Jérémy Faupin : Scattering theory for Lindblad operators

and budget rube now about scattering theory for Lindblad operators so there will be no semi-classical analyzes but there will be non self-adjoint operators here is the outline of my talk so first I will try to motivate the study by recalling some notions and some definitions about the Inlet operators and quantum dynamical semigroups then I will state our main results and finally I would try to give you the main ideas of the proofs so let me start with what is a Lindblad operator and what is a quantum dynamical semi group so let me begin with the physical context that we want to study so typically we are considering one quantum particle that interacts with a quantum system which is supposed to be localized so I will call this localized system target and both the particle and the target interact with the rest of the universe which I call the the environment so typically you can think of the quantum particle as being an electron with spin and the target then may be a magnet or another situation could be that the quantum particle is a neutron and then the target is a nucleus so here’s what we want to to study the situation we want to study how you have localized the quantum system the target and then we send a particle to the target we look at dynamics of the particle and we want to look at what happens typically there are two situations that may Hawker the first one is that the particle is going to interact with the target and then it will be scattered of the target and another possibility that may occur for instance if the particle is a neutron and the target is nucleus is that the target may capture the particle okay so the aim of this study is to try to understand this physical situation by looking at scattering theory for suitable operators so now let me go to mathematics the total elbert space of our system is given as a tensor product of three component HP which is the inward space for the particle HT the it work space for the target and H E the input space for the environment so the total input space is written is it’s just the tensor product of the three Gilbert spaces the total Hamiltonian in a very general form is given as the sum of these four terms so the first one this is just the free energy of the particle HP that acts in its Hilbert space the second one is the free energy of the target HT the third one is the free energy of the environment and of course we had an interaction a meltonian h i that represents the interaction between the three components of our system then we want to study the dynamics of the system so of course h total generates unitary dynamics so i will work in the usual Schrodinger picture the states of the world system are represented by density matrices meaning trade class operators of this notation j-1 stands for the trade class operators on the total is buttface each total and states of course are positive enough trace 1 then the full evolution in the Schrodinger picture is given as usual by this expression if the initially the full system is in the state row then after times T the state in D is in this state so you just conjugate row with the unitary evolution

it was a minus ith total on the left and it was a plus ith total on the right now the problem is that in concrete situation of course the degrees of freedom associated to the target and the environment may be very complicated so that it is all placed to understand the full dynamics of the system so if you are only interested in the dynamics of the particle one way is to define an effective dynamics or reduced dynamics for the particle which is defined as follows so suppose that there is some fixed reference state for the target and the environment which I write here to its row T for target if our environment and r is for the reference state so you can think typically as the state as being the projection onto the ground state of the system composite of the target and the environment assuming it exists then the radius dynamics for the particle is given by this expression so if Rho P is the initial state of the particle so trace cross operator on the ill Burt space for the particle then Rho P of T is given by the partial trace over the degrees of freedom of the target and the environment and then you look at the full evolution with initial state given by Rho P times the fixed reference state so of course the effective dynamics depends on this choice of reference state you look at the full evolution and then you take the partial trace so these defined dynamics and effective dynamic for the particle or Regus dynamics and okay let’s call it dynamical not an effective dynamics of the particle which has the following properties so of course what I wrote here Rho P of T this expression does not define a semigroup right because you take the partial trace or RT and E but using some approximation for instance with coupling limit in some cases it’s possible to to obtain from this expression using an approximation a map dynamic dynamical map lambda T such that for any T lambda T is a bounded map on the space of states for the particle and none that is strongly continuous one parameter semi group which is traced preserving so the traits of lambda T Rho is equal to the trace of Rho for any row and it’s also positive for annual if Rho is positive lambda T Rho is still positive so if you want to understand the