Exact WKB Analysis and TBA Equations for the Stark Effect

We apply the exact WKB analysis to a couple of one-dimensional Schroedinger-type equations reduced from the Stark effect of hydrogen in a uniform electric field. By introducing Langer's modification and incorporating the Stokes graphs, we prove the exactness of the Bohr-Sommerfeld quantization conditions for the Borel-resummed quantum WKB periods in the specific parameter regions of the electric field intensity and magnetic quantum number. It is also found these quantization conditions get modified with an additional suppressed contribution when the parameters vary beyond the specific regions. We also present Thermodynamic Bethe Ansatz (TBA) equations governing the quantum periods in the absence of Langer's modification and discuss its wall-crossing and analytic continuation. Numerical calculations are conducted to compare the complex resonant frequencies from our quantization conditions against ones from the Riccati-Pade method, the TBA equations are also confirmed by comparing its expansions with all-order quantum periods.


Introduction
The Stark effect describing hydrogen under an external electric field [1], is one of the earliest examples considered since the invention of quantum mechanics.The external field leads bound states for hydrogen to resonant states, whose energy becomes complex E = E c + iΓ/2.E c represents the peak of the energy spectra and Γ stands for the width related to the ionization rate.There are many approaches based on perturbation theory to solve this problem [2,3,4,5,6].But the perturbative expansions for the Stark effect are asymptotic, so additional numerical methods such as Borel-Padé technique are necessary.There are also WKB-type approaches that rely on the fact that the three-dimensional Schrödinger equation for the Stark effect is separable under parabolic coordinates.Both of the separated equations take the form of the one-dimensional Schrödinger equation with a linear and inverse potential in addition to a centrifugal term.To satisfy the correct boundary condition at origin, Kramers and especially Langer [7] introduce the so-called Kramers-Langer substitution or Langer's modification used in this paper, see [8].A comprehensive interpretation can be found in [9] in terms of two-notation in an enlightening perspective.The successive research on the WKB method for the Stark problem is almost based on Langer's modification, such as [10,11,12,13].These works mainly rely on the period integrals in the classically allowed region, which will be notated as classical WKB periods.
The main purpose of this paper is to study the Stark effect by exact WKB analysis.This can be achieved by the Borel resummation of formal power series in the framework of resurgence theory [14,15,16,17].The analytic behavior of the Borel-resummed WKB solution is determined by Stokes graphs, and the relation of WKB solutions for different Stokes regions is described by connection formulas [18,19,20].In quantum mechanical problems, one imposes boundary conditions for the wave functions, these deduce the exact quantization conditions (EQCs) for the eigenvalue problems in terms of quantum period [21,22,23,24].It is Borel-resummed quantum WKB periods appearing to connect the solutions with different normalizations.Its discontinuity is captured by the Delabaere-Pham formula (DP formula) [25,26].In this paper, we apply the exact WKB analysis to the Stark effect to obtain the exact quantization conditions.
The development of WKB analysis invoked the study of integrable models and is formed into the so-called ODE/IM correspondence [27,28].ODE/IM correspondence indicates there is a relationship between the Y functions in integrable models and quantum periods from WKB analysis.Specifically, Y systems derived from the Wronskian relations coincide with equations followed by quantum periods from the DP formula [29].These studies on the one hand introduced new techniques into integral models, on the other hand, fed back the research on the ODE side.Especially the ODE/IM correspondence introduces TBA equations for quantum periods, which provides a non-perturbative completion for these asymptotically defined quantities in WKB analysis.This correspondence was summarized explicitly for polynomial potentials [29,30] and also potentials with regular singularity [31].As one analytically continues the parameters in the potential, TBA equations get modified when wall-crossing of quantum periods happens [32].We would like to consider these kinds of TBA equations corresponding to the reduced one-dimensional equations for the Stark effect.This paper is organized as follows.In section 2, we review the fundamentals of exact WKB analysis.Our work starts from section 3, which focuses on the application of exact WKB analysis to the Stark effect.We consider Langer's modification and all-orders quantum corrections and write down the exact quantization conditions (EQCs) from the Stokes graph.It is found the Bohr-Sommerfeld quantization condition for the Borel-resummed periods is already exact for weak field strength but gets an additional term when the electric field becomes strong.Our EQCs are numerically tested against the Riccati-Padé method [33].Section 4 is devoted to finding integral TBA equations for quantum periods.However, these TBA equations are established for bare effective potentials without Langer's modification, which is different from the prescription adopted in section 3. We discussed the wall-crossing of the TBA equations [32] and also the analytic continuation [34,35] against the parameters for the Stark effect.We conclude our results in section 5.

