research article properties of stark resonant states in ...4. stark resonances and their properties...

Research ArticleProperties of Stark Resonant States in Exactly Solvable Systems

Jeffrey M. Brown and Miroslav Kolesik

College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA

Correspondence should be addressed to Miroslav Kolesik; [email protected]

Received 15 September 2015; Revised 23 November 2015; Accepted 29 November 2015

Academic Editor: Emmanuel Lorin

Copyright © 2015 J. M. Brown and M. Kolesik. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Properties of Stark resonant states are studied in two exactly solvable systems. These resonances are shown to form a biorthogonalsystem with respect to a pairing defined by a contour integral that selects states with outgoing wave boundary conditions. Analyticexpressions are derived for the pseudonorm, dipole moment, and coupling matrix elements which relate systems with differentstrengths of the external field. All results are based on explicit calculations made possible by a newly designed integration methodfor combinations of Airy functions representing resonant eigenstates. Generalizations for one-dimensional systems with short-range potentials are presented, and relations are identified which are likely to hold in systems with three spatial dimensions.

1. Introduction

Resonance states have been used to solve a wide range ofproblems in the fields of nuclear physics [1, 2], quantumchemistry [3], nonlinear optics [4–6], and semiconductorphysics [7, 8]. Despite their widely recognized utility, rel-atively little is known about their general properties sincethey do not live in the familiar Hilbert space associated withHermitian quantummechanics [9].The properties and issuesthat are less well-understood than in Hermitian quantummechanics include inner products, normalization and com-pleteness [10–17], complex expectation values [18, 19], andtheir physical interpretation [20]. Despite the mathematicaldifficulties related to their applications, resonance states docontain valuable physical information and it is important toinvestigate systems that could provide some guidance.

Here we add to the present understanding of resonancesystems by analytically calculating a number of useful quan-tities for two exactly solvable quantum systems: the 1DDirac-delta potential and 1D square-well models in the presenceof a homogeneous field. Despite the latter model being atextbook example, and the former being studied and usedin applications for decades (e.g., [21–24]), their resonanceshave so far been studied mainly with numerical tools [8, 25].The quantities of interest, explicitly evaluated for the firsttime in this work, are the normalization factors, eigenvalue

equations, dipole matrix elements, and off-diagonal transi-tion elements that characterize the dependence of resonantbasis states on the external field.

We also generalize our results to more complex modelswith piecewise constant potentials, and for general one-dimensional systems with finite-range potentials.We identifyrelations between the generalized dipole moments and thegradient of the atomic potentials, which resemble similarproperties in systems with self-adjoint Hamiltonians.

Last but not least, all of our new results are based ondirect evaluation of integral expressions, for which we havedeveloped a new integration technique that is applicable tofunctions representing Stark resonances in one dimensionwith a piecewise constant potential.

Beyond developing a deeper understanding of exactlysolvable systems, the additional motivation for this work isin the use of resonance states as a basis for time-dependentSchrödinger evolution, with applications in modeling elec-tron ionization and nonlinear polarization due to a timevarying optical pulse field [5]. Detailed study of exactlysolvable systems with Stark resonant states brings multiplebenefits. First, having explicit expressions for complex-valuedobservables and the ability to study their field and timedependence gives intuition of how one maps these complexvalues and open-system dynamics back to the real expec-tation values and the norm-preserving evolution found in

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015, Article ID 125832, 11 pageshttp://dx.doi.org/10.1155/2015/125832

2 Advances in Mathematical Physics

Hermitian quantum mechanics of a closed system. Such aconnection is crucial for applications in nonlinear optics(e.g., [5, 26, 27]). Secondly, the ability to compare differentresonance systemsmay indicate which properties or relationsare universally valid or common to all resonance systems.For example, we have witnessed in numerical simulations afield-dependent relation connecting the expectation valuesof the gradient of the atomic potential to the resonant statepseudonorm for one- and three-dimensional systems.

2. Non-Hermitian Hamiltonians

In this section, we give some background of the class ofHamiltonians that we want to investigate. We begin with a1D Hamiltonian that is parameterized by the strength of theexternal field 𝐹:

𝐻(𝐹) = −1

2∇2+ 𝑉 (𝑥) − 𝑥𝐹, (1)

where the function 𝑉(𝑥) represents the atomic potential. Tostudy the Stark resonances, one usually assumes outgoingwave boundary conditions at 𝑥 → ∞ and seeks solutionsof 𝐻𝜓

𝑘= 𝐸𝑘𝜓𝑘where 𝐸

𝑘is the eigenvalue. With outgoing

boundary conditions, the system is open as the particlecan escape toward 𝑥 → ∞ and the operator 𝐻 is non-Hermitian. Therefore, energy 𝐸

𝑘, along with many other

observables, is complex-valued.Without Hermiticity, we losemany of the guarantees of Hermitian quantum mechanics,such as conservation of the number of particles, real-valuedobservables, and square integrable wave functions. There isnot yet a full consensus on how to handle and interpretmany of these quantities, including normalization and innerproducts.

