On the spectrum of the local $\mathbb{P}^2$ mirror curve

We address the spectral problem of the normal quantum mechanical operator associated to the quantized mirror curve of the toric (almost) del Pezzo Calabi--Yau threefold called local $\mathbb{P}^2$ in the case of complex values of Planck's constant.


Introduction
The recent progress in topological string theory reveals connections between spectral theory, integrable systems and local mirror symmetry. The results on linkings of some quantum mechanical spectral problems with integrable systems and conformal field theory [6,4], together with the relation of topological strings in toric Calabi-Yau manifolds to integrable systems [2, 20,19,1,17,11,10,13,12], have lead to the conjecture on the topological string/spectral theory (TS/ST) correspondence [8,5]. In many cases, quantisation of mirror curves produces trace class quantum mechanical operators, and, according to the TS/ST correspondence, their spectra seem to contain a great deal of information of the enumerative geometry of the underlying Calabi-Yau manifold, see [16] for a review and references therein.
In the case of toric (almost) del Pezzo Calabi-Yau threefold known as local P 2 , the corresponding operator is of the form (1) O P 2 = u + v + e i /2 v −1 u −1 with invertible normal operators u and v such that (2) uv = e i vu, uv † = v † u.
With various levels of generality, the spectral problem for similar operators has been addressed in [15] from the perspective of exact WKB approximation, in [18,9] using a matrix integral representation of the eigenfunctions, and in [21,3,14] from the standpoint of quantum integrable systems. In this paper, following the approach of [21,14], we address the spectral problem of operator (1) in the case = 2πe i2θ with θ ∈]0, π/2[.

Definitions and notation
2.1. Heisenberg operators. Let x and p be normalised self-adjoint quantum mechanical Heisenberg operators in the Hilbert space L 2 (R) defined by their realisation in the "position representation": Here we use Dirac's bra-ket notation so that for any ψ ∈ L 2 (R), we write if ψ is in the domain of x and if ψ is in the domain of p. One can easily verify the Heisenberg commutation relation (7) [p, x] := px − xp = 1 2πi .

2.2.
Heisenberg-Weyl normal operators. We fix θ ∈]0, π/2[, denote and define the normal Heisenberg-Weyl operators that have the Hermitian conjugates and satisfy the commutation relations Thus, if a and b are arbitrary elements of the algebra generated by u and v, then 2.3. A sequence of polynomials. To any n ∈ Z ≥0 , we associate a polynomial p n = p n (q, E) ∈ Z[q, q −1 ][E] of degree n in E defined by the following recurrence equation Notice the symmetry p n (q, E) = p n (1/q, E). Denoting q n := q n − q −n , the few first polynomials read as follows: Among the properties of these polynomials, one can show that p n (q, 0) = 0 unless n ≡ 0 (mod 3) and (14) p 3m (q, 0) = q −3m 2 (q 2 ; q 6 ) m (q 4 ; q 6 ) m , ∀m ∈ Z ≥0 , where we use the standard q-Pochhammer symbol (1 − xq k ), (x, y, . . . ; q) n := (x; q) n (y; q) n · · · One can also show that where α k,m (x) are polynomials in x of degree m − |k| satisfying the recurrence relations Furthermore, the leading asymptotics of p n at large n is given by the formula It will be of particular interest for us the following two generating series for these polynomials: Taking into account the inequality |q| < 1 and the asymptotics (18), we remark that for any E ∈ C, the radius of convergence is infinite for both of these series and vanishes for the series φ 1/q,E (z).