Exact WKB analysis and connection formulas
In this section, we give a brief review to mention fundamental aspects and fix the notations of the exact WKB analysis applied later.We intensively consult the work of [18,19,20,23,24] and the references therein for this section.

Review of the all-orders WKB expansions
The Stark effect involves a couple of one-dimensional Schrödinger equations as discussed in section 3. Let us first consider the equation for a generic potential Q(x, ) where consists of one classical potential Q 0 (x) , and an additional dependent term Q 2 (x) 2 , usually called quantum potential.The coordinate x is a complex variable, and we regard the reduced Planck constant as an expansion parameter, which is also taken to be complex and set to 1 in the end.One starts by setting a WKB ansatz where is a formal power series.Substitute the above solution for wave function into (2.1), then we arrive at the Riccati equation This equality holds to all orders, matching order by order reduces to the following recursive relations for p n (x) with The odd and even power of P (x, ) was proven splitting by Hereafter we omit dependent in function P (x, ) for short.Then the WKB solution (2.3) can be expressed in terms of P even (x) as where a is a reference point for the integral, and is often exclusively chosen to be a turning point, that satisfies Q 0 (a) = 0.The prescription ± in the exponential signifies the properties of the solutions, exponentially suppressed, exponentially growing, or oscillatory in the given region of x.The generic solution normalized at a consists of the linear combination of two independent components ψ ± a (x) (2.10) One can formally include the all-order quantum corrections into the wave function by expanding (2.9) to orders of Hereafter, we specify p 0 (x) = Q 0 (x).This formal series includes an exponential prefactor and formal power series with its coefficients x-dependent.It is of course divergent in general.The Borel resummation provides a procedure to deal with this sort of divergent series, next we will consider its utilization into formal series (2.11).

Borel resummation and Stokes graphs
We consider a generic form of formal series with exponential prefactor, which appears at the formal expansion of the WKB solution for wave function (2.11) where A is a constant independent of and a n diverges factorially a n ∼ n!.Its Borel transform is which transforms the divergent formal series (2.12) into a well-defined function in the new variable z.This function has a finite convergent radius and can be analytically continued to a domain including the real z-axis, with possible singularities, such as poles.We often call this complex z-plane the Borel plane.If there is no singularity along the positive real line, one can define the Laplace transform of Borel-transformed function B[f ](z) as it is the Borel resummation or simply Borel resum of the original formal series (2.12), and we say the formal series (2.12) is Borel summable in this case.The directional Borel resum can be defined similarly if there is no singularity in angle θ One may note this sort of directional summation is equivalent to writing as which indicates the Borel resum for formal series at the angle θ is equivalent to the resum at angle 0 by rotating to e iθ .We will omit the subscript 0 in S θ when θ = 0.It is much more fruitful to look at the singularities in the Borel plane, let us simply consider there is a single pole along θ ray, then the integral defined in (2.15) makes no sense.But we can bypass the pole by small derivation from above and below, this suggests defining the lateral Borel resum for (2.12) as (2.17) Let us now consider the formal series expansion for wave function (2.11), we will study its Borel singularities and Borel summability.The general Borel resum procedure is applicable, but the expansion coefficients ψ ± a,n are parameter-dependent on x.This parameterdependence promotes the singularities in the Borel plane to be x dependent and then defines curves in coordinate x plane, that is Stokes lines introduced later.At first, We do Borel transform of (2.11) to suppress the divergence Now we obtain a convergent function defined in some domain in the z-plane, it is generally meromorphic with possible singularities and one can further continue the function into the whole z-plane.Its Laplace transform can be defined accordingly unless the integral contour encounters Borel poles.
Figure 2.1 shows the definition of S θ [ψ + a ]( ) for θ = 0. Hereafter we will use ψ ± a (x) to represent Bore-resummed wave functions for simplicity without confusion.It is proved that ±z 0 (x) are singular points of the Borel transform (2.18) [18].The location of ±z 0 (x) depends on the coordinate x, so the integral contour is able to touch the Borel pole z 0 (x) in figure 2.1 by moving in the complex x plane.This condition can be expressed as   For each turning point, one can define the corresponding Stokes lines, and there are three Stokes lines for a simple turning point, i.e. the simple zero of Q 0 (x). Figure 2.2 shows the Stokes lines eliminating from simple turning point a to complex infinity.One Stokes line might start from one turning point, the origin, or infinity and end at another turning point, the origin or infinity.The Stokes line connecting two turning points is a degenerate line and can be resolved by saddle reduction via introducing a small derivation for integral contour θ → θ ± δ.
The Stokes lines divide the coordinate plane into several adjacent regions, usually called Stokes regions.The collection of all the Stokes lines composes Stokes graphs.Within each Stokes region, the formal series for wave functions is Borel summable to obtain the analytic solution for each region.However, if one continues the solution in one region to another adjacent region by crossing a Stokes line, one of the components in ψ ± gets a discontinuity while the other one remains depending on the orientation of the Stokes line.This is quantitatively described by the connection formula in the subsequent section.

