 I would like to thank the organizers very much for being invited. It's a great pleasure for me, in particular in this place which has in fact encouraged my early efforts in the subject many times. I've lectured here around 1980 on the beginnings of the Quartet Oscillator, although I did not belong to this institute. They invited me several times and I'm very grateful for that as well. Today I'm going to speak on a highly speculative subject and I prefer to warn you that there's very little solid results and my goal is in particular to try to awaken topic which I feel has been asleep for over 30 years and I just would not like it to be asleep for 100 years like as in the fairy tales. So if I could if I could awake the interest of researchers on the problems and what to do that, that would be my goal. So I will be mostly telling you a story. It's not going to be solid mathematics, it's going to be speculations leading to conjectures and it will be done on a rather personal note. I would also like to add to finish this general blurb that I've in fact I've been disconnected from resurgence for quite a long time so if you feel that I'm not saying I'm saying incorrect or incomplete things I very much welcome your feedback during the talk. Okay so now this is completely general and I can skip on this very quickly. In quantum mechanics the fundamental parameter is Planck's constant and quantum resurgence is the study of resurgence with respect to the inverse Planck's constant which will be called x in this talk as an asymptotic large parameter complex in general although of course it's a constant in real life. So this is the stationary Schrodinger equation which will be the only focus of our interest here. So we study mostly spectral functions. In this talk a spectral function designates symmetric functions of the eigenvalues and just again to fix ideas I am taking a confining potential which has only positive say positive discrete eigenvalues accumulating to plus infinity like the harmonic or the aquatic oscillator for instance in one or several dimensions. So spectral functions are symmetric functions of the spectra of the eigenvalues e k and they naturally they will depend on Planck's constant because the problem is parameterized by it and the fact is that quite a number of spectral functions can admit asymptotic expansions of this form here that is a formal series in inverse powers of Planck's constant which do not necessarily have to be integer but in fact they will be in practice and times exponentials exponentially small quantities and the known fact is well in cases that have been treated is that the Borel transform of these such functions can be can have a non-zero radius of convergence and can sometimes be endlessly analytically continued to a Riemann surface which with only isolated singularities and then you recover you recover the original functions by Laplace resumption. Now quite often the singularities of the Borel transform can be located and sometimes the germs can be expanded using classical transport equations because the semi-classical correspondence principle in mathematical terms says that singularities are propagated of the solutions. In some cases now that's the really interesting thing the various the germ of various singularities can be explicitly interrelated and that's the resurgence of bridge equations which Iqal has introduced in the 80s. Now after later it happened in 1D that the resurgent description could be translated back into the original space into functional equations which happened to be rich enough so that in fact the Sturm-Leville problems say in one because we're in one dimensional can be quantum integrated but and this is the case now for polynomial potensions in one dimension but this is only in one dimension and in fact the techniques which have been used in quantum resurgence up to now are very much ordinary differential equation techniques pushed to their limits somehow but the real word is not one-dimensional you still have however propagation some singularity propagation principles like the Yegorov theorem for instance in but it's well known that ordinary differential equation techniques break down for the stationary Schrodinger equation in higher dimension than one so we are really stuck and we really think of genuinely multi-dimensional problems problems which are not separable not integrable say real partial differential equation okay so higher dimensional quantum mechanics is a puzzle if you want to extend the what is known by in 1D first of all the one-dimensional treatment of Schrodinger so this is the the sort of black hole here which we don't know what how to attack it so we are we are going to a first survey four possible ways of doing it first of all the one as I alluded to the one-dimensional treatment of Schrodinger equation is a very much ordinary differential equation technique which is called the WKB method in fact and this is well known that it's already inapplicable already in the real domain sorry I have I forgot to say one thing before I go here so this is a sort of a table where we will register our progress our progress or understanding what which we can do I separate I have a special line for real analysis and complex analysis you you may of course we understand that resurgence is about is very much a complex analytic problem however the first thing to find in resurgence is isolated singularities somehow and the fact is that in many problems there are more results known mathematical results in particular under using real analysis than complex analysis so there we have to take a stage where we discuss real problems and indeed if for instance the most striking thing is that if in order to have isolated singularities in the complex domain it is necessary that already on the real line you have isolated singularities otherwise you are stuck so real analysis has plays a role in the in this conjectural discussion