of T , by virtue of [17, Lemma 13.5.1] , we know that P (T +A1)t D(A ) e− ( ) 2 ⊃ H t>0 [ as required in the ﬁrst part of the conclusion. Moreover

1 (T +A1)t A2e− dt< . (9) k k∞ ∞ Z0 In order to show the second part of the conclusion, we use the variation of (T +Aj )t parameters formula. From Lemma 1, it follows that e− 1 for all t> 0. Also, we know that ∈C

t (T +A1)t T t (T +A1)(t s) T s e− =e− + e− − A1e− ds Z0 where the integral converges in q (for t small enough). Since all the improper integrals involved in the followingk·k expression are Riemann integrals and they are convergent in and since the operator A2 is closed, we have k·k∞ t (T +A1)t T t (T +A1)(t s) T s A2e− = A2e− + A2e− − A1e− ds, (10) Z0 (T +A1)(t s) T s see [17, Theorem 3.3.2]. Also, (s,t) A2e− − A1e− is continuous in . Moreover, 7→ k·kq 1 t (T +A1)(t s) T s A2e− − A1e− ds dt Zt=0 Zs=0 q 1 t (T +A1)(t s) T s A2e− − A1 e− dsdt ≤ t=0 s=0 q Z 1 Z t (T +A1)(t s) T s A e− − A e− dsdt ≤ 2 1 q Zt=0 Zs=0 ∞ 1 1

= A2F1(t s) A1F (s) q dsdt t=0 s=0 k − k∞ k k Z 1 Z 1

= A1F (s) q A2F1(t s) dtds s=0 k k t=0 k − k∞ Z 1 Z 1

A1F (s) q ds A2F1(x) dx. ≤ k k k k∞ Zs=0 Zx=0 2 A T In the notation of [17] this is written as j↾D(−T )∈ B(− ) and here we are also invoking loc. cit. Theorem 13.3.1.

11 Here we write F (x) as in the proof of Lemma 1 and

(T +A1)x e− x> 0 F1(x)= (0 otherwise.

The hypothesis and (9), yield that the this double integral is ﬁnite. Hence the second conclusion follows from this, integrating (10).

Corollary 1. Let q 1. Let A be a class q perturbation of the generator T of a Gibbs semigroup.≥ Then PC

(T +A)t T t + e− e− t 0 . k kq ∼k kq →

Proof. Let T2 = T +A with D(T2) = D(T ). Then A is a class q perturbation of T as a consequence of Lemma 2 with A = −A = A . PC 2 1 − 2 If A is both accretive and T -bounded with bound less than one, then T +A is the generator of a C0-semigroup [6, Corollary 3.8]. In Lemma 1, the perturbation A is allowed to be non-accretive, at the cost of being relatively compact (and more). See Remark 2 below.

Lemma 3. Let A be a class q perturbation of the generator T . If 0 Spec A, then the closure of any otherPC closable operator B such that D(B) D(6∈A) is also a class perturbation of T . ⊃ PCq Proof. Let B be the closure of B. The inclusion of the domains and the close graph theorem ensure that B is A-bounded [18, p.191]. Then

T t 1 T t Be− q BA− Ae− q. k k ≤k k∞k k Hence B also satisﬁes the right hand side of (7). Our major objective after this section will be to apply the framework just introduced to the holomorphic semigroup generated by the non-selfadjoint harmonic oscillator and perturbations by potentials. If T is the generator of a bounded holomorphic semigroup on a sector and A is T -bounded with relative bound equal to 0, then T + A + c is the generator of a bounded holomorphic semigroup on that sector for some c > 0, [18, Corollary 2.5, p.500]. If A is ad- ditionally a class perturbation of the generator T of a Gibbs semigroup, as PCq we shall see next, the small t asymptotic behaviour of the q norm is preserved even at the boundary of the sector. C For α, β (0, π ], here and elsewhere we write ∈ 2 ( α, β)= reiω : r> 0, ω ( α, β) . S − { ∈ − } Tτ Let T be an m-sectorial operator. Then, e− is a bounded holomorphic semi- T t group for all τ ( α, β) with suitable α and β. Ife− 1 for all t> 0, then Tτ ∈ S − Tτ ∈C also e− for all τ ( α, β) and τ e− is holomorphic in ( α, β) ∈ C1 ∈ S − 7→ S −

