 First of all, I'd like to thank the organizers, Anton and Maxim, for organizing this fantastic event. And it's really my pleasure to be here, so my first thanks to the organizers. And when preparing this talk, I was thinking what would be the topic which kind of intersects with Samsung's scientific interests. I think I hope that I will show that in many places it touches Samsung's interests, which are very and very interesting. So I'll mention this. And finally, I will say more probably later, but let me mention that Samsung came to Leningrad in 1981, in 1981. And he was a very young fellow, but he had, and he has a very remarkable gift. He can make friends with many people, young and old. I'll tell more, but before I'll start this talk, let me show some of the Samsung friends whom he made in the 80s. Okay, so this was like this ends informal introduction. This is Trinity College Dublin. And now let's start. So the topic originated from local mirror symmetry. And one of the most important papers was by Anganajic, Daigraf, Klem, Marino and Waffen. They studied mirror symmetry for local Toric, Calabio three-folds. And the typical example they consider, and many other people consider, is when the Calabio manifold is a total space of canonical bundle over Toric del Petz's surface. So that's what's written over there. It's a canonical bundle over a surface S. And there are very simple examples of such Toric surfaces, like P2, the basic example. Another example is, here's the Bruch surfaces, which are also very simple. It's P2 blown at 0, 1, 2 or 3 points. Then there are non-smooth examples, which are weighted projected spaces. There are much more, but this would be sufficient. Even in this talk I'll consider only one example, which would be also interesting. But this is the general setup. So this is a non-compact Calabio three-fold, that's why it's end. Then the result is that corresponding mirror manifold is given by precise equation. For this, it can be described explicitly. It corresponds to Landau-Ginsburg model with superpotential W. The equation is written over there. It defines a manifold in a product of C2 times C star 2, because it's natural to consider E of PN, E of Q, where PN Q are just complex coordinates. Then in these examples I mentioned before, the mirror curve is just elliptic curve and can be given by a very simple equation. For instance, for the basic example, when S is the Hilsbrug surface F0, the superpotential has this very simple form. There are two complex parameters, or rather real, say, zeta positive plays the role of physical mass and kappa is the modulus of corresponding elliptic curve. These precise equations were discovered long ago by physicists. They were used by mathematicians like Arou, Katsarkov, Arlov. They were studying these examples very carefully because they are like playing ground where you can test many things. When your delpets surface is a weighted projective space, you have another equation which is also very simple. This was known long ago. But then surprisingly, well, it was said that the idea was in the air for many years, starting from that paper by Agonadzic and other people and her friends. But here, Mariniou had an idea that one need, well, which came from integrable systems and it was probably at the same time when Samson, together with Nikita, were studying quantum integrable systems, Baxter equation, their relation to supersymmetric quantum field theories. At the same time, roughly, Marcus Mariniou and his collaborators, Grassi, Hatsude and other, they had an idea that one needs to quantize this mirror curves basically by replacing classical coordinates P and Q by quantum mechanical operators capital P and capital Q, position and momentum operators. And as a result, so that's the idea, then instead of equation of a quantum curve, we have an eigenvalue problem for the corresponding quantum mechanical operator. And there are many interesting questions, both from physics, from mirror symmetry and from the spectral theorem. So in this talk, I will be concentrating on the spectral properties of these operators as operators in the hybrid space, so that the hybrid space structure will be very important. And so, but first it was discovered by Mariniou and his co-authors that these operators have several remarkable properties. These, well, properties mean conjectures, so they mean the following conjectures, that spectral properties of these quantum mechanical operators on the line is the simplest quantum mechanics with one degree of freedom. They encode in some way the enumerative geometry of the Toric-Colabiau three-fold X, the original manifold we started in the mirror symmetry. More precisely they're related to invariance like Gromov-Witton, Gopakumar-Waffer and other things in some rather complicated way. The second also that the spectral theory of these operators which we'll discuss provides a non-perturbative definition of the topological string theory on an initial manifold X. So that's what studying quantum mechanics on the mirror manifold allows to define non-perturbatively the topological strings on the Colabiau three-fold. Obviously I won't discuss these properties, these things, these are two main problems and they're rather deep and kind of difficult. So I will discuss some very simple things which should be like basic quantum mechanics. Introduction to quantum mechanics, let's put it this way. Okay, so first very briefly let's remind the VAL operators. So our hybrid space