Abstract
We address the spectral problem of the formally normal quantum mechanical operator associated with the quantised 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. We show that the problem can be approached in terms of the Bethe ansatz-type highly transcendental equations.
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1 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 [4, 6], together with the relation of topological strings in toric Calabi–Yau manifolds to integrable systems [1, 2, 10,11,12,13, 17, 19, 20], have lead to the conjecture on the topological string/spectral theory (TS/ST) correspondence [5, 8]. 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 \({\mathbb {P}}^2\), the corresponding operator is of the form
with invertible positive self-adjoint operators \(\varvec{u}\) and \(\varvec{v}\) such that
where \(\hbar \) is a real strictly positive parameter. With various levels of generality, the spectral problem for similar operators has been addressed in [15] from the perspective of exact WKB approximation, in [9, 18] using a matrix integral representation of the eigenfunctions and in [3, 14, 22] from the standpoint of quantum integrable systems.
In this paper, following the approach of [14, 22], we consider the so-called strongly coupled regime for the operator (1) with invertible normal operators \(\varvec{u}\) and \(\varvec{v}\) satisfying the Heisenberg–Weyl relation (2) where
In this case, we show that a realisation of the operator (1) in the Hilbert space \(L^2(\mathbb {R})\) is formally normal so that its spectral problem is still well defined provided the operator admits a normal extension. Our main result is Theorem 2 which describes a set of solutions of our spectral problem in terms of generalised the Bethe ansatz-type transcendental equations.
1.1 Outline
In Sect. 2, we introduce the setup, fix notation and conventions and introduce some objects and constructions that will be used subsequently in the rest of the paper. In Sect. 3, we specify our Hamiltonian as an unbounded operator in \(L^2(\mathbb {R})\), show that it is formally normal, rewrite the corresponding spectral problem as a pair of functional difference equations and discuss a symmetry of the system called F-duality which corresponds to Faddeev’s modularity in integrable systems. In Sect. 4, we describe the possible F-dual asymptotics of solutions of the problem. In Sect. 5, we describe formal power series solutions for each asymptotic behaviour at infinity and establish their convergence properties. In Sect. 6, we develop an analytic tool for the realisation of the formal power series solutions with zero radius of convergence as asymptotic expansions of analytic solutions. This tool is based on the important technical Theorem 1. Section 7 is devoted to explicit constructions of analytic solutions. The specific vector spaces of analytic functions on the punctured complex plane \(\mathbb {C}_{\ne 0}\) specified by linear difference equations play the central role here. In Sect. 8, we formulate and prove Theorem 2, the main result of the paper.
2 Definitions and Notation
2.1 Heisenberg Operators
Let \(\varvec{x}\) and \(\varvec{p}\) be normalised self-adjoint quantum mechanical Heisenberg operators defined by their realisation in the “position representation”
in the complex Hilbert space \(L^2(\mathbb {R})\) of square integrable functions with respect to the standard Lebesgue measure. Here, we use Dirac’s bra–ket notation so that for any \(\psi \in L^2(\mathbb {R})\) we write \( \psi (x)=\langle x|\psi \rangle , \)
if \(\psi \) is in the domain of \(\varvec{x}\), and
if \(\psi \) is in the domain of \(\varvec{p}\). One can easily verify the Heisenberg commutation relation
where in the right-hand side the scalar is assumed to be the multiple of the identity operator.
2.2 Heisenberg–Weyl Operators
We fix \(\theta \in ]0,\pi /2[\), denote
and define the normal Heisenberg–Weyl operators
together with their Hermitian conjugates
These operators satisfy (formally) the commutation relations
so that if \(\varvec{a}\) and \(\varvec{b}\) are arbitrary elements of the algebra generated by \(\varvec{u}\) and \(\varvec{v}\), then formally one has \(\varvec{a}\varvec{b}^\dagger =\varvec{b}^\dagger \varvec{a}\).
2.3 A Sequence of Polynomials
To any \(n\in \mathbb {Z}_{\ge 0}\), we associate a polynomial \(P_n=P_n(q,E)\in \mathbb {Z}[q,q^{-1}][E]\) 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 \(n\equiv 0\pmod 3\) and
where we use the standard q-Pochhammer symbol
One can also show that
where \(\alpha _{k,m}(x)\) 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:
and
The asymptotics (18) implies that for any \(E\in \mathbb {C}\), the radius of convergence is infinite for the series (20), while for the series (19) it is infinite if \(|q|<1\) and zero if \(|q|>1\).