12 eiθT t with respect to . For θ = α or θ = β, the C -semigroup e− might or k·k1 − 0 might not be compact. It is not compact for example, whenever T = T ∗ > 0 π Tτ and α = β = . But, as we shall see in the next section, some times e− 2 ∈C1 for all τ ( α, β) 0 , the maximal sector of analyticity. By applying Corol- ∈ S − \{ } lary 1 to rotations of the operators involved, it is straightforward that class q perturbations preserve this characteristic. PC

Tτ Theorem 2. Let T be the generator of a semigroup e− for all τ ∈ C1 ∈ ( α, β) 0 holomorphic in ( α, β). If A is a class q perturbation of T S − \{ } S − PC (T +A)τ for q < , then T + A is also the generator of a semigroup e− 1 for all τ ∞( α, β) 0 holomorphic in ( α, β). Moreover, for all α ∈Cθ β, ∈ S − \{ } S − − ≤ ≤ (T +A)eiθr T eiθr e− e− r 0. k kq ∼k kq → See also Theorem 3 below.

3 Asymptotic behaviour of the non-selfadjoint Mehler kernel

The numerical range of Hϑ is

iϑ iϑ 1 Num(H )= e− s +e t : s, t R, st ( ϑ , ϑ ), ϑ ∈ ≥ 4 ⊂ S −| | | |

[4, pro. 2.1]. Then Hϑ is m-sectorial and the generator of a bounded holomor- H τ phic semigroup e− ϑ for all π π τ + ϑ , ϑ . ∈ Sϑ ≡ S − 2 | | 2 − | | π i( 2 ϑ ) π π Moreover e ± ∓| | Hϑ are generators of C0-semigroups for all ϑ ( 2 , 2 ). H τ ∈ − Whenever ϑ =0, e− ϑ is continuous in for all τ ϑ 0 . This is not 6 k·k∞ ∈ S \{ } π π the case for ϑ = 0 and τ approaching the boundary of the segment ( 2 , 2 ), iH0t S − because e± are unitary groups for t R. According to the framework of [1, 26],∈ when seen as a family of bounded H τ operators in τ, the holomorphic semigroup e− θ has a bounded extension (in the uniform operator norm) to the maximal semi-modulus π = τ C : τ > 0, arg tanh(τ) < ϑ . Tϑ ∈ ℜ | | 2 − | | ⊃ Sϑ n o This extension is analytic and compact for all τ ϑ, and it is bounded for ∈ T ϑ 2 ϑ 2 all τ ϑ. The operator Hϑ which has Weyl symbol qϑ(x, ζ)=e− ζ +e x , corresponds∈ T to that presented in [26, Example 2.1].

13 H τ We now determine various asymptotic properties of e− ϑ in parts of this maximal region. Let

2τ λ λ(τ)=e− ≡ λ(τ) eiϑ w w (ϑ, τ)=eiϑ = csch(2τ) 1 ≡ 1 1 λ2(τ) 2 − eiϑ 1+ λ2(τ) eiϑ w w (ϑ, τ)= = coth(2τ). 2 ≡ 2 2 1 λ2(τ) 2 − and w 1/2 M (τ,x,y)= 1 exp 2w xy w (x2 + y2) . ϑ π 1 − 2 The classical Mehler’s formula extends to non-real τ [4, Theorem 4.2],