okay so that's one thing it's we cannot apply even the real WKB method cannot will is known not to work in higher dimension for deep topological reasons now another an area where some multi-dimensional results are known in higher the dimension is the integrals like professor Concevich discussed yesterday if you you can write a multi-dimensional integral with a phase function with with what with one over h bar in the exponent and this can be handled by the saddle point method and there is an exact saddle point method which is to largely understood in a higher dimension however it is irrelevant to the Schrodinger equation unless you can effectively develop the method for d equal infinity and well I was very excited I am very excited by professor Concevich's talk because this is a possibility however I'm not aware of any concrete full results that can be obtained currently for the Schrodinger equation again I'm happy to be corrected if I'm wrong there are another possible approach is to use d-dimensional singularity analysis and there the result is there is a limited rigorous result but very important which is the Poisson formula on manifolds and this unfortunately as far as I know again is only known on the real domain for real singularities and for homogeneous operators in a certain sense and finally the the the method that will help us the most will be a singularity analysis using the Balian block approach which I will explain later which can begin to describe a Riemann surface of like we want but it doesn't again as far as I know it is not computationally effective that to the point that it can this fully describe the solution of the problem that you want okay so now we are going I'm going to do a crash survey of d equal one again not in the usual with not using the usual approach because it will not work in one d in higher dimension so I will focus on a type of spectral function which I call theta function because that's the one that has a some degree of generalization to higher dimension so once spectral function for confining potential is the this sum here which I call the trace e to the minus tau h hat or what I should have written t here actually but never mind and a function like this and for the type of spectrum that I've described before this function will be holomorphic trivially sort of holomorphic on the real on the real half plane the real tau positive half plane sorry but this is sort of sort of trivial analyticity which you cannot do very much with for the harmonic potential the spectrum is linear it's it it's the odd integers here the part this partition function is of course known in closed form and there is a miracle it's a meromorphic function so first of all it has an endless analytical continuation and moreover it has only poles and that's this identity here is in fact the generating form of the Poisson summation formula so here is a rough sketch of the singular structure of the of the of the partition function you have simple poles and complete meromorphic structure and that's the Poisson summation formula so in a way this is trivial and immediately settled now the next thing we try we can try is a unharmonic homogeneous potential then the spectrum grows faster than linearly and it therefore it has a vanishing density asymptotically now there is a result of in the thesis of somebody called Vladimir Bernstein in 1930 who showed that if you write this function for a sequence e k such that its asymptotic density is less than delta density asymptotic i'm just saying it in in in rough words then on any sub interval of the imaginary axis of length 2 pi delta there is a singularity of the partition function and the fact that so the fact that the asymptotic density vanishes implies that the singularities are dense on the imaginary axis and therefore this is a natural boundary so you have trivial analyticity but the natural boundary so it's finished you cannot do anything by the way when i'm saying real here i of course i've chosen to rotate time in a sense so for me this can also be considered as a real axis because it's a it's a real subset of the in a way of the imaginary play of course and the most trivial known example is the Jacobi theta function which is obtained for e k equal k k squared and there you it's well known that there's a natural boundary but this is also true here okay so if the problem is the asymptotic vanishing density the trick is one trick is to replace the original operator h hat Schrodinger operator by a power this is a non integer power of the operator and of course the spectrum is the is this and therefore you get a new a new function which has this form then here comes so we are in this sector here the real w k b method to find asymptotically the solutions and the spectrum of this problem and the output for the spectrum is the bosom failed condition spectral condition quantization often called quantization rule spectral formula say i will not i mean i suppose you have all heard about this method i will not explain how it works but the fact is that if you consider this this is the classical action at energy e and the problem is homogeneous so it scales that's is an algebraic help here so you can you can switch to a single variable which we call x which is a combination of inverse Planck's constant and this strange power of the energy then the fact is that there is a complete quantization formula there is a asymptotic formal asymptotic series which begins by with x and with corrections such that the x k which is the value of x for e k the x k are given by sampling this function at half integers that's the statement of the bosom failed condition you then you can invert this you can invert this into an asymptotic expansion of the form x k is k plus a half plus something over k plus a half plus a two over i mean you have a power descend x k is approximately k plus a half but and there are asymptotic corrections i will call this