2.4 Vector Spaces \(F_{p,c}\), \(G_{p,c}\) and \(T^m_{p,r}\)
We let \(H(\mathbb {C}_{\ne 0})\) denote the complex vector space of holomorphic functions \( f:\mathbb {C}_{\ne 0}\rightarrow \mathbb {C}\) and
Let \(c\in \mathbb {C}\), \(p,r\in \mathbb {C}_{\ne 0}\) and \(m\in \mathbb {Z}\). We define the following vector subspaces of \(H(\mathbb {C}_{\ne 0})\):
and
Elements of \(T^m_ {p,r}\) will be called theta functions of order m. For any \(p\in \mathbb {C}_{\ne 0}\) such that \(|p|<1\), one specific theta function is defined by the series
with the defining properties
which imply that \(\vartheta (\cdot ;p)\in T^1_{p,-1}\). Note also the modularity property
where we use notation (8).
Remark 1
Let \(f\in T^m_{p,r}\) and \(z\in U(f)\) (respectively, \(z\not \in U(f)\)). Then, one has \(zp^{\mathbb {Z}}\subset U(f)\) (respectively, \(zp^{\mathbb {Z}}\cap U(f)=\{\}\)).
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 \(\psi _{p,c}\in G_{p,c}\) if \(|p|\ne 1\) and \(\phi _{p,c}\in F_{p,c}\) if \(|p|<1\) so that
and
The elements \(\phi _{p,c}\) and \(\psi _{p,c}\) will be called regular elements of the corresponding vector spaces.
Remark 3
By expanding into Laurent series, it is easily verified that
with the equality if \(|p|^m<1\). In particular, for \(|p|<1\), \(\dim T^1_{p,r}=1\) with the theta function \(\vartheta (-r z;p)\) being a basis element.
Remark 4
One has the identifications
and the inclusions
which for \(n=-1\) become equalities
Remark 5
The multiplication of functions induces a linear map
For example, assuming \(|p|<1\), the product identity
illustrates the special case \(T^1_{p,-1}\otimes T^1_{p,1}\rightarrow T^{2}_{p,-1}\).
Remark 6
Assuming \(1\not \in p^{\mathbb {Z}_{\ne 0}}\), let \(g\in G_{p,c}\). Then, the even part of the product \( g(z)g(-zp) \) is an element of the vector space \(T^0_{p,1}=\mathbb {C}\). Thus, there exists a quadratic form \(\omega :G_{p,c}\rightarrow \mathbb {C}\) such that
In particular, if \(|p|\ne 1\), we have
3 Formulation of the Spectral Problem
Let \(\varvec{u}\) and \(\varvec{v}\) be the normal Heisenberg–Weyl operators defined in (9). Then, the Hamiltonian
with the standard domain for sums of operators
is expected to extend to a normal operator so that the spectral problem consisting in solving the system of Schrödinger equations
in the Hilbert space \(L^2(\mathbb {R})\) is well posed. Below we justify this expectation by showing that \({\varvec{H}}\) is at least formally normal.
3.1 The formal Normality of \({\varvec{H}}\)
Recall that a densely defined linear operator T with the domain \({\mathcal {D}}(T)\) is called formally normal if \({\mathcal {D}}(T)\subset {\mathcal {D}}(T^\dagger )\) and
A formally normal operator T is called normal if \({\mathcal {D}}(T)= {\mathcal {D}}(T^\dagger )\); see, for example, [21]. The notion of a formally normal operator naturally generalises the notion of a symmetric operator when one extends the class of self-adjoint operators to the class of normal operators.
Lemma 1
Let \(\{\varvec{a}_j\}_{j\in J}\) be a finite set of densely defined operators such that \(\varvec{A}{:}=\sum _{j\in J}\varvec{a}_j\) is densely defined, and for any \(j,k\in J\), the operator \(\varvec{a}_j+\varvec{a}_k\) is formally normal. Then \(\varvec{A}\) is formally normal as well.