H τ ∞ e− ϑ f(x)= M (τ,x,y)f(y)dy τ . ϑ ∀ ∈ Sϑ Z−∞ Let r r (ϑ, τ)= [w (ϑ, τ)]. In the next statement, note that j ≡ j ℜ j π ω ϑ cos θ sin ω | |≤ 2 − | | ⇒ | | ≥ | | π (11) and ω = ϑ cos θ = sin ω . | | 2 − | | ⇐⇒ | | | | Lemma 4. The conditions

r (ϑ, τ) > 0 and r (ϑ, τ) r (ϑ, τ) > 0 (12) 2 2 ± 1 hold, if and only if τ . Moreover, as t 0+, ∈ Tϑ → iω 1 1 w (ϑ, e t) = t− + O(1), | 1 | 4 cos(ω+θ) 1 π iω 2 t− + O(1) ω < 2 ϑ r2(ϑ, e t)= | | − | | sin(4θ) t + O(t2) ω = π ϑ ( 3 | | 2 − | | and

1 2 π 1 cos2 ϑ sin2 ω + O(t ) ω < 2 ϑ iω 2 iω 2 = 3 − 2 | | π − | | r2(ϑ, e t) r1(ϑ, e t) 2 t− + O(1) ω = ϑ , − ( sin (2ϑ) | | 2 − | | for ﬁxed ϑ π , π and ω = π . ∈ − 2 2 6 ± 2 Proof. For the ﬁrst part of the lemma we show that π π arg(w w ) < arg tanh τ < ϑ (13) | 2 ± 1 | 2 ⇐⇒ | | 2 − | | and that π π arg tanh τ < θ = arg w < . (14) | | 2 − | | ⇒ | 2| 2

14 Since 1 λ 1 λ tanh τ = − and w w =eiϑ ± , 1+ λ 2 ± 1 1 λ ∓ then arg(w w )= ϑ arg(1 + λ) arg(1 λ)= ϑ arg tanh τ 2 ± 1 ± ∓ − ∓ and hence (13). Suppose that the left hand side of (14) holds true. That is tanh τ . Then also coth τ . By convexity of the sector, also ∈ Sϑ ∈ Sϑ 2 tanh(2τ)= . coth τ + tanh τ ∈ Sϑ Thus, if τ , also 2τ . Since ∈ Tθ ∈ Tθ

w2(θ, τ)= w2(θ, 2τ)+ w1(θ, 2τ), by the equivalence in (13) we get that also (14) holds true. This completes the ﬁrst part of the lemma. In the second part, the proof of the ﬁrst asymptotic formula is straightforward. For the second and third formulas, let a = 2cos ω and b = 2 sin ω. Then cos ϑ sinh 2at + sin ϑ sin 2bt r = 2 cosh2at cos2bt − and cos ϑ sinh at sin ϑ sin bt r r = ± . 2 ± 1 cosh at cos bt ∓ In the following, take into account (11). For the second asymptotic formula, we have cosh2at cos2bt lim − =4 t 0+ t2 → and two possibilities. If ω < π ϑ , | | 2 − | | cos ϑ sinh 2at + sin ϑ sin 2bt lim =2a cos θ +2b sin θ = 2 cos(ω + θ) > 0. t 0+ t → On the other hand, if ω = π ϑ , | | 2 − | | cos ϑ sinh 2at + sin ϑ sin 2bt 2 sin(4θ) lim = . t 0+ t3 4 → This yields the second asymptotic formula. For the third asymptotic formula, taking similar limits gives the following. If ω < π ϑ , | | 2 − | |

2 2 1 4 2 (r r )− = + O(t ). 2 − 1 a2 cos2 ϑ b2 sin2 ϑ − If ω = π ϑ , | | 2 − | | a2b2 r2 r2 = t2 + O(t4). 2 − 1 12 The remaining details in the proof are straightforward.