will be L2 on the line, the very basic. Then we can see the Heisman operators P and Q, quantum mechanical operators. Here I put h bar equals to 1, it will reappear later in a different context. And then we define VAL operators. Usually in quantum mechanics we are taught that VAL operators are unitary operators. There is P and Q, capital P and capital Q are self-adjoint operators in a hybrid space. They define each of them. One parameter group of unitary operators and this unitary operators satisfy Hermann-Veyle commutation relations. And then the one parameter group means the parameter is real on the real line. But in this setup the parameter is pure imaginary. So instead of considering unitary operators bounded, we have self-adjoint unbounded operators. Namely in this setup we have operators U and V which are just exponents of self-adjoint operators. P and Q, they exist as a nice unbounded self-adjoint operators in a hybrid space. And B is a positive constant. It kind of plays the role of a Planck constant. You can absorb it in P and Q, obviously. And then they satisfy on their common domain this Hermann-Veyle-like relation. This is not a true relation everywhere because when you have unbounded operators in a hybrid space you cannot state any algebraic relation between them because domains may be different, you need to be very careful. So this relation is understood formally on a common domain, which is invariant under the action of U and V. And then they obviously satisfy this relation. Some say it's like pre-normal space. Exactly, exactly. For instance, this is the domain which is invariant. Unboundable dimension, dense space, yes. But by no means this is true in general that such relation holds. So this is rather formal. And of course example, using Schrodinger representation we can realize operator U as a shift in imaginary direction in L2 on the line and V as a multiplication by exponent. Obviously V is a nice self-adjoint operator, just multiplication by a function. But it may look weird that a pure imaginary shift is a self-adjoint operator. This is true and this is very simple. Like shift in a real direction is unitary, shift in pure imaginary is obviously self-adjoint. The easiest way to see it is to use a Fourier transform. And then you can describe the domains explicitly. This is very simple to describe domains of these self-adjoint operators. And I like to say that these are self-adjoint in a true mathematical sense. It's not a remission operator which means either symmetric or whatever. It's self-adjoint in precise definition that each operator with dense domain closable has an adjoint and they are equal. This is precisely what we mean by a self-adjoint. So these are the basic operators. And well, that's how we define the inverses. The V is positive, U is also similar to a positive operator. So that's the basic setup. These are just two formulas we need to kind of, we'll be using throughout this talk. And then the functional difference operator, for instance, in this setup, when S is a Hilsberg surface, then we have a family of operators, H of zeta, which is a very symmetric expression. Say, for instance, if zeta is 1, it's like U plus U inverse plus V plus V inverse. Each of the terms is a self-adjoint operator, but in general it's not true. That the sum of two unbounded self-adjoint operators is self-adjoint. There could be some problems. But in this case, one can prove that this is truly self-adjoint operator in a hybrid space. And why this operator is so interesting in kind of the basic, because if we can set zeta to be 1 like a mass, then its symbol has a remarkable symmetry. Its symbol is the sum of two hyperbolic cosines. And this is an analog of a usual quantum mechanical oscillator, whose symbol is psi square plus p square. So if you expand for small bit, you get a harmonic oscillator. Minus 4, I guess. If you subtract 4 and expand, it's a harmonic oscillator. So this operator should have some remarkable properties, as harmonic oscillator does. And this is the operator which just came from my records. It was never studied, neither by mathematicians, in the speckler theory of differential operators, because H formula is a pseudo-differential operator of infinite order. Because if you expand, it's pseudo-differential operator of infinite order. And remarkably, its properties are very close to the second-order differential operator, to the Schrodinger operator. So it came in this way. So there are two, and this is also another remark, kind of interesting, is that you can put, you can discretize Schrodinger operator, and usually people replace second derivative by a second difference. Then this operator with second difference has different properties, because it becomes unitary. But this is a better approximation, replacing by functional difference in imaginary direction. So that's the proper discretization of Schrodinger operator rather of second derivative. Sorry, but before we termed functional difference equations, when you have continuous, there's no difference replacing with those unitary, do you care about geneticity? No, this is, if you have, instead of imaginary shift, you have like x plus b, it will be unitary obviously, it will be unitary, right? It's a shift operator. And then it's kind of interesting that in fact, when the mass zeta is zero, this operator, we should call H of zero, which is u plus u inverse plus v, already appeared in our work with Ludwig Fadeev, when we were discussing quantum