2.4.
Vector spaces F p,c , G p,c and T m p,r . We let H(C =0 ) to denote the complex vector space of holomorphic functions f : C =0 → C and Let c ∈ C, p, r ∈ C =0 and m ∈ Z. We define vector subspaces of H(C =0 ) p,r will be called theta-functions of order m. For any p ∈ C =0 such that |p| < 1, one specific theta-function is defined by the series which imply that ϑ(·; p) ∈ T 1 p,−1 . Note also the modularity property Remark 1. Let f ∈ T m p,r and z ∈ U (f ) (respectively z ∈ U (f )). Then, one has Remark 2. By expanding into Laurent series, one easily checks that the dimensions of F p,c and G p,c are at most 3. On the other hand, the recurrence relation (12) implies that ψ p,c ∈ G p,c if |p| = 1 and φ p,c ∈ F p,c if |p| < 1 so that The elements φ p,c and ψ p,c will be called regular elements of the corresponding vector spaces.
Remark 3. By expanding into Laurent series, it is easily verified that In particular, for |p| < 1, dim T 1 p,r = 1 with the theta-function ϑ(−rz; p) being a basis element.
Remark 4. One has the identifications T m p,r ⊂ T mn p n ,r n p mn(n−1)/2 , ∀m, n ∈ Z, ∀p, r ∈ C =0 , which for n = −1 become equalities T m p,r ⊗ T n p,s → T m+n p,rs , ∀m, n ∈ Z, ∀p, r, s ∈ C =0 . For example, the product identity Remark 6. Assuming 1 ∈ p Z =0 , let g ∈ G p,c . Then, the even part of the product g(z)g(−zp) is an element of the vector space T 0 p,1 = C. Thus, there exists a quadratic form ω : G p,c → C such that In particular, if |p| = 1, we have

Formulation of the spectral problem
Let u and v be the normal Heisenberg-Weyl operators defined in (9). Then, the Hamiltonian is a normal operator, and the spectral problem consists in solving the system of Schrödinger equations in the Hilbert space L 2 (R). In the position representation (3), it is equivalent to the following system of functional difference equations where Ψ(x) := x|Ψ . We are looking for an entire function Ψ : C → C that solves the functional equations (41), (42) and whose restriction to the real axis is square integrable.
Equations (41) and (42) are related to each other by the simultaneous substitutions which correspond to the Faddeev (modular) duality [7] which we will abbreviate as F-duality. For this reason, in what follows, we will write only one equation (containing the variables E and q), but implicitly there will always be a second accompanying equation. In constructing solutions, we will follow the principle of F-duality corresponding to the invariance of the solutions under above substitutions. In this case, it will suffice to check only one equation as the other one will be satisfied automatically.

F-dual asymptotics at x → ±∞
We start our analysis by addressing the problem of asymptotical behaviour of solutions of our spectral problem at large values of x. Following the principle of F-duality, we are looking for possible F-dual asymptotics.
Proposition 1. Let Ψ(x) be a solution of equations (41) and (42). Then, one has the following possibilities for the F-dual asymptotic behaviour of Ψ(x) at large x: Proof. Dividing (41) by Ψ(x), and denoting ρ λ (x) := e λx Ψ(x − ib)/Ψ(x), we obtain a first order non-linear finite difference functional equation with exponentially growing or decaying coefficients Let λ ∈ C be such that there exists a finite non-zero limit value where ∞ means one of ±∞, and there exists an F-dual solution of the finite difference functional equation Then, the corresponding asymptotic behaviour of Ψ(x) is given by f λ (x).
The case x → −∞. Choosing λ = πb, we obtain Thus, one has two possibilities for the limit value The finite difference functional equation (49) admits an F-dual solution of the form f πb (x) = ψ 0 (x) provided ǫ = −1.
Taking into account the remarks in the end of Subsection 2.3, we conclude that χ 2 (z) is holomorphic in C =0 whileχ 2 (z) does not converge to any complex analytic function.
As in the previous case, we obtain a power series F-dual solution where the radii of convergence of the series φ 1/q,E (z) and φq ,Ē (z) are zero and infinity respectively. Under the F-dual substitution is equivalent to the following functional equation on the function χ 0 (z): complemented with the initial value condition (67) χ 0 (0) = 1.
It admits a power series solution χ 0 (z) = ψ q,E (−iz) and its dualχ 0 (z) = ψq ,Ē (iz) with infinite radii of convergence. Thus, in this case, we obtain an entire function Ψ(x). 6. Analytical realisations of the series φ 1/q,E (z) 6.1. First order matrix difference equation. For any f ∈ F q,E , we have a matrix equality Defining (69) L n (z) := L(z)L(zq 2 ) · · · L(zq 2n−2 ) =: which, in particular, implies that Taking the limit n → ∞ in (69), relations (71) imply that where φ q,E is the regular element of F q,E . As we have seen in Subsection 2.3, it can be presented as the everywhere absolutely convergent series (19).
6.2. Wronskian pairing. We define a skew-symmetric bilinear Wronskian pairing Remark 7. One can show that dim F q,E = 3. As the kernel of the Wronskian pairing [φ q,E , ·] contains φ q,E , we conclude that dim[φ q,E , F q,E ] ≤ 2.
6.3. Adjoint functions. For any f ∈ F q,E , we associate the adjoint function .
By construction,f (z) solves the functional equation obtained from the equation underlying the vector space F q,E by the replacement of q by 1/q. This equation admits a formal power series solution φ 1/q,E (z) which has zero radius of convergence. The adjoint functions appear to be analytic substitutes for φ 1/q,E due to the following theorem.
Theorem 1. Let f ∈ F q,E be such that the adjoint functionf (z) is a non-trivial meromorphic function. Then so thatf (z) admits an asymptotic expansion at small z in the form of the series φ 1/q,E (z).
Proof. The proof is based on the matrix recurrence (68). Indeed, the formula (78) det(L n (z)) = z 3n q 3n 2 implies that L n (z) is invertible for any z = 0, and we can write and, taking into account the equality which implies (77) due to the formulae (83) lim n→∞ a n (z) = φ q,E (z/q 2 ), lim n→∞ c n (z) = φ q,E (z), see (72), and the definition of the Wronskian pairing in (73).
Our next task is to construct elements of F q,E with non-trivial adjoint functions.
7. Construction of elements in F q,E 7.1. The vector space V q,α,E . Let α ∈ C =0 . We define a vector space Proposition 2. Let g ∈ G q,E . Consider the linear map where P + is the projection to the even part of a function: Then A g (T 1 q,−α ) ⊂ V q,α,E and the restriction A g | T 1 q,−α is a linear isomorphism between T 1 q,−α and V q,α,E provided ω(g) = 0 (see Remark 6). Proof. Let h ∈ T 1 q,−α and f := A g (h). Denoting u := −1/z 2 , we have Thus, f ∈ V q,α,E .
Thus, h is determined through f : Corollary 1. For any α ∈ C =0 and E ∈ C, one has dim(V q,α,E ) = dim(T 1 q,−α ) = 1. In particular, the function determines a basis element in V q,α,E .
Proof. Indeed, ψ q,E is an element of G q,E with ω(ψ q,E ) = 2 = 0 while ϑ(αz; q) determines a basis element in T 1 q,−α . Remark 8. In the proof of the second part of Proposition 2, we implicitly used an extension of the Wronskian pairing