asymptotic perturbation the so you are close to the harmonic spectrum in a way up to a constant factor but you have asymptotic perturbations now if you write some of e to the minus tau x k this will begin with sum of e to the minus tau k plus a half which is the linear version before here but the effect of the corrections can be understood as applying a series in in powers of inverse differentiation or integration so some you apply some series of this form to the original function and of course applying this to a function which has poles will generate logarithms and then tau times log tau and so on so you will get ramified something at the in the end you you expect to obtain some function where the poles have been replaced by singular logarithmic singularities so here you will have at every singularity expect to have some formal formally holomorphic germ times log of at this origin okay so to understand this a little better we introduce the these two spectral functions the staircase function this data actually unfortunate notation i realized this is the heavy side step function here and it's shouldn't be confused with the other theta but it's standard notation also so theta in terms of the function f of x and this you write again using the Poisson summation formula it's f of x plus sum of exponentials of f of x like this this is the distribution Poisson formula in the variable f and now the next trick is to separate f into the divergent part of asymptotically diverges which is x and you exponentiate the remainder so they call this phi of x then this rewrites as x plus log phi plus powers of phi and phi phi is sorry is an asymptotic power series because this exponentiates well okay now you realize that this spectral staircase is simply the Laplace transform of the theta function of this data function where now again i've replaced as i announced before i replaced the original e k by their powers by this power and so we have this for this formula here translates in the borel plane of the function of the big theta function as in this form of structure where these means just the way the the residues or the ways the leading ways that you have on these singularities so each so this remind you this was the harmonic with poles and the fact that we applied asymptotic perturbations made it generic and the generic result of course is is not meromorphic it's ramified these cuts here are just a graphical way of alluding to the fact that we have singularities of the form some for instance at the origin you have a singularity of the form a n tau n log of tau plus zero to the n sorry something like this so the fact that you this is a distribution of course in the standard notation so we are not really in the complex plane we are in an infinitesimal vicinity of the real axis so this is still a real singularity analysis but i drew very very small cuts we should which you should think of as infinitesimal cuts it's just a graphical trick so the general theta will have poles and logarithmic singularities which are encoded in this formula and when you decode the formula sorry when you decode the formula what you see that in fact at the origin the singularity is the Borel transform of log phi remember phi of x was exponential of 2 pi i B1 over x plus so and at here you have a phi phi squared phi cube these are here so these singularities are strongly interrelated and this is a form of resurgence which should have been could have been found 100 years ago because the constraint is that these should sum up to a staircase function and that's that means that the singularities at the various levels are strongly interlocked now another remark physics is to remark is that the bosom of held quantization condition involves the action around one traversal of the periodic orbit of the problem at energy e this is a contour integral and this is the classical trajectory this is one traversal and it gives the essentially the classical action x the so you can and this x is essentially also the amount by which you have to you have translated to come here to come up here and likewise in the enter in the wkb interpretation of the bosom effect formula you can think of this as the contribution of the repeated of the trajectory repeated twice as three times so you can you can think of these as traces of the periodic orbit repetitions which give you the locations of the singularities of this theta of tau now so that was the real wkb method then uh in fact uh the real wkb method has a complex extension and I've contributed to show that this complex wkb method can be made exact in one for say polynomial potentials and when you do that you break the barrier of this oops you break the barrier of the real axis and you can because thanks to the fact that on the real axis the singularities were isolated you can go beyond the complex wkb method tells you that in that picture that was true on the on this real axis somehow carries over into the complex world and here are see the here is the diagram of the singular structure of this function for the quartic problem where it's essentially the only case where it can be read pretty easily we on a diagram and so you have you do have the real orbits here but now you have branch cuts extending to infinity the function is endlessly analytically continuable and you find new singularities in the complex range where and these correspond precisely to complex periodic orbits of this problem which are the the classical orbits of q4 are elliptic Jacobi elliptic functions and the generating orbits in fact are the shortest orbits of the problem are the are not in the real but they are these two here this one and its complex conjugate so and the algebra that I described you before that here you have the log of a phi here phi phi squared phi cube it goes over into the complex world you have a function analogous to phi but corresponding to each you have a phi one corresponding to this orbit a phi