Due to the non-square-integrable character of wavefunctions 𝜓

𝑘, the standard inner product and normalization

prescriptions do not apply, since the integrals normally usedto calculate them are divergent. Some regularization methodmust be used, and a number of approaches can be found in theliterature [10, 13, 28]. However, it is important to appreciatethat there may not be as much choice as it may seem inhow the Stark resonant states should be normalized. Forexample, if a resonant state expansion of Green’s operatorexists, the eigenstates appear in it with a definite “norm” [29].In what follows we utilize biorthogonality of the Stark reso-nant system and obtain the eigenstates with such preferrednormalization factors.

We consider theHamiltonian𝐻 to act on functions livingon a complex contour C, where the contour follows the realaxis in the vicinity of the atom and then deviates from the realaxis far from the origin. To select the space of outgoing wavefunctions, the contour departs into the upper complex planeas 𝑥 → ∞. The shape of this contour is inconsequential,except its property that it approaches infinity in the sectorof the complex plain in which all outgoing waves, and inparticular the resonance states, decay exponentially. Onepossible example utilizing a piecewise linear path is shownin Figure 1. At the far end of the contour, outgoing wavefunctions that behave as ∼𝑒𝑖𝑘𝑥 → 𝑒−𝑘Im{𝑧} (with positive 𝑘)

a

I{z}

R{z}Θ

Figure 1: An example contour in the complex plane that servesas a “complexified” spatial axis in a model of an open quantumsystem in which a non-Hermitian Hamiltonian (1) acts in the spaceof functions defined along the contour. Both smooth and piecewiselinear contours are admissible for our purposes. In this example,the domain of the Hamiltonian would be specified by requiring that𝑓(𝑎−) = 𝑓(𝑎

+) and that an analogue of Cauchy-Riemann condition,

𝑓(𝑎−) = 𝑒−𝑖Θ

𝑓(𝑎+), is satisfied for derivatives along the contour for

all 𝑓 ∈ 𝐷(𝐻). We also assume that the potential𝑉(𝑥) has a compactsupportwith a “radius” smaller than 𝑎, so that nonanalytic potentialscan be considered.

decay exponentially. Thus, the introduction of the contour isin the spirit of the external complex scaling.

The differential expression𝐻 acting in the space of func-tions defined along the contour results in a non-Hermitianoperator that represents an open system. A 𝑐-product definedas a contour integral

⟨𝜙 | 𝜓⟩def= ∫

C

𝜙 (𝑧) 𝜓 (𝑧) d𝑧 (2)

is the tool that replaces the standard scalar product inworking with non-self-adjoint operators. A formal argumentcan bemade that if two resonant states belong to two differenteigenvalues, then they are orthogonal with respect to theabove 𝑐-product [30], and the latter can serve as a definitionfor a pseudonorm in resonant states.

In this work we aim to avoid any reliance on formal oper-ator properties. Instead, we show by explicit calculation of theunderlying contour integrals that the following orthogonalityrelation holds for outgoing Stark resonances:

⟨𝜓𝑛| 𝜓𝑘⟩

def= ∫

C

𝜓𝑛(𝑧, 𝐹) 𝜓

𝑘(𝑧, 𝐹) d𝑧 = 𝑁2

𝑛(𝐹) 𝛿𝑛𝑘. (3)

In particular, we verify the orthogonality of different func-tions (𝑛 ̸= 𝑘), and we evaluate the normalization factor𝑁

𝑛(𝐹)

explicitly for two exactly solvable models.

3. Motivation

We have recently presented a proof of principle for appli-cation of metastable electronic states to calculate nonlinearresponse in a time-dependent field of an optical pulse [5].We have also demonstrated that the resulting description isextremely efficient in that a single Stark resonance is sufficientto obtain quite accurate nonlinear polarization for realisticmodels of atoms [31]. Here we recap some of the ideas behindtheMetastable Electronic State Approach (MESA) in order toidentify the relevant properties of resonant states.

Advances in Mathematical Physics 3

If we represent a particular quantum stateΨ as a sum overresonance states, plus a “background,”

Ψ (𝑥, 𝑡) = ∑

𝑛

𝑐𝑛(𝑡) 𝜓𝑛(𝑥, 𝐹 (𝑡)) + 𝜓

𝐵(𝑥, 𝑡) , (4)

and ask what is its evolution due to the time-dependent field𝐹(𝑡), then we can find a system of equations describing theevolution of coefficients 𝑐

𝑛with the help of orthogonality

relation (3):

𝑐

𝑛(𝑡) = −𝑖𝑐

𝑛(𝑡) 𝐸𝑛(𝐹 (𝑡)) + ∑

𝑘

𝑐𝑘(𝑡) 𝐹(𝑡) ⟨𝜕𝐹𝜓𝑛| 𝜓𝑘⟩ , (5)

where𝐹(𝑡) is the time derivative of the electric field intensity.Here we have assumed that the expansion states𝜓

𝑛are slaved

to the time-dependent field and are normalized to unity,⟨𝜓𝑚(𝐹(𝑡)) | 𝜓

𝑛(𝐹(𝑡))⟩ = 𝛿

𝑚𝑛, at all times. Here we neglect

the coupling to the “background” 𝜓𝐵which originates from

the continuum contribution contained in various resonantstate expansions. MESA works with a physically motivatedassumption (see [5] for details) that, for systems initiallyin the ground state, the temporal decay of resonant statesrepresents ionization and that the flow of probability out ofthe space spanned by {𝜓

𝑘} manifests as the increase of the

norm ‖𝜓𝐵‖.