15 For x, y R, ∈ [2w xy w (x2 + y2)]=2r xy r (x2 + y2) ℜ 1 − 2 1 − 2 2 2 r1 2 r1 2 2 2 =2r1xy + x x r2x r2y r2 − r2 − − r 2 r2 r2 = r 1 x y 2 − 1 x2. − 2 r − − r 2 2 If (12) holds true, then

H τ ∞ e− ϑ f(x)= M (τ,x,y)f(y)dy τ ϑ ∀ ∈ Tϑ Z−∞ and π w Hϑτ 2 1 e− 2 = | | < τ ϑ. k k 2 r2 r2 ∞ ∀ ∈ T 2 − 1 This is the extension of Mehler’sp formula obtained in [1] for Hϑ. The semigroup property

H (τ+σ) H τ H σ e− ϑ =e− ϑ e− ϑ is valid for all τ, σ ϑ. By analytic continuation this property extends also to τ, σ such that∈τ S+ σ . Hence ∈ Tϑ ∈ Tϑ H τ e− ϑ τ . ∈C1 ∀ ∈ Tϑ

Since ϑ is open, there exists ε > 0 such that (1 ε)τ ϑ for τ ϑ. Then, indeed,T ± ∈ T ∈ T

H τ H (1 ε)τ H (1+ε)τ H (1 ε)τ H (1+ε)τ e− ϑ = e− ϑ − e− ϑ e− ϑ − e− ϑ < . k k1 k k1 ≤k k2k k2 ∞ Moreover, from the asymptotic formulas in Lemma 4 and the periodicity of the hyperbolic functions, it follows the next statement. Recall (11). Lemma 5. For all k Z, ϑ π , π and ω = π ﬁxed, ∈ ∈ − 2 2 6 ± 2 π 1 π iω 1 t− + O(1) ω < 2 ϑ Hϑ(e t+ikπ) 2 8(cos2 ϑ sin2 ω) 2 | | − | | e− = − 2  π√3 2 1 π k k 2 t− + O(t− ) ω = ϑ  8 sin (2ϑ) | | 2 − | | as t 0+.  →

16 4 Perturbations of the non-selfadjoint harmonic oscillator

We now consider locally integrable potentials V : R C satisfying (1). Below we take the maximal domain −→

D(V )= f L2(R): V (x) 2 f(x) 2dx< { ∈ R | | | | ∞} Z and denote with the same letter V the operator of multiplication in that domain. We begin by showing that V is a perturbation of H for suitable r> 1. PCr ϑ π π π Theorem 3. Let ϑ 2 , 2 and ω = 2 . If (1) holds true, then V is a class perturbation∈ of−eiωH for all 6 ± PCr ϑ 2 π (2 α) ω < 2 + ϑ r> −4 | | π | | (15) ( (2 α) ω = 2 + ϑ . − | | | |

Proof. In this proof the constants kj > 0 are independent of n, ω or ϑ, but π might depend on p, r or α. Assume that ω = 2 ϑ . We include full details in this case only as the other one is very similar.| | − | | Our ﬁrst goal is to construct a potential V˜ with the same growth as V such that, for some ε> 0,

H eiωt 1+ε + V˜ e− ϑ = O(t− ) as t 0 . (16) r →

Let n N. Let ∈ χn(x)= χ[2n,2n+1](x). Then n+1 2 2− 2 iω 2 w 2 r2 r1 2 Hϑe t 2 1 2r2y x χne− = | | e− dy e− r2 dx 2 π y R 2n Z ∈ Z n+1 (17) 2 r2−r2 2 w1 √π 2 1 x2 = | | e− r2 dx. √2r n 2 Z2 Let p> 0. Then there exist k1 > 0 such that

p 2 iω 2 w r Hϑe t 1 2 np χne− k1 | | p+1 2− . 2 ≤ (r2 r2) 2 2 − 1

From Lemma 4 it then follows that

iω 2 p+4 Hϑe t np χne− k22− t− 2 t (0, 1). (18) 2 ≤ ∀ ∈

H t Using the semigroup property, then χne− ϑ 2 for all t> 0. Also, note that ∈C H eiωt H eiωt χ e− ϑ e− ϑ < 1 t> 0. n ≤ ∀ ∞ ∞