Connection formulas and quantum periods
We know Borel singularities for wave functions, which are captured by Stokes curves qualitatively, now we want to know the quantitative discontinuities when the wave functions in one Stokes region cross the Stokes line to an adjacent region.In general, this property is concluded by the following connection formula [20,23].
• If crossing a Stokes line anticlockwise at which the ψ + is dominant, ψ + gains a discontinuity controlled by ψ − , while ψ − doesn't change, As aforementioned, the symbol for the Borel resum is omitted.
• If crossing the Stokes line anticlockwise at which the ψ − is dominant, ψ − gains a discontinuity controlled by ψ + , while ψ + remains unchanged, • If crossing a Stokes line clockwise, the formulas differ by flipping the + to − in front of the i, • If crossing a branch cut emerging from a turning point anti-clockwise (clockwise), the dominant and subdominant component exchange (with a minus sign), One usually signs the corresponding matrix to represent this formulation which is often called the monodromy matrix.Then the connection formulas can be stated that the WKB solutions in one Stokes region can be obtained via multiplication of the monodromy matrix by the solutions in the adjacent region, specifically which is represented in figure 2.2 where M = M + .
Furthermore, the two independent wave functions normalized in a turning point a 1 denoted as ψ ± a 1 , and ψ ± a 2 normalized in another turning point a 2 are related by normalization coefficients N ± a 1 a 2 defined as which induces the following connection formula to relate wave functions normalized in a 2 to a 1 (2.28) In the latter context, we will intensively use the following notation where γ is the cycle encircling two turning points a 1 and a 2 .It is usually called Voros multipliers or Voros symbols in the literature.As the same as the monodromy matrix, we can express the above connection formula in a matrix form, which transforms the wave function basis at turning point a 2 to a 1 At the equal significance, we rename the quantity γ P even (x)dx the quantum period or quantum WKB period corresponding to the cycle γ It's now necessary to re-explain notations.Corresponding to (2.1), there exists a classical curve it is called the WKB curve.Geometrically it defines a Riemann surface Σ WKB , and the quantum period is the integral of meromorphic differential P even (x)dx along the one-cycle γ ∈ H 1 (Σ WKB ).The leading order of the quantum period is often called the classical WKB period or classical period which plays an important role in the history of quantum mechanics.The well-known Bohr-Sommerfeld (BS) quantization condition can be written in terms of the classical period as where γ is specified to be one cycle around the classically allowed region.It is widely used for two-turning-point problems by accounting for the classical period integral encircling turning points a 1 and a 2 in the classically allowed region.Here i appears since our notation for p 0 (x) = V (x) − E, which is different from the conventional choice We should carefully note P even (x) is defined asymptotically by recursive relation, so Voros symbols and also quantum periods are divergent formal series of , for instance with its coefficient diverging double-factorially Since the WKB solutions in (2.28) are resummed well-defined functions.One should regard the Voros symbols and quantum periods as Borel-resummed quantities, but we will omit the notation of Borel resummation S for short.
Until now, we have introduced Stokes graphs which are fully determined by the classical potential Q 0 (x), and connection formulas based on the all-orders wave function and its Borel summability.With these ingredients, we are able to analytically continue the WKB solution defined in one Stokes region to another region and rewrite the solutions in one Stokes region as the linear combination of the wave basis for another region.In the content of WKB analysis, these combination coefficients rely on Voros symbols.In practice, proper boundary conditions are imposed for interested problems, which give an additional condition for these coefficients.The equality that fulfills boundary conditions is called the exact quantization condition, which is a functional relation in terms of Voros symbols as follows where V γ i with i = 1, 2, • • • , r are Voros symbols on the relevant one-cycle γ i .For example, the EQC for harmonic oscillator can be written as which is the BS quantization condition (2.34).We intend to drive the EQCs for the Stark problem in the next section by following the above strategy.