dynamics of the particle a natural question of course is to try to write down the generator of this dynamical map we know that it exists right because you have a strongly continuous one parameter of semigroup on this banner space so kozakov ski and in garden kozakowski obtained Sasori and sufficient condition for an operator L to be the generator of such a dynamical map so consider a complex separable Hilbert space hm then an operator L on this space so this is the space of trace class operators that are supposed in addition to be self-adjoint so this is a real bonus basement okay that not does not change muffed much the problem so an operator l is the generator of the strongly continuous trace preserving and positive 1 parameter semigroup on this space if and only if its domain is dance identity minus l is surjective L is dissipative dissipative meaning this inequality for here the sign of Rho is defined things two functional calculus because Rho is self adjoint or it’s well defined and the trace of L Rho is equal to zero so this theorem gives necessary and sufficient condition for an operator L to be the generator of a dynamical map in the sense before but it’s form of course is not explicit noeleen blood you know in a very famous paper in 76 had the following heidi to replace the

assumption that the dynamical map is positive by a stronger assumption that it is completely positive so following lynn blood i will call a quantum dynamical sonic group the map lambda T such that for any T in 0 infinity so lambda T is a bounded map on the space of race class operators such that it is strongly continuous one parameter semigroup it is Trace reserving so it’s the same but now we assume that it is completely positive which means that for any n lambda T times the identity on the space of race class operator on each time CN is positive so it is a stronger assumption but you can use some physical argument to say that it is a natural physical assumption and with this assumption Lindblad proved derived the general form of a linear operator so here is the CRM so H is still a complex and separable Hilbert space then the generator of the norm continuous quantum dynamical semi truck so of course this is a strong assumption here you assume that it is not only strongly continuous it is non continuous then we know that the generator has to be bounded but with this assumption then you can prove all in blood proof that the generator is of the following form so maybe I should write it on the blackboard because I will use this expression several times so L of Rho so any trade class operator is given by the commutator H 0 Rho minus Rho H 0 and then you have minus I divided by 2 so mother J of sISTAR Jessie J Rho plus Rho sISTAR Jessie J and then plus I assume over J of CJ row City FC star J so since L is since the yes I mean it’s it’s a composition right you have a map on the space of three class operators everything is composition here yes so since we have considered norm continuous dynamical similar the generator is bounded so H zero heat is self-adjoint and bounded and all the CJS are bounded in fact in addition the sum over J of C star Jessie J defines the bounded operator so this expression makes sense so you see that to define the generator I use the – I here so that formally I can write sum that is e to the minus i TL this is just a convention to have e to the minus i TL which is close to the usual e to the minus I th which is a unitary dynamics but this is just a convention so this erm is not known in general if you only assume that you have a strongly continuous quantum dynamical semigroup but still we will define a Lin blood operator as being an operator of this form but with h0 on seach j the operator that may be unbounded so if you do that Davis has proven that it is well defined in some dense sense and it indeed generates a quantum dynamical semi-group so more precisely I take h0 to be a self adjoint operator now and bound it CJ to simplify assume that they are bounded and that there are only a finite number of such operators then the operator L with this form it’s an bounded operator yet but you can define it on this domain which is the natural domain on which you can define it I mean ro should preserve the domain of H 0 and the commutator between H 0 on row should

extend to an element trade class operator then you can prove that this generates a quantum dynamical swimming group meaning you strongly continuous one parameter a semigroup which is trade preserving and completely positive now the main assumption of our study now I’m going back to the physical situation we want to consider is that the register AMEX of the particle which interacts with the target and the environment is given by such a Lindblad operator so more Prasad is given by a quantum dynamical semigroup which is associated with such a Lindblad operator ok another simplification I assume that there is only one CJ no so L is the commutator between H 0 and Rho L of Rho minus this expression so it means that if the particle is initially in a state row rave class operator which is positive not trace 1 then after times T the particle will be in the state row of T equal e to the minus i TL Rho in other words it solve this equation which is sometimes called the quantum master equation or you can also find it under the name quantum mechanical fokker-planck equation ok so this will be the