17 Then iω np p+4 Hϑe t χne− k42− r t− 2r t (0, 1),r> 2. r ≤ ∀ ∈

Let ∞ α(n+1) V˜ (x)= 2 χn(x). n=0 X If r and p are such that p +4 p < 1 and > α, (19) 2r r then, for some ε> 0,

iω ∞ p p+4 Hϑe t n(α ) 1+ε V˜ e− k5 2 − r t− 2r k6t− t (0, 1). (20) r ≤ ! ≤ ∀ ∈ n=0 X This confirms (16). 4 Note that the condition (19) is satisfied for 0 α< 2, whenever r> 2 α and p (αr, 2r 4). That is precisely the requirement≤ on α in the hypothesis− above. ∈ − Fix r and p in this range. We now show that A = V˜ is a class r perturbation iω H eiωt PC iω of T = e Hϑ. The operator V˜ e− ϑ has integral kernel V˜ (x)Mϑ(e t,x,y). For all t> 0 fixed,

iω 2 V˜ (x)Mϑ(e t,x,y) dydx< x R y R | | ∞ Z ∈ Z ∈ H eiωt as a consequence of (17) and the deﬁnition of V˜ . Then V˜ e− ϑ 2 and it H eiωt ∈ C is also continuous in for all t > 0. Hence V˜ e− ϑ for all q > 2 also C2 ∈ Cq and it is continuous in the norm of q. This includes q = . Hence for all 2 H eiωt 2 C ∞ f L (R), V˜ e− ϑ f L (R). Thus ∈ ∈ H eiωt 2 D(V˜ ) e− ϑ (L (R)). ⊃ t>0 [ Finally, the fact that 1 H eiωt V˜ e− ϑ dt< 0 r ∞ Z is guaranteed by (20). In order to complete the proof for ω = π ϑ , note that V˜ is invertible | | 2 − | | and that a generic V satisfying (1) with the same α is such that D(V˜ ) = D(V ). Therefore Lemma 3 ensures that V is also a class r perturbation for r in the stated range. PC π Our only additional comment about the case ω < 2 ϑ is that the expo- p+2 | | − | | p+2 nent of t in (18) changes to 2 . This leads to replacing the left of (19) by 2r 2 and this yields r> 2 α . −

18 Remark 2. Here the potential V can be accretive or otherwise. For example V (x)=eix x α where 0 <α< 2 is included in this theorem. | | By combining the above with Theorem 2 it follows that, for ϑ = 0, the non- 6 selfadjoint Schr¨odinger operator Hϑ + V is the generator of a Gibbs semigroup (H +V )τ e− ϑ 1 for all τ ϑ 0 holomorphic in the maximal sector ϑ. Moreover, for∈ Cω π ϑ ∈andS r\{in} the range determined by (15), S | |≤ 2 − | | (H +V )eiωt H eiωt + e− ϑ e− ϑ t 0 . k kr ∼k kr → If ω < π ϑ , this range includes r< 2 and we get from Lemma 5 that | | 2 − | | (H +V )eiωt 1 + e− ϑ t− 2 t 0 . k k2 ∼ → H eiωt From the interpolation inequality in r and the fact that e− ϑ are contraction semigroups, it follows that for r> 2,C

1 π iω iω 2 iω 1 2 t− r ω < ϑ Hϑe t Hϑe t r Hϑe t − r 2 e− r e− 2 e− 2 | | π − | | k k ≤k k k k∞ ∼ t− r ω = ϑ . ( | | 2 − | | Then, 1 π iω O(t r ) ω < ϑ (Hϑ+V )e t − 2 e− r = 2 | | π − | | (21) k k O(t− r ) ω = ϑ ( | | 2 − | | as t 0+. As we shall see next, combining this with (2) leads to asymptotics for the→ eigenvalues of the perturbed operator. Let Spec(H + V )= λ ∞ . ϑ { n}n=1 Denote α = (λ ), n ℜ n i( π ϑ ) iω π β± = e± 2 −| | λ = e λ for ω = ϑ . n ℜ n ℜ n | | 2 − | | iω (Hϑ+V )e t Since e− is compact for all t> 0, it follows that αn, βn± , cf. [9, Theorem 8.2.13]. → ∞

1 Corollary 2. Let V satisfy (1). Then, the resolvent (H + V z)− for ϑ − ∈Cq all q> 1. Moreover, Hϑ + V has an inﬁnite number of distinct eigenvalues and a complete set of root vectors3.