Stark Effect and its quantization conditions
Let us first write down the Schrödinger equation for the stark effect of hydrogen in a uniform electric field F oriented in the z-axis: where atomic units are adopted, and E represents the energy.r denotes the radial coordinate.We remain reduced Planck constant , which is set to 1 in the end.

Langer's modification and boundary conditions
We introduce the parabolic coordinates (ξ, η, ϕ) which are defined by where r = x 2 + y 2 + z 2 , 0 ≤ ξ, η ≤ +∞ and 0 ϕ 2π.If we consider an ansatz for wave function Ψ as the three-dimensional Schrödinger equation (3.1) reduces to a couple of Schrödinger-like equations with centrifugal term where A 1 and A 2 are separation constants satisfying A 1 + A 2 = 1.One can identify the parameters above with that of potential in the standard form (2.1) with to find for the equation corresponding to ξ and η coordinates respectively.In both cases, orbital quantum number and azimuthal quantum number m are related by It is proven that the above equations are appropriate to the WKB analysis including allorder expansions, however, they are not capable to grasp the leading order contribution to the wave function properly.We use the generalization of the Kramers-Langer substitution [8,9] by or in terms of m as Here another parameter i is introduced as an implicit factor, while is an explicit expansion parameter.The different partitions of the centrifugal term as above represent different expansion schemes but give the same physical solution when i = .The WKB expansions for (3.9) do not match term by term except for the first order and infinite order, see [9] for detailed discussion.Since i is not involved in power expansions, we can directly set it to 1 at the beginning.This generalized Langer's modification suggests splitting the centrifugal term into two parts as Now we can set for ξ coordinate and for η coordinate.Let us then check the behaviors of the wave functions at the boundaries.
It is enough to take account of the leading order WKB solution of (2.9) at this stage We temporarily set = 1 for clarity, and consider the ξ-equation at first.Near the origin, the inverse square term dominates the wave function Then the wave function near the origin behaves like where (3.7) is used at the last step.We then obtain one normalizable solution ψ + 1 (ξ), that is proper for the Stark problem, and one divergent solution ψ − 1 (ξ).Now let us turn to consider the behavior of the solution at infinity, the dominant contribution to p 0 (ξ) is It is obvious to find the following asymptotic approximation We get one increasing solution and one exponentially suppressed solution, the latter is what we expect, namely ψ − 1 (ξ) at the asymptotic infinite region.The wave function for the η-equation can be analyzed in the same way.Near the origin, it is entirely the same as that for the ξ-equation.
The boundary condition requires the ψ + 2 component to survive.At large η region, the dominant part for p 0 (η) changes to then the wave function asymptotically behaves like This time we obtain one outgoing wave ψ + 2 (η) and one ingoing wave ψ − 2 (η), the former satisfies the required boundary condition.
Then we are going to apply connection formulas in section 2 to relate the wave functions from one Stokes region to another region and impose the above boundary conditions, which generates a functional relation for the quantum periods and gives us exact quantization conditions for the Stark problem.