Proposition 3. The multiplication of functions induces a linear map
q 2 ,q 2 α and h := f g. We have Thus, h ∈ F q,E .
7.2. Adjoint functions revisited. Let f ∈ V q,α,E , g ∈ T 1 q 2 ,q 2 α . Then, the adjoint function of the product f g takes the form where, in the last expression, by abuse of notation, we extend the Wronskian pairing to include the space V q,α,E The inclusions of vector spaces in (33) specified to T 2 q 2 ,q/α become (97) T 2 q 2 ,q/α ⊂ T 2n q 2n ,q n 2 /α n , ∀n ∈ Z, which imply that and one has the equalities where (100)f := f /φ q,E .
By adjusting the normalisation of f , we can write an equality where s = s(α, q, E) is a fixed zero of [φ q,E , f ]. We conclude that

Solution of the Schrödinger equations
Based on two possible asymptotics at x → +∞, the most general Ansatz for the common eigenfunction of our spectral problem is of the form and Ξ ′ ∈ C.
Remark 9. As the bar-operation is eventually the complex conjugation, the two functions are related as follows That implies that if |Ξ ′ | = 1, then An important additional condition for this Ansatz, to be called Requirement(I), is that the functions Ψ 1 (x) and Ψ 2 (x) should share one and the same set of poles.
Putting everything together, we obtain (114) Ψ(x) = e 2πic b x e πix 2 f (z)φq ,Ē (z) + Ξe −2πi(ζ+2 sin θ)xf (z)φ q,E (z) ϑ( z s ; q 2 )ϑ( zsq α ; q 2 ) with substitutions (111) which determines a discrete set of solutions for the variable σ, while the corresponding eigenvalues of the Hamiltonian are given through the implicit function E = E(ζ, θ, σ). Given the fact that the parameter ζ is an auxiliary one, we conjecture that the eigenvalues of the Hamiltonian as well as the eigenvectors are independent of ζ. This is confirmed by numerical calculations.