starting point of our study and now we want to study scattering theory for such operators so that there are not much works in the mathematic literature about this subject at the beginning of the H is Davis then Aliki and Alec and figure will did some work but the subject so basically what do you want to prove we assume that the free dynamics is generated by this part h0 Rho minus Rho H 0 so this generates a group which is an isometry and and on the space of third class operator so typically we would like to prove something some statement like that suppose that the interaction between the particle and the rest of the universe is not too strong then we’d like to prove that for any initial state role which is not bound state so you have to first understand what is bound state in the setting which is not obvious then there exist a scattering state Rho Plus such that the full dynamics applied to Rho as time T goes to infinity is close to a free dynamics applied to this scattering state and this equality should also for the norm of Trey’s class operators okay so let me emphasize that this is not a group it is only a semi group and it’s not even an isometry okay but this this statement is sometimes called weak asymptotic completeness and it is in fact equivalent to proving the existence of a wave operator so more generally we’d like to study the following two waves operator so the first one is the usual outgoing wave operator defined by this expression so T must go to plus infinity here for T going to – I think minus infinity it may be not well defined of course because it was a – ideal then is not a contraction some group and Omega – is this operator so since e to the minus i TL 0 the three dynamics is isometric proving the existence of this Omega – is equivalent to proving this weak asymptotic completeness as usual and then in this context assuming that everything is well-defined the scattering operate is given by the composition of Omega minus and Omega plus okay so let me know state our main results so I just recalled here what we are studying so L of Rho this is the same expression as in the blackboard but with only one of the operator CJ which is supposed to be bounded and H zero is self-adjoint okay so you can rewrite the expression of

airflow if you want by this H Rho minus Rho H Star Plus this term with edge given by H zero minus I divided by two sister see when particular H is a dissipative operator acting on the hilbert space and one of the main ingredient for studying scattering theory fall in blood operator is to study scattering theory for dissipative operators in Ilford space you will see why in in a moment so the first crm concerns the existence of Omega plus of ll0 which is given by this strong limit so in in general it’s more easy to prove existence of this operator than the other one right because you have the three dynamics on the right so you start by applying the three dynamics which may be explicit in examples I will give you examples in a moment so this existence is more easy okay so the first result which was proven by Davis in a tea in a slightly different context but it’s not very difficult I mean you suppose the directive students sub-state subsidy in the elbert space such that for any you in this dense subset you have this integral converges so this is the usual assumption that you do when you want to apply cooks method to study the scattering theory for h0 on edge okay if you do this assumption then it’s two lines to prove that the wave operator associated with H 0 and H exist the statement of the sir M is that with this assumption you can also prove that the scattering operator associated with L and L 0 exists so it’s not completely trivial but still it’s not very difficult basically you have to apply cooks method and used at some points a sickly CT of the trace okay now let me go to the more difficult point proving weak asymptotic completeness or if you prefer the existence of this operator Omega minus of L 0 L which is by definition equal to this strong limit so one point of our work was to recognize that very convenient assumption in this setting to study scattering theory is to consider kaito smoothness estimates as assumptions which are well-known in concrete examples of to derive such assumptions so suppose the direct is the positive constants C 0 which is less than 2 so the interaction should not be too strong in this sense such that we have this estimates so it means that C is relatively smooth with respect to H 0 in the sense of cattle with a constant less than 2 so L 0 is the free dynamics as before then with only this assumption you can prove that the two waves operator exists on on the space of trait class operator on the wall space G 1 of H okay so here again we only need an assumption that the yield at space level okay this is an assumption that concern only see an H zero and with this assumption you can prove this statement and if you strengthen a little bit the constraint of the on the constant C zero so if you assume that it is less than 2 minus square root of 2 the statement is becomes more precise now the two waves operator are invertible as maps as bounded maps on the space of trade class operators and the two operators l and l 0 are similar so let me make a few comments about this assumption in red so as I mentioned before assuming the direct positive constant such that this holds following the well-known paper by Kato it corresponds to centers that C is H 0 smooth and it is well known that it is equivalent to assuming that the