Liouville model on the lattice. This was the trace of local monodromy matrix, then this operator appeared in representation theory of quantum group SL2R and so on. And I remember that when around that time Dan Friedan was visiting Soviet Union, he was in Moscow and in Leningrad, and when he came to Steklov Institute, I think something was there, we discussed with Fadeev, with Dan Friedan, because he was interested obviously in Liouville theory at that time it was, and of course this is, that's where this operator came from. Then later it was studied extensively by Kashiev, because if one quantizes, considers kind of deformation, quantization of the Taishner space, considered as a symplectic manifold with vile Peterson's symplectic form, then this operator H plays the role of a dent twist operator, that's what Renat Kashiev was discussing. So this has a long history, but surprisingly the spectral theory of this operator H, which is very simple, have not been done, although many things were discussed. Now that's another set of operators for weighted projective spaces, they have very nice also property, it's also a self-adjoint operator. And then let me just mention results about this operator, and then I'll discuss them. So first of all when the operators H zeta, for positive zeta with massive case, and this operator H mn can be proved that they have a pure discrete spectrum, the inverse operators are of trace class, and you can compute the asymptotic for the eigenvalue counting function. You can prove the analog of Hermann-Veyle law, and instead of area, like for Schrodinger, you get log square with precise constants. So these are theorems, and where the constant c is of this form. So the story is that we, this is in our paper with mathematicians, Laptev and Schimmer, and who got really interested in this operator, the experts in functional analysis, and people there never seen operators of this type, of this form. And so let me spend maybe five minutes to comment on the proof of these results, which are rather general. For instance, let me just mention that these asymptotics for the eigenvalue counting function says that the eigenvalues are very sparse. For instance, n's eigenvalue grows like exponent of square root of n with some coefficient. So they are not like, but of course if you expand, then, yeah, so. So the main idea of the proof is very simple. It based on another physics method using coherent state transform, which maps L2 on the line into L2 on the plane. This is not, of course, this is a partial isometry, it cannot be isometry, defined by this form, and then there is some kind of remarkable identity which reflects integrability of this operator, and this single identity allows to prove all these results. Namely, if you compute quadratic form of operator H in terms of the coherent state transform, then quadratic form exhibits the symbol of the operator. The symbol of the operator appears in a quadratic form with some constants d1 and d2. That's the, and if you, and using this thing and the standard techniques estimating risk sums, risk means which are written over there, it's a sum function of lambda, which is the sum over all eigenvalues less than lambda, lambda minus lambda j summation over all j such that lambda j is less than lambda, and then you get the identity, and then one can estimate it from above and below by the same constant using Jensen's inequality, convexity, and this representation, using two different types of representation. So this is basically very simple, and the key is this formula for the quadratic form of the operator. And of course it's also non-difficult, rather easy to prove that these operators have a discrete spectrum. And then from this, from the Vale law, it follows that they are of the trace class, inverses of the trace class, and these are all positive operators. And then, so that's what was discussed before, and of course the main thing, which the conjection stated by Marini and co-authors is that in fact these operators are of trace class, inverses of these operators are of trace class, one can define Fredgol determinant of these operators as entire function, and then it's asymptotic expansion of the Fredgol determinant, or traces of the all positive powers of the inverse operator in code enumerative invariance of the Calabria of three-fold. And this conjection of Marini is very complicated because it uses Nikrasov-Shatashvili function, it uses some, a lot of ingredients, some Nikrasov-Shatashvili limit, some kind of quantum theta functions, it's really very involved. But these simple operators seem to carry this information. But before this, we study the massless operator. And there's also another interesting story where Samson was involved, because in 2012 we invited Fadeev to the Simon Center, and he came probably for a month or maybe for four weeks or whatever, and then he was, Fadeev was working with this operator for quite a while, and then, but somehow when we asked him what is the spectral properties, what is the resolvent of this operator, he kind of, they didn't study it. So I started to study this, and then we developed the full, it's a very simple operator, and we developed the scattering theory for this operator, like it's precisely like a Schrödinger operator with a potential that grows exponentially at one end and decays very fast at the other end. And this can be done very nicely, but there is some interesting feature of this analysis, which I like to outline, is the