Proof
The case with \(j=k\) implies that the operator \(\varvec{a}_j\) is formally normal for any \(j\in J\). This immediately implies the inclusion of domains
where the last inclusion is due to the inclusion of operatorsFootnote 1\( \sum _{j\in J}\varvec{a}_j^\dagger \subset \varvec{A}^\dagger . \)
For any \(j,k\in J\), by taking into account the inclusion of operators \(\varvec{a}_j^\dagger +\varvec{a}_k^\dagger \subset (\varvec{a}_j+\varvec{a}_k)^\dagger \) as well as the formal normality of \(\varvec{a}_j\) and \(\varvec{a}_k\), we have the following equivalent characterisation of the formal normality of \(\varvec{a}_j+\varvec{a}_k\):
where
Indeed, this is verified as follows:
Thus, for any \(x\in {\mathcal {D}}({\varvec{A}})\), we obtain
\(\square \)
Lemma 2
Let \(\mathsf {b}=e^{i\theta }\) with \(\theta \in ]0,\frac{\pi }{2}[\). Then, the set of three operators
satisfies the assumptions of Lemma 1.
Proof
For any \(j\in \{1,2,3\}\), the operator \(\varvec{a}_j\) is obviously normal. Let us check that \(\varvec{a}_1+\varvec{a}_2\) is formally normal.
By using notation (8), recall that Faddeev’s quantum dilogarithm
is a meromorphic function of \(z\in \mathbb {C}\) which satisfies the functional equation
and the unitarity condition
and it has neither zeros nor poles in the strip \(|\mathfrak {I}(z)|<\cos (\theta )\). By using these properties, we write
where \(U{:}=\Phi _{\mathsf {b}}(\varvec{p}-\varvec{x}) \) is a unitary operator and we used the following equalities:
which hold true due to the fact that the complex shifts in the argument of \(\Phi _{\mathsf {b}}(z)\) are within the strip free of zeros and poles:
Thus, \( \varvec{a}_1+\varvec{a}_2\) admits a normal extension, and in particular, it is formally normal. The formal normality of \(\varvec{a}_1+\varvec{a}_3\) and \(\varvec{a}_2+\varvec{a}_3\) is verified similarly. \(\square \)
We conclude that Lemma 1 in conjuction with Lemma 2 implies that the Hamiltonian (39) is a formally normal operator.
3.2 The Difference Equations and F-Duality
In the position representation (4), the spectral problem (41) is equivalent to the following system of functional difference equations
where \(\Psi (x){:}=\langle x|\Psi \rangle \). We are looking for a holomorphic function \(\Psi :\mathbb {C}\rightarrow \mathbb {C}\) that solves the functional Eqs. (55), (56) and whose restriction to the real axis is square integrable.
Equations (55) and (56) are related to each other by the simultaneous substitutions
which correspond to the Faddeev (modular) duality [7] and which we abbreviate as F-duality. For this reason, in what follows, we will write only one equation (containing the variables E and \({\mathsf {q}}\)), 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 solutions under above substitutions. In this case, it will suffice to check only one equation as the other one will be satisfied automatically.
4 F-dual Asymptotics at \(x\rightarrow \pm \infty \)
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 \(\Psi (x)\) be a solution of Eqs. (55) and (56) such that there exists a finite nonzero limit of \(e^{\lambda x}\Psi (x-i\mathsf {b})/\Psi (x)\) when x tends either to \(+\infty \) or to \(-\infty \) for some \(\lambda \in \mathbb {C}\). Then, one has the following possibilities for the F-dual asymptotic behaviour of \(\Psi (x)\) at large x:
and
Proof
Dividing (55) by \(\Psi (x)\), and denoting \(\rho _\lambda (x){:}=e^{\lambda x}\Psi (x-i\mathsf {b})/\Psi (x)\), we obtain a first-order nonlinear finite difference functional equation with exponentially growing or decaying coefficients
Let \(\lambda \in \mathbb {C}\) be such that there exists a finite nonzero limit value
where \(\infty \) means one of \(\pm \infty \), and assume that there exists an F-dual solution of the finite difference functional equation
Then, the corresponding asymptotic behaviour of \(\Psi (x)\) is given by \(f_\lambda (x)\).
The case \(x\rightarrow -\infty \). Choosing \(\lambda =\pi \mathsf {b}\), we obtain
Thus, one has two possibilities for the limit value
The finite difference functional equation (63) admits an F-dual solution of the form \(f_{\pi \mathsf {b}}(x)=\psi _0(x)\) provided \(\epsilon =-1\).