imaginary part of the free result and H 0 is uniformly bounded for said in C for that outside the real axis if you put the weight C on sister on the left and on the right now one very useful observation in in the context of dissipative operators is that C is always relatively smooth with respect to H so in this respect it is more it is easier than scattering theory for self-adjoint operators I mean in in several examples you have to work in order to prove such an estimate if H for instance is H 0 plus V where V is a perturbation of H 0 it’s not obvious and you have to to prove some resolve and estimates for instance in the context of dissipative operator of this is obvious I mean you just have to differentiate something it’s one line to prove that this this old so this assumption this is can this property is very convenient but you see that here we have constant one in front of normal u square and in fact we can state the same theorem as before but with a slightly different condition okay so I suppose that still sees relatively smooth with respect to H zero but not with a small constant with any constant so if there is a positive constant that I called C tilde 0 strictly less than 1 such that this estimates holds so see relatively smooth with respect to H maybe I should have written what H is working right here so H this is the dissipative operator in the inward space edges 0 minus I divided by 2 sisters so we know that the estimate is always true with since tilde 0 equal to 1 but if we assume that it is true with C tilde 0 strictly less than 1 then we have the same results the two wave operators exist on the wall space J 1 of H and if in addition we assume that G tilde 0 is less strictly less than 1 else then the two wave operators are invertible and the operators L and L 0 are similar no I did not change the sign yes okay yes it is possible to define the inverse semi gross but they are not contraction semigroups of course but it is possible to define yes it’s possible to define e to the minus i TL with t- ok no generally it’s not possible but here I have I mean L is L 0 which is a group plus a bounded operator so of course you can define yes yes and in fact this is the dairy market here assuming that this old these estimates all’s we see tilde 0 strictly less than 1 it is in fact equivalent to assuming that it was ith so here I changed the sign e to the i th is not contraction so we grow the priori it could blow it could blow up on some initial state but if I assume that she killed a co a strictly less than 1 it’s equivalent to assuming that it cannot blow up it is a uniformly bounded semi group in L of H okay ok now let me go to some examples so I assume that the particle that I am considering is non relativistic particle with elbert space l2 of our free times H where H is the ill Botsford internal degrees of freedom so to simplify that miss suppose that it’s a finite dimensional hilbert space and the effective dynamics of the particle is supposed to be generated by such an expression so the interaction is the

same as before and l0 the free dynamics is supposed to be generated by minus laplacian plus a matrix which corresponds to the internal degrees of freedom so for instance if you have an electron with spin you should think of this operator I mean this elbow space H is just 1/2 degrees of freedom for the spin of the electron I know a problem in this framework is that the explicit form of the effective dynamics so the explicit form of the Lindblad operator may be very complicated in general if you have a complicated target also you have to deal with the degrees of freedom of the environment so he goes derivation of the effective dynamics is in general an open problem but in some cases you can define it using a coupling limit and this is also famous paper by Davis where he obtained the form of the linear operator for a finite dimensional system coupled to a free heat bath but talk in general it’s it’s an open problem but using some heuristic argument from physics for instance if you assume that the interaction of the particle with the environment induces the coherence in position space then it seems reasonable to assume that the interactions see is of this form G of x times an operator and the finite dimensional hilbert space so you can think of s if you want as being the spin operator if you have an electron or it can be the identity it doesn’t matter but G H should be a function with sufficiently fast decay which is related to the assumption that the target is localized in space so with this form of see it’s easy to obtain Kohala reads of our theorem for this example so here’s the color if you assume that the composition of C with multiplication by the norm of X is less than 2 pi minus 1 alpha then the two wave operator exists on the full space of trade class operators and if it is less than 2 minus square root of 2 P to the minus 1 alpha then in addition there are invertible and the two operators L and L 0 of similar so this is an obvious consequence of our sir and together with this Kato smoothness estimates which has been proven by cattle and and cattle yahshimabet I mentioned Simon here because he showed that the optimal constant for this estimate is equal to PI if I use this together with our serum