two corresponding to this orbit here and each time you go up this in this direction you gain a power of phi one in the resurgence and each time you advance on this direction you gain a power of phi two when I say power I should have said of course the powers are in the multiplicative representation in the boreal plane they are convolution powers and all that can be described you see the weights can also be described they are a combinatorial weights essentially so you can and the additional thing which we know thanks to the complex wkb method is that we are also able to find the similar singular analysis for all the for these generating functions and then by convolution for all the functions that occur in every possible cut so and that's really what the complex wkb method in one dimension is able to tell us and that we won't have in higher dimension so before I go to several dimensions I just give you an illusion of what works in general in general the real wkb method works the bosom affair spectral formula works now we have a non-homogeneous problem so it's a little more complicated but what saves our life is that in one dimension you can you can devise a function f of h hat of the Schrodinger operator such that the spectrum is quasi linear in in the homogeneous case remember it was h hat n plus 2 over 2n the power power of h hat in the general homogeneous case it doesn't work but what you can do is you take the action as function of the energy classically you have sum of pdq defines action which is a function of the energy on which you integrate of the of the curve on which you integrate and it is well known that that's again the bosom affair rule that this this gives you a quasi linear spectrum that is a spectrum xk which is k plus a half plus corrections and therefore the the general picture I drew on the on the real world goes over and as far as the complex wkb method is concerned it has been thanks to later workers and it has been rather rigorously established for say the polynomial any polynomial potential now now I am aware and that there seems to be one loophole in the proof of resurgence so and which fam has repeatedly quoted in the 90s and he said when he retired that he would work on it and then he didn't do it and I'm not aware I'm not sure that that a certain that one loophole in the proof has is still present or not but as far as computations are concerned we can't do them and they work very nicely and so at least for polynomial potentials it works I would like to have a word a sad word for a colleague who actually deceased in a very young age tatsuya koike because he also made very very valuable contribution to extend the exact wkb method to a rational to two potentials with poles and we miss him a lot okay now the next well the following page so this I'm finished with 1d the next thing is now we enter the higher dimensional world for the now the only result about location of singularities for a spectral function that I know of is a series of results due to chazarin 2-stem at gillman first chazarin working on the Laplace operator on a compact romanian manifold he said oh but you must take the spectrum of the wave of the square root of the Laplacian to form the wave what we call the wave group the wave propagator and take its trace as a distribution this is a well-defined distribution on the real line so we are in reality which will be imaginary tau later and this is the function you consider it is a distribution and the only the singular support is the set of lengths of real periodic geodesics zero is considered as a length it's a very important singularity but you might not think of it but it has to be included the generalization of this result is that for by the semat and gillman is that you can replace the Laplacian by elliptic pseudo differential operator of any order but it has to be homogeneous order m greater than zero m may need not be an integer and so it is you again have this positive spectrum and now you take the you have to take the spectrum of f of p equal p to the one over m again this has that this is similar to what we did in one day that we you take a function to unwarp the spectrum somehow and then the the whole point is that this operator is of all the one it's very important homogeneity of all the one is the very important property because it implies the propagation of singularities so you can follow the singularities and when you take the trace you find them so this function again is now singular at the periods of closed big or periodic by characteristics that's the result okay now let me dispel one possible confusion in higher dimension it is not possible to invent a transformation of the operator which will make the spectrum quasi linear so this here while superficially it looks like what we did in one dimension it does something completely different it does it it will it's only possible because you have homogeneity and so we are here in fact at the moment we are here okay so very simple first let me begin with a trivial example if you take the one again if you take the one-dimensional Laplacian then we are here the the eigenvalues are k squared so the transformation of Deux-de-Math or Chazarin is simply you this becomes k so you we fall back on here so that's here the overlap is perfect but in one d another example next more interesting is the two-dimensional sphere then the eigenvalues are lambda k lambda j equal j j plus one with multiplicity two j plus one now it's pretty nasty if you take a square root of this an obvious idea is to replace this by lambda k by lamb by to say that x k squared is lambda j plus one fourth and that's j plus one half and then if you take tracer tracer sum of e to the minus tau x j you obtain essentially you obtain essentially the same thing as as for the linear spectrum you you obtain but because of the multiplicities you apply this differentiation here so you obtain something which I will still