For the purposes of this work, (5) identifies the quan-tities that we aim to calculate. First, we need the complex-valued energies 𝐸

𝑛. Second, we require normalization and

orthogonality relation (3) to be satisfied by all resonancestates. To evaluate the induced polarization, we also need thegeneralized dipole moments

𝑑𝑛𝑘

= ⟨𝜓𝑛| 𝑥 | 𝜓

𝑘⟩ = ∫

C

𝜓𝑛(𝑧, 𝐹) 𝑧𝜓

𝑘(𝑧, 𝐹) d𝑧, (6)

and lastly we must calculate the coupling terms ⟨𝜕𝐹𝜓𝑛| 𝜓𝑘⟩,

which describe the change of the resonant state basis as itevolves slaved to the external field.

The coupling terms can be related to the dipole momentmatrix elements with the help of the following argumentutilizing the parametric dependence of the Hamiltonian on𝐹 [32, 33]:

𝜕𝐹⟨𝜓𝑛| 𝐻 (𝐹) | 𝜓

𝑘⟩ = (𝜕

𝐹𝐸) 𝛿𝑛𝑘

= − (𝐸𝑛− 𝐸𝑘) ⟨𝜕𝐹𝜓𝑛| 𝜓𝑘⟩

− ⟨𝜓𝑛| 𝑥 | 𝜓

𝑘⟩ .

(7)

Moreover, for the normalized resonances the couplingterms are antisymmetric in indices 𝑛, 𝑘, since 𝜕

𝐹𝛿𝑛𝑘

= 𝜕𝐹⟨𝜓𝑛|

𝜓𝑘⟩ = ⟨𝜕

𝐹𝜓𝑛| 𝜓𝑘⟩+⟨𝜓𝑛| 𝜕𝐹𝜓𝑘⟩ = 0.Thismeans that we have

⟨𝜕𝐹𝜓𝑛| 𝜓𝑘⟩ = −⟨𝜓

𝑛| 𝜕𝐹𝜓𝑘⟩ and for 𝑛 = 𝑘 the self-coupling

vanishes ⟨𝜕𝐹𝜓𝑘| 𝜓𝑘⟩ = 0 as a consequence of the 𝑐-product

symmetry.Thus, the evolution system (5) can be alternativelywritten with the substitution

⟨𝜕𝐹𝜓𝑛| 𝜓𝑘⟩ =

{{

{{

{

0 𝑛 = 𝑘

−⟨𝜓𝑛| 𝑥 | 𝜓

𝑘⟩

(𝐸𝑛− 𝐸𝑘)

𝑛 ̸= 𝑘.

(8)

In the appendix, we present a new integration technique thatallows calculation of the coupling terms directly, withoutreliance on the formal derivation underlying relation (8).

4. Stark Resonances and Their Properties

In this section we outline the properties of resonant wavefunctions for two different systems, and we find associatedresonant eigenvalue equations. We also calculate explicitnormalization factors for all Stark resonances using theorthogonality relation (3). Having established these tools, wecontinue to calculate dipole moment matrix elements andcoupling factors.

In this work we assume that the potential 𝑉(𝑥) has acompact support contained in (−𝑑, +𝑑). Thus, the asymptoticform of both the conventional and resonant wave functionscan be obtained as a combination of Airy functions, 𝑐

1𝐴𝑖 +

𝑐2𝐵𝑖. The requirement that the solutions are regular for 𝑥 →

−∞ dictates 𝑐2= 0. For 𝑥 > 0 one can use combinations

𝐶𝑖±(𝑥) = 𝐵𝑖(𝑥) ± 𝑖𝐴𝑖(𝑥) which behave as outgoing (+)

and incoming (−) waves at 𝑥 → ∞. Representation of theeigenstates of the original𝐻 (i.e., the one that acts on the realaxis) which is particularly suitable for our purposes can bewritten as

𝜓𝐸(𝑥) =

−2𝐴𝑖 [𝛼 (𝑥 + 𝐸/𝐹)]

𝑈√𝐷+ (𝐸)𝐷−(𝐸)

𝑥 < −𝑑,

𝜓𝐸(𝑥) =

𝑖

𝑈

√𝐷−(𝐸)

𝐷+ (𝐸)𝐶𝑖+[𝛼 (𝑥 +

𝐸

𝐹)]

−𝑖

𝑈

√𝐷+(𝐸)

𝐷− (𝐸)𝐶𝑖−[𝛼 (𝑥 +

𝐸

𝐹)]

𝑥 > +𝑑,

(9)

where 𝛼 = −(2𝐹)1/3 and 𝐷±(𝐸) are sought expressionsrepresenting the eigenvalue equations for outgoing (+) andincoming (−) wave functions. The fact that the originaloperator (i.e., the one acting on the real axis) is self-adjointguarantees that the above states can be normalized to a deltafunction in energy:

∫

∞

−∞

𝜓∗

𝐸(𝑥) 𝜓𝐸 (𝑥) d𝑥 = 𝛿 (𝐸 − 𝐸) . (10)

It is sufficient to examine the asymptotic behavior of thesestates to verify that this normalization is obtained with 𝑈 =22/3

𝐹1/6. While a particular normalization is not crucial for

us, the above form of eigenstates allows us to infer the formof the resonant functions sought below.