I get this Kowal re and you can do the same thing if you want by using other cattle Hestia mates for instance you can use this well-known cattle smoothing I mean showing a smoothing effect which has been again proven by different authors in constant time so and Bernards inclined Amin and again Simon has proven that the optimal constant is equal to P divided by 2 or another example is to use to make the assumption that you have a whole nick potential for C meaning that this norm is bounded and then it’s easy to prove such an estimate so the point here is that you can get explicit condition in examples with our CRM which are possibly easy to verify in a concrete setting okay no for the moment what I described do not allow for understanding the the phenomenon of capture I mentioned at the beginning that it may be possible that the particle is sent to the target and then the target captured the particle so if you want to understand it you have to do something different to define in particular have a modified wave operator so how do we do that no I assume that the interaction between the particle and the rest of the universe do not only produce this operator CJ in the effective dynamics it’s it’s also of the consequence of adding self-adjoint part

to the free dynamics given by this operator of V here so for to simplify assume that V is relatively compact with respect to H 0 then now my Lindblad operator is the same as before but I have replaced H 0 by H 0 plus V it is still a well-defined closed operator on the space of trace class operators so you can write the same expression as before but now H is not only H 0 minus something which is as a positive or negative imaginary part you also have this perturbation V so I will write H V for the operator H 0 plus V now I have to make some assumptions on V so here are the assumptions so again in a concrete setting its known out to verify such an ass so you should still think of ages evil as being – laplacian and V as being a potential if you want then I assume that the spectrum of h0 is purely absolutely continuous the singular continued spectrum of h0 plus V is empty and HT is supposed to have at most finitely many eigenvalues with finite multiplicities then the two waves operator the well-known operators w + on – associated to hv and n G H 0 and H 0 an edge V so of course you have to project onto the absolutely continuous part of HP they are supposed to exist and to be a sum asymptotes are complete in the sense that the range of the incoming and outgoing wave operators is equal to the absolutely continuous spectral subspace of H V which coincide with your thuggin all complement of the pure point spectral subspace of HP and the two order wave operator have a range equal to H okay so this is an assumption that we know out verified in concrete example I will give you conditions in the case of Schillinger operators okay no before defining the modified wave operator following Davis I have to introduce a few subspace is associated to H and H star the dissipative operator H and H star so I call HB of H the closure of the vector space generated by the set of eigen vectors with real eigen values of H so in fact we know that if u is an eigen vector of H associated with a real eigen value it will also be an eigen vector of H 0 plus V right its I mean it has to be in the kernel of C and I also define HD of H and end LG of H star as the set of vectors in the but space such that the norm of e to the minus ithu goes to zero and the same for H star but then you have to change the sign of time then the modified wave operator considered by Davis is the following I let P be the orthogonal projection with Colonel H P of h plus h D of H so we should think in this context as HB of h plus h g of h as the set of bounded states okay so the projection projects outside the set of bounded set know the modified operator is given by this expression so it’s the same as before but you project on the left and on the right with this projection this orthogonal projection on your thermal complement of the set of bound States now here is the theorem that we proved now I assume that now C is relatively smooth with respect to H V so H V it’s H 0 plus V if the self adjoint operator but with the perturbation V we assume that we have certain estimates so of course here we have to project on the absolutely continued spectral self subspace if you have a bound state for H V then this cannot all unless the eigenstate is in the kernel of C but in

general you have to project onto the absolutely continuous part and we assume that we have such an estimate with C V less than 2 so again we know out verified such an assumption and in the case of running operators for instance know L 0 is the the free dynamic so it’s still the same so she did with H 0 then the modified way operator defining the previous slides exists on the world space and if you take an initial stage role so today’s class operator which is positive and of trace one then you have that this quantity the trace of Omega tilde – applied to Rho is between 0 and 1 and interpretation is that this number gives the probability that the particle initially in the state row eventually escapes from the target so one minus this quantity gives the probability that the particle will be captured by the target okay so albert example so i consider again and under here atavistic particle as an example so