put into the linear spectrum which is exactly the same pictures here except that you replace the simple poles by double poles okay but so that's fine the problem again is that we need to be homogeneous another remark is so now in higher dimension there is one and only and only one asymptotic series that we I know of with that which we can handle and that's the so-called minak shisundaram playl expansion or sometimes yeah and that takes the say the original theta function that is the trace of the e of the laplace of the exponential of the laplace in itself and very old result is that for the laplace operator and other differential operators all as well it has an expansion for t tending to zero now this again remember this is only homomorphic in the on the real side on the right side of half half plane and therefore let's say we take t tending to zero from above on the real axis and then you have a complete asymptotic expansion with ration with rational powers something which looks something like this okay now so but we are puzzled because dois smart I mean chazarin tells you that you take the square root and for the square root you don't have such a result so what's the what's the connection between between this and this okay let me just draw a picture here I'm the picture that chazarin gives you resembles the picture I showed for the bosom afl rule in one dimension that is you have a singularity as a singularity at zero and then you have singularities at the periodic orbit at the length of periodical bits here and so I exactly as before we are in the real domain so I draw infinitesimal cuts to rip to say that here you have you have singularity expansion possible but in the reals in the terms of real distributions now these I'm going now to describe what happens at for the main singularity at tau equals zero at at tau equals zero in this thing here so let's let's consider some auxiliary spectral functions which are also very important in our own right the resolving trace here which is the Laplace transform of the theta of the original theta function and therefore it has an expansion a related expansion here in this expansion is for large lambda the coefficients are related and often this expansion is called the vile the herman vile series now the zeta function the spectral zeta function is simply the sum of the powers of the eigenvalues and this is obtained by performing a melin transform of the theta function and dividing my gamma the it's a standard result of melin theory that an asymptotic expansion like this translates in the thing into a meromorphic structure for eta of s that is the exponents give you the location of the poles and the coefficients give you the residues you divide by gamma and you get the similar meromorphic structure of zeta of s and now what we do is let's look at this at the function theta of tau sum from zero to infinity e to the minus tau lambda to the now the eigenvalues raised to the power one over m the trick to understand this is to go through through its zeta function because the zeta function of this is trivially little zeta of s over m so what we do is that we take the singularity description of zeta of s over m we and we go up again we multiply by gamma and we go back now we do the inverse operation and and that means that we can get the asymptotic description of theta of tau at zero and that this is what you get you get first the finite sum of polar terms the strongest singularity is a pole of all the d where d is the dimension and then all the other terms give you this expansion where you have t two integer powers of t so this is a formal series of a holomorphic germ formally and you have log t and plus the problem is that when you undo when you you go up backwards you care the coefficients of the regular terms are not generally available they are values of the zeta function at non-integer places in general and then so that means that the regular part this function has a singular part which is in one sort of one one correspondence with the original expansion but the regular part is not accessible it's transcendental and now the important thing is that in the in the coefficients of the of the singular part you have this factorial n here and that means if you that in effect this series here that is the series of the singularity here is the borel transform of w n because in w n you have gamma you have this gamma factor which is equivalent to one over gamma of one over of this gamma by the functional relation for the gamma function so up to little trivial algebraic transformations you should think of of this thing here as the borel transform of the vile series and that's that's the that's the only thing which is analogous to the 1d case remember in 1d case we had shown that the singular in the bosom afl problem the the singularity at the origin was the borel transform of yes of f of the borel of the bosom afl series so i don't understand this part this is an exponential of it and this all this we have it for real t yes that's real and i i'm just using i'm working on i'm using a different variable no but these things is a kind of poisson transformation it's in both sides you take the poisson formula gives but there is no poisson formula here because for the sphere it's not about it's a poisson relay it's called poisson then what is the collection between this formula and that thing means because this t is real here tau is real here and there i know but the connection is at this level the the tau you write it equal tau here this or here it's just a change of variable they probably sign there because this gives sign and this gives sign yeah because the i mean i'm working in the tau variable well it's a problem of course you it's a problem of choice of notation i it's more convenient for some reason to have a cuts more as horizontal so i i chose to work in the tau variable but if you want you can rewrite everything in the t variable this formula