Quantization conditions
The quantization conditions can be derived from the Stokes graphs incorporating boundary conditions discussed in the last section.Let us now focus on the Stokes graph for the ξ-equation in the weak F region, its Stokes graph is shown in figure 1(a).To derive the quantization condition, we have to know the relation of wave functions between origin and infinity, which can be accomplished by using connection formulas in section 2. One can continue the solutions at infinity to the origin along the green arrow depicted in figure 1(a), this procedure can be written as follows.
1. We start from the wave function basis in Stokes region I ∞ normalized at a, it is necessary at first to cross the branch cut which exchanges the exponentially suppressed and growing solutions, this progress can be written as where the operator ι maps the function in one square sheet to one in another sheet.
2. Now we are working in the same sheet, we continue the wave functions ψ ± a,I∞ (ξ) to cross the Stokes line labeled by + anticlockwise to get basis in the intermediate region I m normalized at the same turning point a by the multiplication of the monodromy 5. We finally pull back the normalization from b to a for comparison.
Accounting for all these transformations, it's apparent the two bases are related by the matrix multiplication as here V γ is the Voros symbol, and γ is the trajectory enclosing turning points b and a.This expression relates the wave function basis at positive infinity to that around the origin.We note that this formula is very similar to the Q-function in the ODE/IM correspondence [27] and also the numerical method used in [5,6].The boundary conditions for (3.15) and (3.17) require ψ − a,I 0 and ιψ + a,I∞ to vanish, which deduce the following quantization condition We can then write down the Bohr-Sommerfeld-type quantization condition where n ξ denotes the quantum number corresponding to ξ coordinate.Let us emphasize that the left side quantum periods should be understood as its Borel resum.This quantization condition was presented for the Stark problem a long time ago such as [12], where the leading order contribution is considered.Here we derive it rigorously and show it is exactly satisfied when taking quantum correction into consideration.Then this Bohr-Sommerfeld quantization condition for the Borel-resummed quantum periods provides the exact condition.
Next, we turn our attention to the Stokes graph for ξ-equation with a strong F .The Stokes graph changes its topology when F is enlarged, as described in figure 1(a) and 1(b).We follow a similar process as discussed before.the connection of the solutions from infinity to the origin for 1(b) is given by the matrix multiplication Then the boundary condition requires which is the exact quantization condition for the large F region.where we use the fact We can write the quantization condition of the modified Bohr-Sommerfeld type from (3.31) as Here Π γ ( ) and Π γ 1 ( ) are also Borel-resummed quantities.It was found that V γ 1 gives an exponentially suppressed contribution, so the BS quantization condition still gives an approximation in the small limit.The additional logarithmic term plays a role of nonperturbative correction, which becomes significant when F is large.During this process, turning points b and c are connected by one Stokes line at an angle ϕ with 0 < ϕ < δ.And quantum period Π γ ( ) exhibits singularities in its Borel plane at direction ϕ, the discontinuity of the quantum period is given by the Delabaere-Pham (DP) formula [25] exp here S ± are two lateral Borel summations for θ = 0 defined in section 2. The DP formula shows the EQC or complex resonant frequency is continuous during the transition of the Stokes graphs.Furthermore, it gives the resurgent structure of the quantum periods which indicates the discontinuity of the perturbative contribution is governed by the non-perturbative term.See [36,37] for a similar discussion.
The derivation of quantization conditions for η-equation is parallel to the above analysis, the critical condition for the mutation of the Stokes graph is given by which divides parameters in the potential into the weak F region and the strong F region accordingly.We can similarly denote F η,c (m) as the critical value of the transition for fixed m, an estimation for m = 1 implies F η,c (1) ∈ (0.5, 1).The boundary condition sorts out the ψ + (η) component at origin and infinity, and there is no relevant branch cut to cross.
One can write down the relation of the wave functions by matrix multiplication for small F and for large F from Stokes graphs figure 1(c) and 1(d) respectively.We use M −1 + because of the clockwise continuation.So the quantization condition can be written as , for small F region, (3.38) , for large F region, (3.39)where n η takes values in non-negative integers.The quantum periods are Borel-resummed as well.Similarly, the last term contributes exponentially small, it can be neglected in the semi-classical limit.The continuity of the two EQCs comes from (3.34) by a similar argument.