Proof. By adding to Hϑ + V a suﬃciently large constant, without loss of generality we can assume that ϕ(H + V )= 1 and take z = 0. The inverse Laplace ϑ − transform identity (3) for T = Hϑ + V gives

1 ∞ (H +V )t 2 (H + V )− f = e− ϑ fdt f L (R). (22) ϑ ∀ ∈ Z0 3We follows the standard terminology here, meaning that the set of ﬁnite linear combina- tions of all the root vectors has zero as orthogonal complement.

19 From (21) and the assumption on the uniform growth bound, it follows that

∞ (Hϑ+V )t e− qdt 0 k k Z 2 ∞ (Hϑ+V )t (Hϑ+V ) (Hϑ+V )(t 1) e− qdt + e− q e− − dt ≤ 0 k k 2 k k k k∞ Z 2 Z 1 ∞ t k t− q dt + k e− dt< . ≤ 6 7 ∞ Z0 Z2 Then, the integral in (22) is absolutely convergent in and so the associated k·kq operator belongs to q. The second andC last statements are classical. A concyse proof is achieved by means of a direct application of e.g. [22, Corollary 4.10]. Indeed, taking V = 0 in the ﬁrst statement just shown, yields that Hϑ has “order”, in the sense of loc. cit. p.918, any constant less than one. We know that Hϑ is m- γπ sectorial with angle ϑ = 2 for γ < 1 and V is Hϑ-bounded with bound zero. That is “completely subordinate” in the terminology of loc. cit. p.910, so the hypotheses of the mention corollary are satisﬁed.

As we shall see next, lower bounds on the asymptotic behaviour of αn and βn± can be derived from Lidskii’s inequality. Corollary 3. Assume that ϑ =0. Let V satisfy (1). Then there exist constants K > 0 and n N such that 6 0 ∈ 1 α Kn and β± Kn 2 n n . n ≥ n ≥ ∀ ≥ 0 Proof. Recall that

(H +V )eiωt eiωλ t Spec(e− ϑ )= e− k ∞ 0 . { }k=1 ∪{ } 4 Let r> 2 α . For t = rs, −

∞ i i e ωλ s r (H +V )e ωs r 2 + e− k e− ϑ = O(s− ) s 0 . | | ≤k kr → Xk=1 Then ∞ ± ∞ ± β t β s r 2 2 + e− k = (e− k ) = O(s− )= O(t− ) t 0 . → kX=1 kX=1 Assume that the eigenvalues are ordered so that βn± is non-decreasing (with possibly diﬀerent orders for the two cases ). Then ± n ± β±t β±t ∞ β t ne− n = e− n e− k . ≤ Xk=1 Xk=1 Hence there exist a constant k8 > 0 such that

± β t k8 ne− n 0

20 where t0 > 0 is small enough, this for all n N. Take n0 N large enough such 1 1 ∈ ∈ that ±

1 2 ne− k (β±) n n . ≤ 8 n ∀ ≥ 0

This ensures the validity of the claim for βn±. The conclusion for the case of αn is achieved with a similar argument noting that the asymptotic changes to α t 1 + ne− n = O(t− ) for t 0 . → The estimate above is optimal for αn, as it should hold true for V = 0. Since

∞ n1/2t ∞ x1/2t 2 + e− e− dx t− t 0 , ≥ ∼ → n=1 1 X Z 1 we know that the exponent 2 for βn± above is also optimal, given the asymptotic behaviour of the r norm of the semigroup. However, it is not clear that the exponent in the latterC is optimal for potentials satisfying (1). That is, we do not know if the exponent for t in the formula (21) is optimal. From general principles, it follows that the ε-pseudospectrum