the same as before but now I have minus appellation plus the internal meltonian plus a perturbation given by V of X so L 0 is the free dynamics associated with – laplacian plus the internal a meltonian and I have this perturbation means conjugate or is V of V of X plus disappear enters so they are supposed to be real valued of course so what about the conditions on V that we can make him wear for instance you can suppose that V of X decays like minus 2 minus epsilon for some positive epsilon and you add the assumption that 0 is neither an eigenvalue no resonance of HV then using results that have been proven by Bernard Klein oh man I mean with this assumption but before by Rochelle Jensen and cattle we know that there exists a positive constant C 1 which depends on V of course such that you are such a cattle smoothness estimate with weight equal to 1 / X to the power 1 plus Epsilon so this also in you and then again I can apply this estimate and use our abstract theorem to deduce that under the previous conditions on V and if you assume that C is sufficiently localized meaning that C composed with X x plus 1 plus epsilon is bounded with norm less than to c1 minus 1 where c1 is the constant in the previous cat to smoothness estimates then we can deduce that the modified wave operator Omega tilde minus exists on the wall space of trace class operator ok now let me finish the talk with giving you some ideas the main ideas of the proof ok so this was the first result using this this cook assumption to apply cooks method so I already mentioned it I I just recalled here the form of the Lindblad operator that we are considering then if we make this assumption we can prove that this operator exists on j1 of H so the idea of the proof follows a paper by Davis you apply cooks method and then you use secrecy T of the trace so let me go to the more difficult part where you want to prove existence of Omega minus of L 0 and L so you start by applying the four dynamics and of course it’s more complicated so we say that with this assumption the wave operator exists so here is the idea of the proof let me define L H of role as being equal to H Rho minus Rho H star where remember that h h 0 minus this imaginary term then we decompose we want to prove that this quantity as exists as a strong limit on the worldspace of thread class

operator so we decompose this term by adding e to the minus I TL h with this generator and then of course we have the difference so let’s first consider the first term so it’s given by this expression but you can rewrite that as a composition in this way right and then if we know something about the scattering theory for dissipative operator meaning if we know that this term strongly converge on the ill both space level then we can deduce that this will go to omega minus rho omega minus star where omega minus is the wave operator associated with H 0 and H assuming we can prove it exists but with this assumption in fact we can prove that it indeed exists because here in fact I do not need the assumption that C 0 is less than 2 for any positive C 0 D it would still be true because we have this Gatos smoothness estimate and we know that C is always relatively smooth with respect to H ok this was the estimate that I showed you before so we know in fact that Omega that W sorry W minus exists on H so this is fine now the more difficult part is to prove the existence of this term as a strong limit so what can you do here well you can write the difference of the two dynamics as an integral so you will get this integral from 0 to T so there’s something maybe a little bit complicated but you can see that you have e to the I T minus l t minus s l 0 e to the minus i t minus s LH so it’s the same as before so formally if I let T go to infinity then I will get the integral from 0 to infinity and I will have this term Omega minus C and etc my gamma in star so all the problem I mean formally it’s it’s easy but the world problem is to Jase justify this limit using some version of the dominated convergence theorem and in order to justify this limit you have to study scattering theory for dissipative operators in Ilford space okay so you have in particular to understand this operator of w- that appears but also they’re useful to understand the usual outgoing wave operator w+ so scattering theory for dissipative operators has been studied by many authors in in many context but in these abstract settings let me mention these papers by martin Mochizuki davis Simon and more recently kadowaki okay so I will come back I will come back in in two minutes but this this part which is really the key the key point of our proof so let me just before mention this result remember if I assume that in these estimates the constant is strictly less than 2 minus square root of 2 then the result is more precise in the sense that the two wave operators are invertible and the two operators l and l g who are similar so how do you prove that when you can again introduce the successor re dynamics and then the point is to estimate the dies and series that you can you can obtain from expanding using UML’s formula and iterating and then you can estimate the isin service with this assumption so to conclude let me give a few words about scattering theory for dissipative operators in Albert’s face in this abstract setting okay so the first result which I mean it’s not very difficult but we have we had to prove this because we didn’t find it in the literature under this assumption so assume here that C is relatively smooth with respect to H zero but with any constant C zero then with H given by this expression you can prove that the two wave operators exist and moreover that W plus of h h 0 is injective and w- as a dense range of