To obtain the remaining portion(s) of energy eigenstates,one has to “fill in” the wave function in the central regionof −𝑑 to 𝑑 and in doing so satisfy whatever conditions agiven potential imposes on them. In both cases treated in thiswork, this means to find functions that are continuous and torequire continuity of derivatives in the square-well case and a“cusp condition” (13) in the Dirac-delta case. This procedurereveals the concrete form of expressions 𝐷±(𝐸) for a given𝑉(𝑥).


The asymptotic form of the energy eigenstates as shownin (9) indicates the form of the resonance wave functions. Forexample, if we find a complex-energy root of 𝐷+(𝐸) = 0 theincoming part of the wave function 𝐶𝑖− will be eliminated.At the same time a pole will appear in the projection ontothe outgoing wave function. This tells us that the resonancebehaves as 𝐴𝑖 and 𝐶𝑖+ for large negative and positive 𝑥,respectively.

We present our final results, including those on resonantstate normalization, in the form that is independent of howone chooses to parameterize the eigenfunctions. Wheneverwe show intermediate results, it is for the wave functionswritten as follows. For the Dirac-delta model, we take theunnormalized ansatz for the outgoing resonance in the form

𝑔 (𝑥, 𝐹) =

{

{

{

𝐶𝑖 (𝛼𝛽)𝐴𝑖 [𝛼 (𝑥 + 𝛽)] 𝑥 < 0

𝐴𝑖 (𝛼𝛽)𝐶𝑖 [𝛼 (𝑥 + 𝛽)] 𝑥 > 0.

(11)

For the system with a square-well potential of width 2𝑑 anddepth 𝑉

0we take

𝑔 (𝑥, 𝐹)

=

{{{{

{{{{

{

𝜅0𝐴𝑖 [𝛼 (𝑥 + 𝛽)] 𝑥 < −𝑑

𝜅1𝐴𝑖 [𝛼 (𝑥 + 𝛽

)] + 𝜅

2𝐵𝑖 [𝛼 (𝑥 + 𝛽

)] −𝑑 < 𝑥 < 𝑑

𝜅3𝐶𝑖 [𝛼 (𝑥 + 𝛽)] 𝑥 > +𝑑

(12)

with coefficients 𝜅𝑖to be fixed to ensure continuity of wave

function value and derivative across well boundaries. Forthese functions to become resonant states the energies in 𝛽 =𝐸/𝐹 and 𝛽 = (𝐸 − 𝑉

0)/𝐹must be solutions to the eigenvalue

equation(s) we present next.

4.1. Eigenvalue Equations. For the Dirac-delta potential𝑉(𝑥) = −𝐵𝛿(𝑥), where 𝐵 is the depth of the potential, and theeigenstate representations (9) are valid with 𝑑 = 0. The deltafunction potential imposes a boundary condition on thewavefunction’s value and derivative:

𝑑𝜓 (0+)

𝑑𝑥−𝑑𝜓(0−)

𝑑𝑥= −2𝐵𝜓 (0) . (13)

This “cusp condition,” when applied to the above eigenstateparameterization, leads directly to thewell-known [4] expres-sion for the eigenvalue equation for resonant energies:

𝐷±(𝐸) ≡ 1 −

2𝜋𝐵

(2𝐹)1/3

𝐴𝑖(−2𝐸

(2𝐹)2/3

)𝐶𝑖±(

−2𝐸

(2𝐹)2/3

) . (14)

Complex-valued solutions to 𝐷+(𝐸) = 0 determine theresonant energies of the outgoing Stark functions.

Longer calculations are required to obtain the analogousequation [25, 34] for the square-well potential. One needs toconnect the outer regions with a linear combination of Airyfunctions (12) and eliminate the unknown coefficients. Theresult reads

𝐷±(𝐸) ≡ (𝐴

0𝐴

1− 𝐴

0𝐴1) (𝐵2𝐶

3− 𝐵

2𝐶3)

− (𝐴0𝐵

1− 𝐴

0𝐵1) (𝐴2𝐶

3− 𝐴

2𝐶3) ,

(15)

−0.5 0.50.0Ereal

−0.6

−0.4

−0.2

0.0

0.2

Eim

ag

(a)

0.0 0.5−0.5Ereal

−0.6

−0.4

−0.2

0.0

0.2

Eim

ag

(b)

Figure 2: Outgoing resonance eigenvalue equation landscapes forDirac-delta ((a) 𝐵 = 1) and square-well potential ((b) 𝑑 = 4 and𝑉0= −0.5) systems in the external field 𝐹 = 0.03. To visualize the

locations of the energy eigenvalues, we evaluate (14) and (15) over arange of 𝐸 in the complex plane and convert |𝐷| to a height map viathe formula (1 − (1+ |𝐷|0.3)−1 + 𝜖)−1, so that its roots are representedby poles that are easy to locate (𝜖 ≈ 0.1).

where we utilized shorthand notations to compress the other-wise long expression.𝐴,𝐵, and𝐶 stand for the correspondingAiry functions, and primes denote derivatives. For a well withdepth 𝑉