this is for real t but here but but i i i'm we are not using this except to identify that here you have a singularity which is the borel transform of how do you define zeta function then it hurts zeta function is for for it because zeta function does not use t it's here here t t is real this is this is holomorphic in the original theta of t is holomorphic here and undefined here and you integrate on the on the real t line you make a laplace transform for this and the melin transform for the zeta function hmm okay now the you can you can likewise but it's more complicated compute the singular expansions that you obtain at the various lengths of periodic orbits and the codeword for this is the method of quantum berkhoff normal forms for instance see zeldic's work you can compute those series you there are specter the coefficients are certain spectral invariants it's horribly complicated and it's i think it's a formal computation i'm not sure anybody knows whether those series are borel summable for instance but in principle you have some excess order by order to the singular expansions here again the this theory only says something about the singular parts it says nothing about the regular parts those are unknown now next example with quartier in the 90s we looked at the spectrum of the compact hyperbolic surface and two dimension where you can use you have at your disposal a very convenient tool which is called the selberg trace formula the selberg trace formula is essentially says is a generalization of the Poisson summation formula which says that if you first it says don't look at so the spectrum is lambda k we don't look at lambda k but we look at lambda k minus one fourth this minus plus in the case of the sphere this was we looked at square root of minus delta plus one fourth and this one here is the is the Gaussian curvature in fact so in the selberg trace formula the object of interest is this and so we write x k equals square root of lambda k minus one fourth as so it's not exact we don't we make a little change with respect to chazarin who take the square root of the Laplacian we have a little shift here but again this um x k and lambda k and square root of lambda k are asymptotic perturbations of each other so we are in the we are not really making anything harmful and this replacement is going to make things extremely simple the selberg trace formula which is a generalization of Poisson trace formula is a sort of machine which which is a spectral function evaluator you give you give it a spectral function that is a symmetric function of x k and it tells it gives you a formula for it in terms of the periodic geodesic length so you it does the work for you in fact and when you you have to work a little bit but when you apply it to this theta function you have a so a rather explicit formula and from the on the formula you can read that this function theta of tau has in fact a meromorphic continuation for all complex tau which is singular at the length of periodic geodesics as predicted by chazarin uh first of all the two there are two differences with chazarin that here is the diagram on the left that first of all the singularities are poles and not branch cuts so it's really a meromorphic function and that comes from the very special nature of the problem it's a little bit like the harmonic oscillator for the for these negative curvature problems and the other thing that the formula selberg trace formula uncovers for you is that you have singularities in the when you do the the meromorphic continuation you discover that you have singularities at minus two pi minus four pi minus six pi and this have a very natural interpretation that as professor concevich already said yesterday that the the periodic geodesics on the negatively curved surface when you continue them into the imaginary time direction they become geodesics on a on a real sphere and those geodesics are periodic like the geodesics of the sphere and so you're not surprised that well first of all we are very happy because we have a complex extension of the here you obtain something in this domain for the selberg trace formula and the you obtain what you would expect as a physicist that is the real result has a natural extension to the complex domain so we are very at this point we're very happy but again this is an exceptional problem with homogeneity and which and moreover with exceptional meromorphic structure here in fact i should have plotted not the harmonic oscillator but the spectrum of the sphere but again there is a very simple relation between the two what why are there no geodesics in the quadrants well it's i think the i to my knowledge that's all there is again i'm not sure but in any way the selberg trace formula only gives you singularities here that's clear the here the singularities here on the imaginary line are simple poles because and that that's what chazarin predicts at the leading order because the geodesics are isolated on the other hand the periodic geodesics on the sphere are degenerate and they give you double poles exactly like they gave in the in the case for the sphere so so we think so now there comes another thing we would like a result like chazarin and distermat gillman for again for homogeneous problems perhaps but not not exceptional that would involve that would give you singularities and in complex periods as well and this is a big question mark for me so that's the first question mark can we extend the chazarin and distermat gillman results into the complex domain to obtain hopefully resurgence resurgent big theta functions in there are some positive signs for quantum billiards which are similar to uh compact manifolds in some sense very health studied high order of vice series and on certain very strange shaped billiards they obtained that the high order of vice series were governed by certain complex periodic orbits so there is a hope to find something here