Computation and results
Now we utilize the quantization conditions in the last subsection to compute complex resonant frequencies and compare them with the results from the Riccati-Padé method [33] that shows very high precision.We first establish the classical periods in terms of elliptic integrals.
Classical periods Let us parametrize the potential Q 0 (ξ) in the ξ-equation as where the coefficients in the potential are related to Stark parameters as (3.11).We consider the relevant period corresponding to one-cycle γ which appears in the quantization condition.The integrals for one-cycle γ 1 can be obtained by the rotation of roots a → b, b → c and c → a.The classical period is where (3.42) K(k), E(k), and Π α 2 , k are the elliptic integrals of the first, second, and third kind [38].
The numerical calculation is a bit subtle since the turning points and integral contours are all in the complex coordinate plane, one should be careful about the branch cuts and sheets.Similarly, for the η-equation, we parametrize the potential Q 0 (η) in (3.12) as The classical period is evaluated as here u 0 = − F 4 is negative, which gives an overall imaginary unit i.These formulas have already appeared in the literature like [12], but the conventions are different from an imaginary unit i.It is almost impossible to compute quantum period by direct integration, yet there is a systematic method based on the Picard-Fuchs equation to evaluate these quantities.It can be sketched by introducing the following fundamental periods.

Fundamental periods
We first define the differential We use x to represent ξ or η uniformly for simplicity.It was found that all p 2n (x) can be expressed by the combination of the above fundamental differentials up to a total derivative The expansions for the quantum periods can be evaluated by determining the Picard-

Fuchs coefficients c
(n) i and the fundamental periods, which are defined as And the second row is for the Riccati-Padé method.
[a] means 10 −a , this notation is also used later.
It is straightforward to calculate the complex energy by simultaneously solving quantization conditions for ξ-and η-equations.We show the numerical result for F = 0.03, 0.1, 10 and m from 0 to 3 with half unit in table 3.1.For the m = 0 case, the turning point b collapses to 0, and the elliptic integral of third kind Π(α 2 , k) becomes zero.We adopt BS quantization conditions (3.29) and (3.38) for F = 0.03 and F = 0.1 and modified BS quantization conditions (3.33) and (3.39) for F = 10.At this stage, we only consider the first-order contribution, namely Π (0) .For convenience to compare with other methods, both quantum numbers n ξ and n η are set to 0. In fact, it is the worst case for the application of the WKB method, since it approximates much better for large n states.We find even the first-order WKB analysis gives good estimation for a wide range of parameter choices.As a matter of fact, F = 0.03 is already a strong field intensity in the atomic unit.
If one examines for very weak field F , the result shows high precision as in [12].The calculation indicates even for a very strong field, for example, F = 10 here, the approximation is also efficient.
We also find the effect of the logarithmic term in modified BS quantization conditions (3.33) and (3.39).Table 3.Here only contributions from classical periods Π (0) are included.
One of the new results in this paper is to include the quantum corrections to Π (0) .Table 3.3 exhibits the results for F = 0.005 by including the first several orders of quantum periods.The energy level for the ground state is really precise, the imaginary part of complex energy for this case is very small3 and neglected here.However, the improvement for m = 3 is not so satisfactory.One quick reason may be that the quantum corrections are very large compared with classical periods for this case, where the numerical approximation performs not well even the Borel-Padé technique is adopted.It is not expected in general that the present method entirely matches with standard perturbation theory results for each order of except for the first order.Since our method is based on Langer's modification which is just a kind of expansion scheme.Different schemes give different results, but they all coincide to include all-orders expansions.In practice, we use the first several leading orders to approximate the exact Borel-resummed quantum WKB periods as a numerical approach.