0and half-width 𝑑, the subscripts indicate on which

sides of the well walls the arguments of the functions areevaluated, with 0, 1, 2, 3 representing 𝛼(𝑥 + 𝛽) at 𝑥 = −𝑑 − 𝜖,−𝑑+𝜖,+𝑑−𝜖, and𝑥 = +𝑑+𝜖, respectively.The value of𝛽 is alsodependent on where the functions are evaluated. For regionsoutside of the well 𝛽 = 𝐸/𝐹 and for regions inside the well𝛽 = (𝐸 − 𝑉

0)/𝐹. Thus, the subscripts 0 and 1 represent Airy

functions evaluated just outside and inside the left boundaryof the well, while subscripts 2 and 3 represent arguments atthe right boundary of the well.

It is helpful to visualize the resonance energy “landscapes”illustrated in Figure 2. Although the eigenvalue equation


for the square-well (15) is more complicated, it is apparentthat the two systems share some common properties. Themain differing feature is the possibility of multiple boundstates (highlighted in red) in the square-well potential, whilethe Dirac-delta system only supports a single bound state.However, there exist two other infinite families of resonances.The “right” family has eigenvalues located along the real axisand corresponds to longer living states, while the “left” family,with energies along the ray arg(𝑧) = −2𝜋/3, are fast decaying,short-lived states. This resonance structure is most likelya generic feature at least in case of short-ranged attractivepotentials.

4.2. Stark State Normalization. To establish formulas fornormalization and to verify the orthogonality relation (3) forboth systems, we make use of the formula (VS 3.50) in Valleand Soares [35] giving a primitive (antiderivative) functionfor a square of arbitrary combination of Airy functions.

To calculate the contour integral(s), the correspondingprimitive functions are evaluated at points of discontinuitiesof the potential and at both ends of the contour, and this iswhere the choice of the contour is important. The resonantwave functions decay exponentially for 𝑥 → −∞, and thecorresponding boundary terms vanish. On the other side ofthe contour, the asymptotic behavior of the primitive functionis dominated by

𝐶𝑖+[𝛼 (𝑧 + 𝛽)]

≈

𝑒−𝑖𝜋/4exp [+𝑖 (2/3) [−𝛼 (𝑧 + 𝛽)]3/2]

√𝜋 [−𝛼 (𝑧 + 𝛽)]1/4

,

(16)

where 𝛼 = −(2𝐹)1/3, 𝛽 = 𝐸𝑛/𝐹, and 𝐸

𝑛is a root of

the eigenvalue equation. It is straightforward to verify thatasymptotically along the contour, where 𝑧 ∼ 𝜌𝑒𝑖Θ, this func-tion decays for arbitrary fixed 𝛽 as 𝜌 → ∞. Thus, theboundary terms brought by this end of the contour alsovanish. Moreover, since the integrands are in all cases entirefunctions (containing no singular points), the precise shapeof the integration path does not affect the outcome. As aresult, in any piecewise constant atomic potential 𝑉(𝑥) itis only the special points of 𝑉(𝑥) that give rise to nonzerocontribution(s).

Thus, direct integration along the contour, followed bysimplifications making use of the eigenvalue equation andthe Wronskian for Airy functions, yields the following nor-malization factor for the Stark resonance in the Dirac-deltamodel:

𝑁2=𝐴𝑖 (𝛼𝛽)𝐶𝑖

(𝛼𝛽) + 𝐴𝑖

(𝛼𝛽)𝐶𝑖 (𝛼𝛽)

𝛼𝜋

=1

𝛼𝜋[𝜓(𝑥 = 0

+) + 𝜓(𝑥 = 0

−)] .

(17)

To calculate the normalization factor for the square-wellsystem, a similar but more complicated procedure to evaluate

(3) using formula (VS 3.50) results in a surprisingly simpleexpression for the normalization factor:

𝑁2

=𝑉0

𝐹[(𝐴2𝐶

3− 𝐴

2𝐶3)2

𝐴2

0− (𝐴

0𝐴1− 𝐴0𝐴

1)2

𝐶2

3]

=𝑉0

𝐹[𝜓 (𝑥 = −𝑑)

2− 𝜓 (𝑥 = 𝑑)

2] .

(18)

To the authors’ knowledge, this is the first time these resultshave been presented in the explicit form.

To verify the mutual orthogonality with respect to (3) fordifferent resonant states, we use formula (VS 3.53), togetherwith the fact that the complex energies satisfy the eigenvalueequation(s).

To conclude this subsection, we note that our direct verifi-cation of the orthogonality relation (3) and the explicit calcu-lation of the corresponding normalization factors means thatthere exists an infinite dimensional space of functions thatcan be expressed as superpositions of Stark resonances. Inthe spirit of [29], one should ask if this gives us the preferredexpansion. Indeed, one can alternatively use the self-adjointHamiltonian eigenstates (9) and find the projector onto agiven resonant state as the residue of 𝜓

𝐸(𝑥)𝜓𝐸(𝑦) at the pole

in the complex plane that corresponds to its energy 𝐸 = 𝐸𝑛.