following. So like, let's consider, we think of the operator of Schrödinger type, namely u plus u inverse replaces the negative second derivative operator, and v is a potential. And then, like we know from elementary quantum mechanics or functional analysis, that the Green's function for the second derivative operator is continuous on a diagonal, and its first derivative is discontinuous. And therefore, the delta function, which appears in the equation for the Green's function, comes from Heavisite formula. The derivative of a jump is a delta function at this point. Here the resolvent, let's consider the unperturbed, the free operator H0, which is u plus u inverse. It's an integral operator. It can easily be solved by Fourier transform, then integrating back its integral operator of this form. Let me, of this form, integral operator where this is a nice, smooth function everywhere is smooth at x equals zero. And then how come that this resolvent satisfies the equation H of r equals lambda times r plus delta function? What is the mechanism how the delta function appears? And it turns out that it's kind of interesting. It appears because of Sochotsky-Plemel formula, so which one can think of as a smooth version of Heavisite formula. Namely, it's easy to prove, well, it's elementary, that this property follows from Sochotsky-Plemel formula, where this you can consider as a smooth version of a Heavisite function. Also, it may be important, maybe not, but Polikov observed that this expression can be interpreted as one partition function for both the Einstein statistics. This is, of course, trivial, because you expand denominator into geometric series and you get both the Einstein partitions function. But maybe it's important, maybe not. But anyway, this is the Sochotsky-Plemel formula from which the whole thing follows. And then we have a free resolvent. Then one can study the solution of the Eigenvalue problem, scattering solution, where lambda is parameterized by two hyperbolic cosine of 2 pi bk, sorry. And then, actually, in this case, one can think of this problem as a Schrodinger operator on a half line from 0 to infinity. Then it is known that it has Yoast solutions and a scattering solution, which is 0 at the origin. And then the scattering solution here can be written nicely as Fourier transform of a product of two Fadeev's quantum dialogarin functions. And Yoast solutions can be also defined. And then we have precisely the same formula as for the Schrodinger equation, that the scattering solution is the linear combination of the Yoast solution, where this will define the plancherial measure. This will define the spectral function. The modulus square inverse defines the spectral function for this operator. This is very simple, but this is one can think of this operator as a Q-analog of the Schrodinger operator with potential e to the x, whose solutions are modified Bessel functions of the second kind. And then, orthogonality and completeness relation, spectral decomposition for this operator is classic result in theory of special functions. It's called Kontarovich-Lebedev transform. So this can be considered as a Q-analog of Kontarovich-Lebedev transform. Then the resultant for the perturbed operator has a very simple formula. It's the same type, like for the Schrodinger operator, and then starting. And then the last operator, U, which is a unitary operator from L2 on the line to L2 on the half line with the spectral measure, defines the spectral decomposition because operator UH, U inverse, is a multiplication operator. So this finishes the spectral theory for this operator, for operator H. This is very simple, so I'll just mention it. Now what happens with the operator, with our operators when the mass is non-zero? So, and then there is a whole story, which is rather involved because there is a conjecture. Starting from Nekrasov-Shetashvili paper, we know that total lattice, the periodic total lattice is integrable and one can find some exact quantization conditions using Nekrasov-Shetashvili function, which is a young, young potential for this theory. And this Nekrasov-Shetashvili function is obtained as a super potential for some super symmetric gauge theory in a rather complicated fashion. So that's the story. In particular, for the simplest Schrodinger operator, negative second derivative plus hyperbolic cosine, there is some quantization condition, which is impossible to check because it uses Nekrasov-Shetashvili function. So there is some interesting story going on, even with the classical, with modified Mathieu equation, when you replace ordinary cosine potential by a hyperbolic cosine. This gives you results from Nekrasov-Shetashvili theory. But then the claim is made by Marini and Hatsuda and other people, it's a conjecture that for all these integrable operators from mirror curves, there is a certain quantization condition, exact, which uses two ingredients. One ingredient is Nekrasov-Shetashvili function, which should be used for the operators and the dual operators. I forget to mention that we have a pair, u and v, and then there is a dual pair, u tilde say, and v tilde, which formally commute with u and v, only formally, they don't commute as self-adjoint, but formally they commute, and they have corresponding q tilde. It's like b goes to 1 over b of this sort. And then for one q you construct one, Nekrasov-Shetashvili function, for another q tilde you construct function for q tilde, and then the quantization