TBA equations and analytic continuation
The quantum period is the fundamental ingredient in the WKB analysis, however, it can be established from another approach based on the ODE/IM correspondence.Since Langer's modification is not necessary for this analysis, we will not consider the modification in this section.But both setups are the same when m = 0.

Singular potentials and quantum periods
Let us now revisit the Stark problem without Langer's modification to reconsider (3.4), which is uniformly written as (3.5).We will consider quantum periods from WKB expansions at first.We use the same notations for Q 0 (x), Π( ), and γ which should not be confused with the prescription adopted in the last section.The classical and quantum periods can be written in terms of elliptic integrals.
Classical periods Denote two turning points e 1 and e 2 of Q 0 (x) in (3.5): One can define three one-cycles on the Riemann surface defined by y 2 = Q 0 (x), which encircle e 1 and e 2 , 0 and e 1 , 0 and e 2 and are denoted as γ, γ1 and γ2 respectively.Then the classical periods corresponding to these three cycles are represented by the elliptic integrals as where m = e 2 e 1 and k = e 2 e 1 − 1.These periods are related by their definitions 3) The quantum periods are evaluated by the Picard-Fuchs method as well, which is composed of a combination of fundamental periods.
Fundamental periods Define the fundamental differentials first for Q 0 (x): and the associated fundamental periods They are represented as As the same as (3.46), there is a similar relation Moreover, the higher-order expansions of the Y function for large θ are consistent with all-order quantum periods.When the mass parameters m 1 and m are complex, we set then the driving term in (4.15) becomes m 1 e θ = |m 1 |e θ+iφ 1 and me θ = | m|e θ+i φ.We shift These equations hold when which defines a region in the space of parameters u 0 , u 1 , and u 2 .It is called the minimal chamber.If we rotate to redefine a new constant ˜ = i , then the equation for η coordinate becomes

.20)
We found the parameters for the ground state with m = 0 lie in the minimal chamber now.We solve these TBA equations iteratively via the Fourier discretization by taking m 1 and m as the initial seed and compute the first several order expansions by the large θ expansion for log Y (θ) where α 1 ( ) is the zero of 1 − e 2πi Ŷ (θ) in the |Im(θ)| < π 2 strip, and is determined by

.25)
The algorithm for solving these TBA equations refers to [35].In this case, expansions of the quantum periods are also modified by α 1 ( ).
However, it seems the parameters u 0 , u 1 , and u 2 for > 0 are already outside the minimal chamber.But we can test the relationship (4.23) for some fictitious value, u 0 = 1, u 1 = −3, u 2 = 1 and = 1 5 to confirm this correspondence holds on with correction.This is not covered in [35].We list the numeric results in table 4.2 to compare with those evaluated from the Picard-Fuchs method.See also [40] for further discussions.
As increases steadily, the second singular point for the Y function gets involved, and the enters into the second modified region 0.28 0.5.The TBA equations for this region are where α 1 ( ) and α( ) are determined by In this case, quantum periods are modified by α 1 ( ) and α( ) as follows  [5]i − 3.62538588259 [4] Table 4.4: The first two order quantum periods with F = 0.03, m = 1 2 for ξ equation.
We consider a specific case F = 0.03 with m = 1 2 .The turning points for (4.32) are ± √ e 1 and ± √ e 2 with e 1,2 the turning points for (4.14).Then the classical periods are defined as q,1 = 2 √ e 1 Q q 0 (x)dx, for η variable, (4.33) here Q q 0 (x) = 4u 0 z 4 + 4u 1 z 2 + 4u 2 is the quartic potential.The higher orders can be evaluated from the Picard-Fuchs method as well.let us now consider the TBA equations for (4.32).For parameters in η variable, the corresponding TBA equations are in the (4.34) The TBA equations for the ξ variable are now in the maximal chamber.Equation (5.21) in [29] gives the explicit form of these equations, the mass parameters are related by   In this section, we consider TBA equations related to the Stark problem without Langer's modification.These equations on the one hand confirm the validity of ODE/IM correspondence and give an alternative computational approach for quantum periods on the other hand.Unfortunately, we have not found consistent EQCs in terms of these quantum periods without Langer's modification.