While it is beyond the scope of the present paper, we notethat the two approaches in fact lead to the same expansioncoefficients.

4.3. Dipole Matrix Elements. Next we calculate the gen-eralized dipole matrix elements, both diagonal and off-diagonal. Expressedwith the help of unnormalized resonanceeigenfunctions 𝑔[𝛼(𝑥 + 𝛽

𝑚)], the contour integrals we need

to evaluate read

⟨𝜓𝑚| 𝑥 | 𝜓

𝑛⟩

=1

𝑁2𝑛

∫C

𝑔 [𝛼 (𝑥 + 𝛽𝑚)] 𝑧𝑔 [𝛼 (𝑥 + 𝛽

𝑛)] d𝑧.

(19)

Note that these quantities differ from their Hermitian coun-terparts as there is no complex conjugation in the integrand,and the result is complex-valued [19].Herewe assume that thecontourC is chosen such that it only starts to deviate from thereal axis for𝑥 > 𝑑, that is, outside of the potential support. Forboth the delta potential model and the square-well potentialsystem, we integrate over each distinct interval of constantpotential𝑉making use of the formula (VS 3.51). As the shapeof the contour ensures vanishing contributions from its ends,one only needs to evaluate the primitive functions at 𝑥 = 0for Dirac-delta and 𝑥 = ±𝑑 for the square-well system.


For the diagonalmatrix element in the Dirac-delta poten-tial model, we obtain the following expression in terms ofAiry functions:

⟨𝜓𝑛| 𝑥 | 𝜓

𝑛⟩𝐷=

1

𝑁2𝑛

⋅𝐴𝑖 (𝛼𝛽

𝑛) 𝐶𝑖 (𝛼𝛽

𝑛)

3𝛼2[𝐴𝑖(𝛼𝛽𝑛) 𝐶𝑖 (𝛼𝛽

𝑛)

− 𝐴𝑖 (𝛼𝛽𝑛) 𝐶𝑖(𝛼𝛽𝑛)] +

2𝐸

3𝐹𝑁2𝑛

⋅1

𝛼[(𝐴𝑖(𝛼𝛽𝑛) 𝐶𝑖 (𝛼𝛽

𝑛))2

− (𝐴𝑖 (𝛼𝛽𝑛) 𝐶𝑖(𝛼𝛽𝑛))2

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

−𝑁2

𝑛

.

(20)

These results can be further simplified using the formulafor the normalization factor𝑁2 together with the eigenvalueequation (14) and are found to be related to the change in theeigenvalue with respect to the field 𝐹 as follows:


𝑛⟩𝐷=

1

6𝜋2𝛼𝐵𝑁2𝑛

−2𝐸

3𝐹= −𝜕𝐹𝐸𝑛, (21)

where the second equality can be verified by differentiatingthe eigenvalue equation (14) with respect to 𝐹.

For the off-diagonal dipole matrix elements, we use (VS3.54). Utilizing 𝐶

𝑛,𝑘= 𝐶𝑖(𝛼𝛽

𝑛,𝑘) and 𝐴

𝑛,𝑘= 𝐴𝑖(𝛼𝛽

𝑛,𝑘) to

shorten the notation, we find that


𝑘⟩𝐷=

1

𝑁𝑛𝑁𝑘

2

𝛼5 (𝛽𝑛− 𝛽𝑘)3[𝐶𝑛𝐶𝑘𝐴

𝑛𝐴𝑘

− 𝐶𝑛𝐶𝑘𝐴𝑛𝐴

𝑘− 𝐴𝑛𝐴𝑘𝐶

𝑛𝐶𝑘+ 𝐴𝑛𝐴𝑘𝐶𝑛𝐶

𝑘]

+1

𝑁𝑛𝑁𝑘

2

𝛼4 (𝛽𝑛− 𝛽𝑘)2[𝐶𝑛𝐶𝑘𝐴

𝑛𝐴

𝑘

− 𝐴𝑛𝐴𝑘𝐶

𝑛𝐶

𝑘] .

(22)

Grouping terms in order to identify Wronskians allows us tosimplify the expression down to


𝑘⟩𝐷=

𝐹

2 (𝐸𝑛− 𝐸𝑘)2[𝑁𝑛

𝑁𝑘

+𝑁𝑘

𝑁𝑛

] , (23)

which in turn can be written solely in terms of the wavefunction properties at 𝑥 = 0,


𝑘⟩𝐷

=−𝐹

𝛼 (𝐸𝑛− 𝐸𝑘)2[𝜓

𝑛(0−) 𝜓

𝑘(0−) − 𝜓

𝑛(0+) 𝜓

𝑘(0+)] ,

(24)

with 𝜓 standing for the derivative of the normalized Starkwave function.