condition is for the sum of these two functions f plus f tilde, and then this is a quantization condition for the eigenvalues of the discrete spectrum, and this is remarkable that usually when in defining young-young superpotential from supersymmetric gauge theories, you don't usually pay attention whether your solution is, is it in L2 or in some other space, but in this case this quantization condition defines a true L2 function, which is solution of the eigenvalue problem, which is kind of mysterious, and it's really, if it really holds, it's a miracle. But there is a second part of the story, that this quantization condition doesn't, the energy, the eigenvalues are not directly involved in this quantization condition, it's rather a certain function of these energies, and this certain function is obtained as a quantum A period of the mirror curve, and this is the object which nobody knows how to compute, the quantum A period of a mirror curve. Mirror curve is elliptic curve, it has A and B periods which can be computed, and they are used to obtain semi-classical asymptotics of the eigenvalues, but the quantum A period, whatever it means, there is some conjecture by Waffer's digraph, but it's not clear what it really means. This quantum period is allowed to determine how the lambda eigenvalue depends on the parameter T in Nekrasov-Shatashvili function. So there are two conditions involved. One is a pure quantization condition and another is change of variable. And so whether it's very useful or not, it's not clear because for potential Kosh X, for Schrodinger operator, there is also good swiller quantization condition, which is also exact. Probably good swiller condition in this case is a special case of Nekrasov-Shatashvili, but this would be interesting to check. I think maybe Tashner checked it, maybe not. It's not clear. Yeah, but either it's exactly that yo is good swiller condition. Exactly. Yeah, but good swiller in Schrodinger's work, because good swiller was the first who proved that the quantum, periodic quantum total lattice is completely integrable and he gave exact quantization condition. Later it was explained by separation of values. There are some formulas that if you take same function W and differentiate with respect to zeta, it centers as a parameter, then that would give the energy, evaluating on the solutions of discrete spectrum. Right. And bad things that protect Nekrasov. Right, but it would be interesting to check correspondence with good swiller. Okay, so and then the idea was that since this exact quantization conditions are very difficult in the conjecture that there is another way to treat the eigenvalues. It's considered the trace identity. Namely, you have Friedholm determinant, which is entire function of a parameter lambda. Its zeros are precisely the eigenvalues. Then one can study either its behavior when parameter lambda is zero, just Taylor expansion, but then one can consider its asymptotics for large lambda. And asymptotics for large lambda contain a lot of information about eigenvalues. They kind of reflect some of eigenvalues, some of squares of the eigenvalues regularized. And this is the thing which one can do. So I'll finish by giving some ideas how the trace identities work. Because that's the approach that kind of related to the Schrodinger operator and which probably can be done nicely in this case. So let me briefly review the history of the trace identity for Schrodinger and this type of operator. So that's what I said, that you have a regularized determinant of the operator. I write it in this form. It should be regularized at the Friedholm determinant. And then you consider the asymptotics of this operator for large values of parameter lambda. And express it in terms of the coefficients. So first, let's start with the basic example. It's the Schrodinger operator with a potential v of x, either on the interval with some boundary conditions or on the whole line, half line, anywhere, with appropriate boundary conditions. First, let's consider the Stroum-Lewin operator on zero pi, say, with smooth potential v of x. This is, of course, celebrated Gal-Font-Leviton-Dickey trace identity. That's a beautiful formula of Gal-Font-Leviton where you can see the Stroum-Lewin operator on the interval, say, with zero Dirichlet boundary conditions. Then some of the eigenvalues obviously diverge. You regularize it. There is a canonical way. Then it equals to the integral of a potential. So that's Gal-Font-Leviton, 1953 or something like that. And then, of course, this was developed further by Gal-Font-Leviton-Dickey and there is the whole beautiful machinery of trace identities. Schrodinger operator on a half line. Then on the interval, there is no condition on a potential. Just smoothness. It's sufficient. On a half line, then, of course, potential should be rapidly decreasing. Then there is what is called Busleif-Fadier trace identities, which also express regularized sum of eigenvalues plus a contribution from a continuous spectrum in terms of the potential. It's some integrals of potential, v of x, its derivatives, values of v at x equals zero, its derivatives, and so on. There is a complete characterization. Then when you consider Schrodinger operator on a line with rapidly decreasing potential, then you can generalize Busleif-Fadier. It will be Fadier-Zaharov trace identities, which everybody knows because you use them for complete integrability of the KDV equation, which express precisely