course the problem here is that the two waves operator are not isometric right because e to the minus I th is a swimming group of contraction so it’s not an isometry and it could happen that the range of W plus is not closed and in fact this is the main problem that we have to deal with we had to find conditions such that the range of this operator is closed which may not be true in general okay but then it’s fine if C 0 is less than 2 then the two wave operators are bijective and in fact the this last statement which activity in the case where the constant C 0 is less than than 2 is a result which is close to what cutter did in in 66 in in fact in a more general context but in our setting it’s it’s easier and also well this is a small point but we do not need the assumption that the resolve n’t is uniformly bounded okay you remember this assumption is equivalent to saying that the imaginary part of the resultant h0 minus dead is uniformly bounded and in its paper kato used in addition the assumption that the wall resolved n’t is uniformly bounded in fact you do not need that you can do a proof completely time-dependent you do not need to use the stationary arguments here okay so the last result is the following so this is for the case of capture where I added this part V to H 0 and then I had this assumption the scatter smoothness inequality of C with respect to edge V but if I project on the absolutely continuous part then we can prove that this operator is injective and in this case its range is indeed close because we can prove that it is equal to Y orthogonal complement of HB of h plus h d of h now so HB of h it was the space of eigenvectors associated with real eigenvalues and HD of H star is the state such that e to the i th star u goes to 0 so if you take the orthogonal complement you get the range of W so in particular of course it is closed so let me remark that in the case of dissipative Schrodinger operator with small imaginary part there is a recent result by 1 on zoo where they use some global limiting absorption principle with some weights on the left and on the right one point here is that we do not need such global limiting absorption principle I mean because again because of this fact that C is always relatively smooth with respect to H and O course ok no of course one of the main improvement of such result would be to relax the assumption that you have constant width which is not too large it may be possible to do that in some concrete examples by marking that you have in fact this equality the range of w+ is also the set of hue such that the inverse semigroup stays uniformly bounded so the point would be to prove that this is closed and there is a paper by Golder where where it considers reading the alternator’s with complex potential but there are some implicit condition that the operator does not have real resonance on the real axis and also other condition but in this case clearly this is indeed verified so it could be possible to prove such a thing without smallness condition okay well I thank you for your attention so questions forces you my question with a local contribution

you can think of using some propagation is teammate in order to get the sum integer by eighteen time a large sign Millia showed yes it is possible actually to write a density map each version of propagation estimates have you thought about this no no but yes I mean this is certainly what we plan to do for a future future work if you want to go beyond such a result you have to do some concrete analysis in you have to fix some concrete model that you want to study for instance based on shredding evaporator but know for the moment this was really an abstract setting because it could be very it may be possible and we could also use some version of more theory for this apart evaporator as a junior junior Hawaii did or Galena and beside there are some darshan that that we could use yes from this is something we would like to to try to do another question oh it’s not an atom so it’s clear that comparing the two is the most difficult oh my commands yes yes usually dissipated when proving this one one imposed some kind of Rajat Turin for the dissipative semigroup no you impose some condition about the group exponent minus ID is zero so if you try to prove this on because so you start by another ketosis of course this is not so easy but usually this could be obtained it having some assumption that implies that you have registering the Rajat theorem is just the same so that you change h 0 by H and C by is another thing this is my question what is the question if I if I the question is if you could prove the existence of Omega – yes we’ve called passing by this argument given constant zero said well my argument would impose some condition about the integral what is involved that’s the rate of semigroup Raja Terry know that that’s possible that I don’t know we didn’t try it that’s possible that it works yes but I don’t know we would have to look at it precisely and as our quick comment nothing thank you again