Let us proceed with the dipole moment calculations forthe square-well potential. The matrix elements can be foundby again making use of (VS 3.51) and (VS 3.54).Thanks to thecontinuity properties of the wave function, many terms that

arise in the course of this calculation cancel, and the resultingdiagonal terms are


𝑛⟩𝑆

=𝑉0

3𝐹[(𝑑 − 4𝛽)𝜓 (−𝑑)

2+ (𝑑 + 4𝛽)𝜓 (𝑑)

2]

+2𝑉2

0

3𝐹2[𝜓 (−𝑑)

2− 𝜓 (𝑑)

2]

+2𝑉0

3𝐹𝛼[𝜓(−𝑑)2− 𝜓(𝑑)2] ,

(25)

which is simplified further using the normalization factor(again as with the delta model) to


𝑛⟩𝑆=2𝑉0− 4𝐸𝑛

3𝐹

+𝑉0𝑑

3𝐹[𝜓 (−𝑑)

2+ 𝜓 (𝑑)

2]

+2𝑉0

3𝛼𝐹[𝜓(−𝑑)2− 𝜓(𝑑)2] .

(26)

It can be checked numerically that the diagonal elements arealso equal to −𝜕

𝐹𝐸𝑛, just as in the delta potential model.

Now we calculate the off-diagonal elements. When using(VS 3.54), taking into account that continuity of the wavefunction allows us to ignore terms that do not involve 𝑥 andthe sum 𝛽

𝑛+ 𝛽𝑘, as a result every other term cancels, and the

resulting equation is surprisingly simple:


𝑘⟩𝑆

=𝑉0

(𝐸𝑛− 𝐸𝑘)2[𝜓𝑛(−𝑑) 𝜓

𝑘(−𝑑) − 𝜓

𝑛(𝑑) 𝜓𝑘(𝑑)] ,

(27)

and one should note the similarity with its counterpartformula for the Dirac-delta model. We thus arrive at theconclusion that all dipole matrix elements can be expressedin simple formulas which only depend on the values of thewave functions (and their derivatives) at the special pointsgiven that characterize the potential. It will be interesting tosee if these results can be generalized for arbitrary systemswith piecewise constant potentials.

4.4. Coupling Matrix Elements. We now turn our attentionto the terms ⟨𝜕

𝐹𝜓𝑛| 𝜓𝑘⟩ identified in (5). These quantities

mediate the connections between Stark resonances of a givensystem at different values of the field 𝐹 and we call themaccordingly coupling terms. They are needed to describe theevolution of the system in a time-dependent 𝐹(𝑡). Instead oftrusting the formally derived relation (8), we compute thesequantities through direct integration. Thus, our result can bealso interpreted as a direct verification of (8) for two modelsystems. Moreover, the procedure and the particular repre-sentation of the results lead us to a generalization for arbitraryone-dimensional systems with short-range potentials.


To calculate the coupling terms, we first differentiate thewave function with respect to the field 𝐹 and then integrate

∫

∞

−∞

(𝜕𝐹𝜓𝑛) 𝜓𝑘d𝑥 = 1

𝑁𝑛𝑁𝑘

∫

∞

−∞

(𝜕𝐹𝑔𝑛) 𝑔𝑘d𝑥

−(𝜕𝐹𝑁𝑛)

𝑁𝑛

∫

∞

−∞

𝜓𝑛𝜓𝑘d𝑥

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝛿𝑛𝑘

.

(28)

It is clear that the diagonal (𝑛 = 𝑘) coupling terms betweennormalized states vanish, and we only need to calculate off-diagonal elements and therefore can disregard the secondterm, leaving only the integral ⟨𝜕

𝐹𝑔𝑛| 𝑔𝑘⟩. For the model

with Dirac-delta potential, the integral is

⟨𝜕𝐹𝑔𝑛| 𝑔𝑘⟩𝐷

= Δ𝐶

𝑛𝐶𝑘∫

0

−∞

𝐴𝑖 [𝛼 (𝑥 + 𝛽𝑛)] 𝐴𝑖 [𝛼 (𝑥 + 𝛽

𝑘)] d𝑥

+ Δ𝐴

𝑛𝐴𝑘∫

∞

0

𝐶𝑖 [𝛼 (𝑥 + 𝛽𝑛)] 𝐶𝑖 [𝛼 (𝑥 + 𝛽

𝑘)] d𝑥

+ Δ𝐶𝑛𝐶𝑘∫

0

−∞

𝐴𝑖[𝛼 (𝑥 + 𝛽

𝑛)] 𝐴𝑖 [𝛼 (𝑥 + 𝛽

𝑘)] d𝑥

+ Δ𝐴𝑛𝐴𝑘∫

∞

0

𝐶𝑖[𝛼 (𝑥 + 𝛽

𝑛)] 𝐶𝑖 [𝛼 (𝑥 + 𝛽

𝑘)] d𝑥

−2

3𝛼2

⋅ 𝐶𝑛𝐶𝑘∫

0

−∞

𝑥𝐴𝑖[𝛼 (𝑥 + 𝛽

𝑛)] 𝐴𝑖 [𝛼 (𝑥 + 𝛽

𝑘)] d𝑥

−2

3𝛼2

⋅ 𝐴𝑛𝐴𝑘∫

∞

0

𝑥𝐶𝑖[𝛼 (𝑥 + 𝛽

𝑛)] 𝐶𝑖 [𝛼 (𝑥 + 𝛽

𝑘)] d𝑥,

(29)

whereΔ = 1/(3𝜋2𝛼3𝐵𝑁2), and𝐴𝑘,𝐵𝑘,𝐶𝑘stand for the corre-

sponding Airy functions or their combinations evaluated at𝛼𝛽𝑘(i.e., at 𝑥 = 0).For the model with the square-well potential we have a

combination of integrals of similar types; namely,

⟨𝜕𝐹𝑔𝑛| 𝑔𝑘⟩𝑆= ∫

−𝑑

−∞

𝜅

0𝛾0𝐴𝑖 [𝛼 (𝑥 + 𝛽

𝑛)]