that certain spectral data in terms of integrals of potential, of polynomials of potential v of x and its derivatives, which trace identities represent all local commuting integrals for the KDV on a real line with rapidly decreasing case. There are also integrals in periodic boundary condition, but that's a different story I won't discuss it. So, it's periodic boundary condition also can be considered. Lambda should go to negative infinity. Thank you. Negative. Yeah, yeah. It's still in the formula. Right. Exactly. Exactly. Right. So, now, this is just what I said before. We can see the pure discrete spectrum. What is determinant of H is difficult to define because it needs a family, but if you have a family, you can define and normalize. Determinant by this, either by Adamar product, if the operator H inverse is of trace class, then the sum of the eigenvalues or rather one of eigenvalues converges. This is Adamar product. Or you define if your operator is not, if H inverse is not of trace class, but its power is of trace class, you define it somewhat differently. If H inverse minus epsilon for some small epsilon, you define it like that. So, basically, that's the object. Our operators from mirror curves, they belong to the first class. They belong to this class. And this is basically Fredholm determinant. Or you have to replace lambda by one over lambda. It doesn't matter. This is Fredholm determinant. Just ordinary determinants. Characteristic determinant of an operator. So, we have these properties. And then, well, there is another simple... Now, how to obtain asymptotics of this determinant when lambda goes to minus infinity? And how are they related to the distribution of the eigenvalues? Well, we define the operator zeta function to H, operate H. We define the operator zeta function, which is given by this series for real S large enough, you know, okay, greater than one, even one-half depends. And then, if, since the operator zeta function admits a miramorphic continuation to the whole S plane, then we can just by milline transform, you can express log of this determinant as an integral of the operator zeta function with these factors. Where sigma, you integrate along the vertical line in a complex plane. And sigma is large one. Large, greater than zero. And then, to get asymptotics of this determinant when lambda goes to minus infinity is very simple. You shift the contour of integration to minus infinity. And how far you shift? You get asymptotic, you get terms of your expansion plus an error term. And if you shift it to infinity, you get asymptotic series. Otherwise, you get exact formula with a remainder term. So, that's standard representation, like everywhere. And so, that's one way of obtaining the asymptotics. So, and then, of course, here we see, like, naively that since zeta function admits another continuation, poles are at, when S is an integer, say, when S is a positive integer, well, you don't go to the right. You go to the left. When S is, so S is a negative integer, then you get inverse powers of lambda. Inverse powers of lambda. And values of zeta and zeta of S at S equals to negative integer. So, you get regularized. You get regularized sum of eigenvalue squares and so on by zeta function, and you get this expansion. So, that's the standard part. Now, the non-trivial part is to obtain the same expansion using the potential v of x to get a different representation for this asymptotic series which is obtained from this formula. What I said is just a way of defining the sum of the eigenvalue, sum of square, how to regularize this. And then, there is a way which I put it here in rather cryptic form, but I can explain. How can one do it for Schrodinger? Now, we are discussing Schrodinger equation with growing potential. Growing potential of such that the spectrum is discrete. How to get a trace formula for this class? This was not really considered. But then, there is a way of, if operator is positive, you can see the asymptotics when lambda goes to negative infinity. It's like doing WK beam method without turning points. And this was known long before WKB. It's called Louisville Green method when there is no turning points. And in some cases, which one should be very careful, Louisville Green method gives uniform asymptotics for large negative lambda, its large negative energies, and uniformly in X. Then, doing this, so you can see the solution of this equation for any lambda negative. So it cannot be an eigenvalue. And then, you can see the solution which decays at one end, say at plus infinity. And then, you can see these asymptotics by this Louisville Green method when lambda goes to negative infinity. And these asymptotics should be uniform in X. So that's the leading term. When then subtracting these asymptotics, you can reduce your Schrodinger equation to Riccati equation. But you need to know what to subtract. This Louisville Green says precisely to Riccati equation where instead of a potential, will be some combination of potential in terms of this Louisville Green formula. So it depends how fast potential grows. To infinity involves certain integration. And this can be done explicitly. And then you can get Riccati equation and you get coefficients in the asymptotic expansion. Here, I didn't write it because here you expand this in inverse powers of lambda with coefficients. And then you can get these coefficients recursively by using Riccati equation. So you get a bunch of given potential even if