Discussion and conclusion
In this paper, we have applied the exact WKB analysis to the well-known Stark problem to resolve the energy frequency and ionization rate simultaneously.This method has several advantages.
• At first, it gives the complex resonant frequency including E c and ionization rate at the same time, and it applies to a wide range of parameter choices for F and m.
• Second, the analysis is exact as the term EWKB implies, we have derived the exact quantization conditions by utilizing Stokes graphs and found the BS quantization condition is already exact in weak field strength, which clarifies the validity of the WKB approximation and quantization conditions in [11,12,13].
• Third, the method presented in this paper is fully analytic in principle, except for the final numerical calculation, which makes it possible to investigate the asymptotic behavior in the limit of the parameters.
• At last, this method can be systematically extended to higher orders, which is crucial to know the large-order behavior of the asymptotic expansions of the quantum periods even its non-perturbative contributions.
lines eliminating from a turning point a.Moreover, the real part tells us which one in ψ ± a (x) is growing or decreasing, so we attach each line + or − to specify the behavior of solutions.It's determined by the sign of the following quantity Re 1 x a p 0 (x)dx.(2.21)The positive value indicates ψ + (x) increases exponentially, while ψ − (x) decreases exponentially.If it's negative, ψ − (x) increases while ψ + (x) decreases.

Figure 2 . 1 :
Figure 2.1: This is the Borel plane for Borel transform of ψ + a (x).The arrow represents the integral contour for θ = 0.The right red point denotes the corresponding Borel pole.

Figure 2 . 2 :
Figure 2.2: This figure shows the Stokes lines eliminating from a simple turning point a. ± labels the orientations of Stokes lines.The arrow states the continuation path from Stokes region I to II and the red wavy line is branch cut.

Figure 3 . 1 :
Figure 3.1: Stokes graphs and its transition.The orange cross denotes three turning points a, b, and c respectively.The blue point is the origin.The red wavy lines are branch cuts and the green arrow represents the continuation path.γ, γ 1 , and γ 2 are one-cycle encircling turning points, dashed lines indicate the contour enters the second sheet after crossing a branch cut.The Stokes lines are represented by black lines with + and − indicating the orientation.I 0 , I m , and I ∞ are adjacent Stokes regions.

.23) 3 . 4 .
We obtain the basis in the region I m normalized at a, to continue the procedure along the green line, it's necessary to arrive at the basis normalized at b for the same Stokes region I m .It is accomplished by the normalization matrix As the same as the second step, by crossing a Stokes line labeled by + anticlockwise, one gets the basis in the region I 0 normalized at b

Table 3 . 2 :
2 shows the complex energies for a strong field intensity F = 5 and m = 1, 2, 3, which belong to the strong F region for both ξ-and η-equations.It is obvious that quantization conditions including the logarithmic term give closer results to exact values.Our observation further implies the imaginary part gets corrected significantly by the additional modified term.BSQC 2.32739 − 7.03055i 3.54168 − 9.13333i 4.58412 − 10.9252i 1st-order EQC 1.70365 − 5.72785i 3.08635 − 7.85116i 4.22348 − 9.67094i RPM 1.87776 − 5.82185i 3.24425 − 7.91809i 4.37684 − 9.72372i This shows the complex resonant frequencies for F = 5 and m = 1, 2 and 3.The first and second row represents the results from BS quantization conditions and modified quantization condition including the additional logarithm correction respectively.

5 : 1 2
The first two order expansions of the quantum periods with F = 0.03, m = for η-equation.

Table 4 .
2: First four order expansions of the quantum periods for fictitious parameters

Table 4 .
4shows the numerical calculations for ξ-equation with F = 0.03 and m = 1 2 , where corresponding TBA equations are in the maximal chamber, and table 4.5 gives results for η-equation, with its TBA equations (4.34) in the minimal chamber.Numerical calculations confirm the correspondence (4.35) very well.