but i'm not aware of any any uh any result okay that now we uh the valiant block uh representation of quantum mechanics is what will help us the most they uh well let me skip the details because i'm beginning to be late uh the big idea of valiant block was to take the Schrodinger problem and to make a Laplace transform with respect to one over Planck's constant so you you take x you take psi of x equals sum of e to the minus x over h bar x uh tau sum psi tilde uh if psi is the solution i mean you you transform everything into the uh formally into uh the say the boreal plane of one over h bar they didn't although they didn't study boreal summation per se unfortunately but what they found is of course the the great thing in that transformation is that the Schrodinger equation becomes um partial differential operator which is homogeneous so we gain we for the general problem we gain homogeneity a sort of homogeneity which we we did not have before uh remember now we are we are asking what's going on here uh so uh they replace the uh Schrodinger equation by an integral equation for the for the resolvent kernel and from this they can find they found that uh singularities propagate because it's a homogeneous problem and they they can locate singularities uh at real and complex classical trajectories so uh in particular if you take the trace which gives the the trace of the resolvent which leads to the spectral density you find by applying the saddle point method essentially that the the singularities shall be located at the actions of real and and complex classical trajectories of energy e so but that that's a formal but very helpful result it gives you the hope that you can uh that you can describe the singular structure of something the other ingredient is something which physicists like a lot which is the good trace formula the good trace formula is uh was an attempt to generalization of the bosom afeld formula in one day remember the in the bosom afeld formula we had a spectral staircase and we could write the Poisson summation formula decomposition of the spectral staircase into exponentially things here the spectral staircase is more complicated it depends on on the spectral variable and Planck's constant and Goodzfiller found that there is a he wrote a formal periodic sum over real periodic orbits where here you have a sum of asymptotic formal asymptotic expansions in power of h bar multiplied sorry multiplied by exponential of the action of the periodic orbit the problem of this this has been very popular this formula but in somehow it's very ill-defined because all summation diverge these you expect already in one day you know that these will be asymptotically factorial divergent series and sometimes they are not even Borel summable because you have singularities on the real axis and the sum of a periodic orbit is also also very badly diverges in multidimensional problems because there are too many orbits now if we invoke by the banier block representation this cures the pathology of the formula formally and the cured formula suggests a resurgent structure in high-dimensional quantum mechanics that's the message the whole the main message for the last part of the talk that you combine the two that is you write it's exactly like before you write the spectral staircase function but now you write it you have to write it for as n of e h bar with a new variable spectral variable theta of x minus x k of e and what ballet and block tells you tell you is that the correct spectral variable is x one because that's the one with respect to which you go into the Borel plane so the spectrum you look at is the spectrum of x at fixed e which is a parameter okay and so we make this transformation which is more complicated than simply taking a function of the operator we have a parametric problem now and when you formally completely formally write the Goodfiller trace formula here what you obtain is a picture like Shazarin except that this function that is function theta of e which is now the sum of this sum here has singularities on real time which is imaginary tau at the actions at the i times the actions of real periodic orbits so we have a conjecture this is unfortunately a conjecture but it's a possible result here which asks for being proved unfortunately because this is a generalized eigenvalue problem i have no idea of what technique to use now combining this ballet and block rewriting of the Goodfiller trace formula here and the encouraging results obtained for the hyperbolic problem in two dimensions i proposed the conjecture that this theta e of tau could possibly be extended perhaps partially but or completely i don't know into the complex tau plane and the singularities should lie at the at the i times the action of real and complex classical periodic orbits of energy e so that's a completely conjectural thing and of course we are still very far so this here are two question marks we are very far from complete resurgent description because we have no idea how to compute these singularities and how to compute the singularities of the singularities and whether there are they are interrelated as in 1d okay thank you very much sorry i just remarked just in the case of chazaran it's it's exactly what yesterday proposed when i have no potential just of course yeah complete picture what will be yeah yes i was afraid you were you were going to give my talk yesterday at some point yeah yeah i see but okay but again i that's the only possible entry point i see at the moment in several dimensions that's why i would like to push it to for people to look at it but there may be others but i just don't know how to how to even to describe them in words not even speaking of proving it was paid for a few days ago but i'm treated by don zegir and somebody else about i think it's actually the relation i'd be very interested to know that i mean i've not been aware of that thank you very much