Spec (H ) z + seiω : z Num(H ), 0 s ε, ω π ε ϑ ⊂{ ∈ ϑ ≤ ≤ | |≤ } for all ε > 0. In fact, Specε(Hϑ) is known to obey the following more precise enclosures for ﬁxed 1 < q 3 < q < . Write 1 ≤ 2 ∞ R = r + rqeiω : r 0, ω ϑ . q { ≥ | | ≤ | |}

For all ε1 > 0 there exists E1 > 0 such that

(E + R ) Spec (H ), 1 q1 ⊂ ε1 ϑ see [4]. But for all γ, E2 > 0 there exist ε2 > 0 such that

∞ Spec (H ) (E + R ) z C : 2n +1 z γ ε ε , ε ϑ ⊂ 2 q2 ∪ { ∈ | − |≤ } ∀ ≤ 2 n=1 [ see [20]. See also [19, 11, 28]. Then, according to Corollary 3, asymptotically the eigenvalues of Hϑ + V lie way inside Specε(Hϑ) and the distance from 1/2 ∂ Specε(Hϑ) to λn grows (at least like n ) as n . The following result gives an indication of the→ shape ∞ of the pseudospectra of Hϑ + V . It implies that the distance from the real axis to z ∂ Specε(Hϑ + V ) is o( (z)) as z . ∈ ℜ → ∞ Corollary 4. Assume that ϑ =0. Let V satisfy (1). Then 6 iϑ 1 lim (Hϑ + V e± ρ β)− =0 ρ →∞ k − − k for all β R. ∈

21 Proof. By rotating the operator and directly applying Theorem 1, the conclusion follows for all β suﬃciently negative. We use Corollary 3 and a spectral decomposition similar to that in [6, 2.2] to show the property for all β R. Fix γ R. By virtue of Corollary§ 3, there exists N N such that ∈ ∈ ∈

λ ∞ ( ϑ , ϑ )+ γ. { n}n=N+1 ⊂ S −| | | | Let C be a simple Jordan curve such that only λ N are in its interior. Let { n}n=1

1 1 P = (z H V )− dz 2πi − ϑ − ZC be the corresponding Riesz projector. Let

= P [L2(R)] D(H ) and = (I P )[L2(R)]. M1 ⊂ ϑ M2 − The subspace D(H ) is ﬁnite-dimensional, L2(R) = + and M1 ⊂ ϑ M1 M2 1 2 = 0 . Generally there is no orthogonality between 1 and 2. M Let∩M { } M M [Hϑ + V ]j = (Hϑ + V ) ↾ : j D(Hϑ) j Mj M ∩ −→ M denote the corresponding restriction operators. Then

N Spec([H + V ] )= λ and Spec([H + V ] )= λ ∞ . ϑ 1 { n}n=1 ϑ 2 { n}n=N+1 Since are invariant subspaces for the resolvent, then they are also invariant Mj under the action of the C0-semigroup (this is guaranteed from the fact that the latter commutes with the resolvent). The restriction operators are generators of the corresponding Gibbs semigrous on the subspaces, that is [6, Theorem 2.20]

[Hϑ+V ]j τ (Hϑ+V )τ e− =e− ↾ : j j Mj M −→ M for all τ ϑ 0 holomorphic in ϑ. Since the restriction to 1 is bounded, we have ∈ S \{ } S M 1 lim ([Hϑ + V ]1 z)− =0. z k − k | |→∞ i( π ϑ ) According to Theorem 1 applied to T =e± 2 −| | [Hϑ + V ]2, we have

ϑ 1 lim ([Hϑ + V ]2 e± ρ β)− =0 β < γ. ρ →∞ k − − k ∀ Since

1 1 1 (H + V z)− ([H + V ] z)− + ([H + V ] z)− , k ϑ − k≤k ϑ 1 − k k ϑ 2 − k the conclusion indeed follows for all β<γ. We complete the proof by choosing γ arbitrarily large.

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