The case \(x\rightarrow +\infty \). We can choose either \(\lambda =-2\pi \mathsf {b}\) or \(\lambda =4\pi \mathsf {b}\) with the limit values
The corresponding F-dual solutions of (63) are given by \(f_{-2\pi \mathsf {b}}(x)=\psi _1(x)\) and \(f_{4\pi \mathsf {b}}(x)=\psi _2(x)\). \(\square \)
Our further goal is to try to solve the eigenvalue Eqs. (55) and (56) in three asymptotic regimes (58)–(60) by using the substitutions \(\Psi (x)=\psi _i(x)\varphi _i(x)\), \(i\in \{0,1,2\}\) and looking for functions \(\varphi _i(x)\) having finite nonzero limiting values in the corresponding asymptotic limits.
5 Solutions in Terms of Power Series
5.1 The Case \(\psi _2(x)\)
Assume that the asymptotic behaviour (60) takes place. If we write \(\Psi (x)=\psi _2(x)\varphi _2(x)\), then the eigenvalue equation (55) is converted into the equation
which we complement with the limit value condition
Under the F-dual substitution
(67) is reduced to the equation
which we complement with the limit value condition
It admits a power series solution
The dual function \({\bar{\chi }}_2(z)\) is obtained by the substitutions \({\mathsf {q}}\mapsto 1/\bar{\mathsf {q}}\) and \(E\mapsto {\bar{E}}\) with the result
Taking into account the remarks in the end of Sect. 2.3, we conclude that \(\chi _2(z)\) is holomorphic in \(\mathbb {C}_{\ne 0}\), while \({\bar{\chi }}_2(z)\) does not converge to any complex analytic function.
5.2 The Case \(\psi _1(x)\)
Assume that the behaviour (59) takes place. If we write \(\Psi (x)=\psi _1(x)\varphi _1 (x)\), then the eigenvalue equation (55) is converted into the equation
which we complement with the limit value condition
As in the previous case, we obtain a power series F-dual solution
where the radii of convergence of the series \(\phi _{1/\mathsf {q},E}(z)\) and \(\phi _{\bar{\mathsf {q}},{\bar{E}}}(z)\) are zero and infinity, respectively.
5.3 The Case \(\psi _0(x)\)
The substitution \(\Psi (x)=\psi _0(x)\varphi _0 (x)\) converts the eigenvalue equation (55) into the equation
which we complement with the limit value condition
Under the F-dual substitution
(77) is equivalent to the following functional equation on the function \(\chi _0(z)\):
complemented with the initial value condition
It admits a power series solution \(\chi _0(z)=\psi _{\mathsf {q},E}(-iz)\) and its dual \({\bar{\chi }}_0(z)=\psi _{\bar{\mathsf {q}},{\bar{E}}}(iz)\) with infinite radii of convergence. Thus, in this case, we obtain an entire function \(\Psi (x)\).
6 Analytic Realisations of the Series \(\phi _{1/\mathsf {q},E}(z)\)
6.1 First-Order Matrix Difference Equation
For any \(f\in F_{\mathsf {q},E}\), we have a matrix equality
Defining
we have
which, in particular, implies that
Taking the limit \(n\rightarrow \infty \) in (83), relations (85) imply that
where \(\phi _{\mathsf {q},E}\) is the regular element of \(F_{\mathsf {q},E}\). As we have seen in Sect. 2.3, it can be presented as the everywhere absolutely convergent series (19).
6.2 The Wronskian Pairing
We define a skew-symmetric bilinear Wronskian pairing
Remark 7
One can show that \(\dim F_{\mathsf {q},E}=3\). As the kernel of the Wronskian pairing \([\phi _{\mathsf {q},E},\cdot ]\) contains \(\phi _{\mathsf {q},E}\), we conclude that \(\dim [\phi _{\mathsf {q},E},F_{\mathsf {q},E}]\le 2\).
6.3 Adjoint Functions
For any \(f\in F_{\mathsf {q},E}\), we associate the adjoint function
(see (21) for the definition of the domain) defined by the formula
By construction, \({\tilde{f}}(z)\) solves the functional equation
obtained from the equation underlying the vector space \(F_{\mathsf {q},E}\) by the replacement of \({\mathsf {q}}\) by \(1/\mathsf {q}\). This equation admits a formal power series solution \(\phi _{1/\mathsf {q},E}(z)\) which has zero radius of convergence. The adjoint functions appear to be analytic substitutes for \(\phi _{1/\mathsf {q},E}\) due to the following theorem.