potential is, you get some integrals of these derivatives or if potential is given like e to the x or hyperbolic cosine, you get a bunch of numbers. Let's consider a couple of examples. This brings us to the 19th century actually. So first, the trivial example. The harmonic oscillator. You will all know that it's kind of discrete spectrum, but in fact all solutions are important. They give special functions, parabolic cylinder functions called Weber functions. Then this is the determinant. It's inverse gamma function with k-efficient. And then trace identities obtained by Louisville Green method gives you the sterling formula for gamma function. So trace identities are very meaningful because this is a... they combine information from two different points. So this gives sterling formula for zeta function, for gamma function. So there's one example where some... you cannot make a simple example. Then the second case, as I mentioned before, essentially is e to the 2x on the half line with directly boundary conditions at the origin. Solutions are modified basically function of the second kind. Then determinant is just the... this k is a McDonald's function, k function, evaluated at point 1, because log 1 is 0. And then it's function k of i square root of lambda. So again, you get using asymptotic expansion of this at lambda goes to infinity. There is some... by Green-Lewville method. You obtain formulas for the sum of the zeros of Bessel function. There are squares, regularized sums. Actually, just as a side remark, this is a toy operator which Poyer was using when he discussed approximation, some naive approximation to remain zeta function. It's not important. Now, the last case. The last case is very... excuse me. The last case is rather involved, but there is similarity. Since operator, although H is now pseudo-differential operator of infinite order, it behaves like a second-order differential operator. And one can also develop the analog of Louisville-Green method and obtain the corresponding trace identities. I hope to tell about it maybe next time at Samsung's birthday, maybe next year. But some details are rather involved. But this can be considered as a constructive way of obtaining some invariance like Gopakumar-Waffer and others from this trace identities. Because this potential is just hyperbolic cosine. Therefore, coefficients and trace identities will give us a bunch of numbers doing some obtained by some weird integrals. And then one can compare this bunch of numbers with exact invariance like Marini and his co-authors were doing. It's not done completely, but that's a way to avoid these quantization conditions and quantum A periods, because this is not really because it's very difficult. Not doable. Not doable, right. And get some information. And the key, of course, is that this remarkable operator is as nice as harmonic oscillator. So, of course, we cannot get a formula for the eigenvalues, but we can completely characterize its eigenfunctions or maybe the general solution like parabolic cylinder functions in terms of some integral representation. What we mean? We have special functions. Meaning that we have a function like Bessel function. We have some serious expansion which doesn't help much, but we have an integral representation which really is the thing which allows us to connect behavior at large and at small values of parameters. Same should be here because this is a remarkable operator which should be even better than harmonic oscillator. So probably this is like the bottom line of this talk. That one should look at this operator very carefully. And I think, yeah, it's time to stop. So, that's... Happy birthday Samson. So, you gave some examples of those kind of planning operators with exponential potential and with the shifts which we have as second-order operators. Is it clear what is the class kind of right? There were many examples, but is there some more general class of operators which became similar to second-order differential operators? Well, I think operator U plus U inverse, you can add anything. You can add capital V, you can add small V of X would be of the same class. U plus U inverse just shifts in plus and minus imaginary directions. That's the true analog of a second... And then an arbitrary potential. Or arbitrary reasonable... Right, right, exactly. Yeah, there is also the harper operator. Can you use it to get some information? Yes, exactly. That's a very good point. When we consider the case when U and V are self-adjoint. If U and V are unitary, that's the famous half-stadter harper model. But this is a totally different harper operator. It has a periodic spectrum. It's like Mathieu's equation with cosine X and hyperbolic cosine. Spectral theory for one doesn't correspond to spectra. They're kind of like some analytic continuation. Yes, so yeah. Do you drink? Yes. Does the data function verify functional equation? I don't know. I'm asking this because for the harmonic oscillator the spectrum is integer so you get the classical... Right, right, right. You know, there is some weird equation Bender, you know, road, yeah but it's not true. You know Bender. Do you think that we try to consider elliptic difference analog of these equations because it's like few difference operators when the front can try... More complicated. Locals we can want to... Not to worry, but need itself. You don't know. No, I don't know. It's like the baby example. Right, but it's very... Let's thank the speaker again.