Theorem 1
Let \(f\in F_{\mathsf {q},E}\) be such that the adjoint function \({\tilde{f}}(z)\) is a non-trivial meromorphic function. Then
so that \({\tilde{f}}(z)\) admits an asymptotic expansion at small z in the form of the series \(\phi _{1/\mathsf {q},E}(z)\).
Proof
The proof is based on the matrix recurrence (82). Indeed, the formula
implies that \(L_n(z)\) is invertible for any \(z\ne 0\), and we can write
so that
and taking into account the equality
we obtain
which implies (91) due to the formulae
see (86), and the definition of the Wronskian pairing in (87). \(\square \)
Our next task is to construct elements of \(F_{\mathsf {q},E}\) with non-trivial adjoint functions.
7 Construction of Elements in \(F_{\mathsf {q},E}\)
7.1 The Vector Space \(V_ {\mathsf {q},\alpha ,E}\)
Let \(\alpha \in \mathbb {C}_{\ne 0}\). We define a vector space
Proposition 2
Let \(g\in G_{\mathsf {q},E}\). Consider the linear map
where \(P_+\) is the projection to the even part of a function:
Then \( A_{g}(T^1_{\mathsf {q},-\alpha })\subset V_{\mathsf {q},\alpha ,E}\) and the restriction \(A_{g}\vert _{T^1_{\mathsf {q},-\alpha }}\) is a linear isomorphism between \(T^1_{\mathsf {q},-\alpha }\) and \(V_{\mathsf {q},\alpha ,E}\) provided \(\omega (g)\ne 0\) (see Remark 6).
Proof
Let \(h\in T^1_{\mathsf {q},-\alpha }\) and \(f{:}{=}A_g(h)\). Denoting \(u{:}{=}-1/z^2\), we have
Thus, \(f\in V_{\mathsf {q},\alpha ,E}\).
Assuming \(\omega (g)\ne 0\), we solve the equality \(f=A_g(h)\) for h as follows:
Thus, h is determined through f:
\(\square \)
Corollary 1
For any \(\alpha \in \mathbb {C}_{\ne 0}\) and \(E\in \mathbb {C}\), one has
In particular, the function
determines a basis element in \(V_ {{\mathsf {q}},\alpha ,E}\).
Proof
Indeed, \(\psi _{{\mathsf {q}},E}\) is an element of \(G_{{\mathsf {q}},E}\) with \(\omega (\psi _{\mathsf {q},E})=2\ne 0\), while \(\vartheta (\alpha z;{\mathsf {q}})\) determines a basis element in \(T^1_{{\mathsf {q}},-\alpha }\). \(\square \)
Remark 8
In the proof of the second part of Proposition 2, we implicitly used an extension of the Wronskian pairing
and the identity
Proposition 3
The multiplication of functions induces a linear map
Proof
Let \(f\in V_{{\mathsf {q}},\alpha ,E}\), \(g\in T^1_{{\mathsf {q}}^2,{\mathsf {q}}^2\alpha }\) and \(h{:}{=}fg\). We have
Thus, \(h\in F_{{\mathsf {q}},E}\). \(\square \)
7.2 Adjoint Functions Revisited
Let \(f\in V_{{\mathsf {q}},\alpha ,E}\), \(g\in T^1_{{\mathsf {q}}^2,\mathsf {q}^2\alpha }\). Then, the adjoint function of the product fg takes the form
where, in the last expression, by abuse of notation, we extend the Wronskian pairing to include the space \(V_{{\mathsf {q}},\alpha ,E}\)
The inclusions of vector spaces in (33) specified to \(T^2_{{\mathsf {q}}^2,{\mathsf {q}}/\alpha }\) become
which imply that
and one has the equalities
where
By adjusting the normalisation of f, we can write an equality
where \(s=s(\alpha ,{\mathsf {q}},E)\) is a fixed zero of \([\phi _{{\mathsf {q}},E},f]\). We conclude that
8 Solution of the Schrödinger Equations
Define the notation
Theorem 2
Let \((\zeta ,\sigma ,E,{\bar{E}})\in \mathbb {C}^4\) be such that
where
and
Then, the function
is a holomorphic solution of the system of difference Eqs. (55), (56).
Proof
Based on two possible asymptotics at \(x\rightarrow +\infty \), the most general ansatz for the common eigenfunction of our spectral problem is of the form
where
and \(\Xi '\in \mathbb {C}\). \(\square \)
Remark 9
If the bar operation is the complex conjugation, the two functions are related as follows
That implies that if \(|\Xi '|=1\), then
An important additional condition for this ansatz, to be called Requirement(I), is that the functions \(\Psi _1(x)\) and \(\Psi _2(x)\) should share one and the same set of poles.
If one chooses nonzero \(h=f g\) and \(\bar{ h}=\bar{ f }\bar{ g}\), where
for some \(\alpha ,{\bar{\alpha }}\in \mathbb {C}_{\ne 0}\), then the denominators simplify to \( [\phi _{\mathsf {q},E},f] \) and \( [\phi _{\bar{\mathsf {q}},{\bar{E}}},{\bar{f}}]\) [(see eq. (110)] to become elements of 2-dimensional vector spaces \(T^2_{\mathsf {q}^2,\mathsf {q}/\alpha }\) and \(T^2_{\bar{\mathsf {q}}^2,\bar{\mathsf {q}}/{\bar{\alpha }}}\), respectively. By adjusting the normalisations of f and \({\bar{f}}\), we can assume that
where
are some chosen zeros of \([\phi _{\mathsf {q},E},f]\) and \([\phi _{\bar{\mathsf {q}},\bar{ E}},\bar{ f}]\), respectively.
Remark 10
By solving Eq. (130) for \(E=E(s)\) and \(\bar{E}=\bar{ E}(\bar{ s})\), one can think of s and \(\bar{ s}\) as independent variables.
In order to fulfil Requirement(I), to rewrite \([\phi _{\mathsf {q},E},f](z)\) by using the substitutions
and the modularity property (27)
Thus, Requirement(I) is fulfilled if we substitute in \([\phi _{\bar{\mathsf {q}},\bar{ E}},\bar{ f}]\)
In this way, the first two relations in (119) become equivalent to (130).
Putting everything together, we obtain
with substitutions (131), (133) and \(\bar{ z}\mapsto \bar{ p}_x\), and the parameter
that we choose by the condition of cancellation of the pole of \(\Psi (x)\) at \(x=\sigma \) with the result
equivalent to (121) under the substitutions \(s\mapsto p_\sigma \) and \(\bar{ s}\mapsto \bar{ p}_\sigma \).
Remark 11
Equality (136) implies that all the poles of \(\Psi (x)\) at \(\sigma +i\mathsf {b}m+i\mathsf {b}n\), \(m,n\in \mathbb {Z}\), are cancelled as well. The proof is based on the relations (114) and their complex conjugate counterparts.
Now, the last step in the proof is to fulfil Requirement(II) which consists of cancelling the remaining poles of \(\Psi (x)\). Due to the equivalence (117) and Remark 11, Requirement(II) boils down to a single equation
which determines a discrete set of solutions for the variable \(\sigma \), while the corresponding eigenvalues of the Hamiltonian are given through the implicit function \(E=E(\sigma )\).
In conclusion, we would like to remark that the variable \(\zeta \) in Theorem 2 is an auxiliary parameter whose role is unclear to us. We conjecture that the eigenvalues of the Hamiltonian as well as the eigenvectors are independent of \(\zeta \). This is confirmed by numerical calculations.
Notes
By inclusion of operators \(S\subset T\), we mean that \({\mathcal {D}}(S)\subset {\mathcal {D}}(T)\) and \(S=T\vert _{{\mathcal {D}}(S)}\).
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Acknowledgements
We would like thank Vladimir Bazhanov, Vladimir Mangazeev, Marcos Mariño and Szabolc Zakany for valuable discussions. We also would like to thank the anonymous referee whose remarks were very helpful for improving the mathematical rigour of our presentation, especially the part that concerns the formal normality of the Hamiltonian. The work is partially supported by the Australian Research Council and Swiss National Science Foundation.
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Communicated by Jan Derezinski.
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The work is partially supported by Australian Research Council and Swiss National Science Foundation.
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Kashaev, R., Sergeev, S. On the Spectrum of the Local \({\mathbb {P}}^2\) Mirror Curve. Ann. Henri Poincaré 21, 3479–3497 (2020). https://doi.org/10.1007/s00023-020-00960-y
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DOI: https://doi.org/10.1007/s00023-020-00960-y