⋅ 𝐴𝑖 [𝛼 (𝑥 + 𝛽𝑘)] d𝑥

+ ∫

𝑑

−𝑑

(𝜅

1𝐴𝑖 [𝛼 (𝑥 + 𝛽

𝑛)] + 𝜅

2𝐵𝑖 [𝛼 (𝑥 + 𝛽

𝑛)])

⋅ (𝛾1𝐴𝑖 [𝛼 (𝑥 + 𝛽

𝑘)] + 𝛾2𝐵𝑖 [𝛼 (𝑥 + 𝛽

𝑘)]) d𝑥

+ ∫

∞

𝑑

𝜅

3𝛾3𝐶𝑖 [𝛼 (𝑥 + 𝛽

𝑛)] 𝐶𝑖 [𝛼 (𝑥 + 𝛽

𝑘)] d𝑥

+𝛼

3𝐹∫

∞

−∞

(3𝐸

𝑛+ 𝑥 − 2𝛽

𝑛) 𝑔[𝛼 (𝑥 + 𝛽

𝑛)]

⋅ 𝑔 [𝛼 (𝑥 + 𝛽𝑘)] d𝑥 +

2𝛼𝑉0

3𝐹2∫

𝑑

−𝑑

𝑔[𝛼 (𝑥 + 𝛽

𝑛)]

⋅ 𝑔 [𝛼 (𝑥 + 𝛽𝑘)] d𝑥,

(30)

where 𝛾 and 𝜅 are the coefficients that guarantee continuity of𝑔𝑛and 𝑔

𝑘, respectively, at the well boundaries. These factors

are expressed in terms of Airy functions and depend on𝐹 through 𝛼, 𝛽

𝑛, and 𝛽

𝑘. Their primed notations represent

derivatives with respect to 𝐹.The above integrals for both models contain terms of

two kinds. The first group can be evaluated making useof known, previously published Airy integrals. These havethe form ∫𝑏

𝑎𝐴𝐵 d𝑥, where 𝐴, 𝐵 are a linear combination

of Airy functions with arguments 𝛼(𝑥 + 𝛽), where 𝛽 isdifferent in 𝐴 and 𝐵. Thus, the first two lines in (29) and firstthree lines in (30) can be dealt with, although each integralrequires lengthy computations, especially in the square-wellcase. Then there are integrals of the types ∫𝑏

𝑎𝐴𝐵 d𝑥 and

∫𝑏

𝑎𝑥𝐴𝐵 d𝑥. No known formulas are available for these, and

we have developed a new integration technique that weoutline in the appendix. As a result, analytic results can beobtained for both systems.

For the Dirac-delta model, the individual integrals on theRHS of (29) simplify pairwise. The first pair is calculatedusing (VS 3.53), while the last two pairs use the identitiesderived in the appendix. The intermediate expression reads

⟨𝜕𝐹𝑔𝑛| 𝑔𝑘⟩𝐷

=Δ

𝛼2 (𝛽𝑛− 𝛽𝑘)[𝐴

𝑛𝐴𝑘𝐶𝑛𝐶

𝑘− 𝐴𝑛𝐴

𝑘𝐶

𝑛𝐶𝑘]

−Δ

𝛼2 (𝛽𝑛− 𝛽𝑘)[𝐴

𝑛𝐴

𝑘𝐶𝑛𝐶𝑘− 𝐴𝑛𝐴𝑘𝐶

𝑛𝐶

𝑘]

−2

3𝜋𝛼5

1

(𝛽𝑛− 𝛽𝑘)2[𝛽𝑛𝐶𝑛𝐴𝑛− 𝛽𝑘𝐶𝑘𝐴𝑘]

−4

𝛼7

1

(𝛽𝑛− 𝛽𝑘)3[𝐴

𝑛𝐶𝑛𝐴

𝑘𝐶𝑘− 𝐴𝑛𝐶

𝑛𝐴𝑘𝐶

𝑘] .

(31)

The first two lines can be simplified using the Wronskianand normalization factors, while the third can be rewrittenwith the help of the eigenvalue equation. Combining theseshows that the first three lines sum up to zero, leaving only


the last term which we write in terms of derivatives of thewave functions at the origin:

⟨𝜕𝐹𝜓𝑛| 𝜓𝑘⟩𝐷

=𝐹

𝛼 (𝐸𝑛− 𝐸𝑘)3[𝜓

𝑛(0−) 𝜓

𝑘(0−) − 𝜓

𝑛(0+) 𝜓

𝑘(0+)] .

(32)

Comparing this expression to (24), it is clear that the rela-tion between coupling term and off-diagonal dipole matrixelements (8) is valid.

Coupling terms (30) for the square-well system are alsocalculated in analytic form but due to excessive numberof terms comprising the result they are not listed here. Inprinciple, a similar procedure to simplify the square-wellcoupling term (30) should work. However, the resultingexpression is extremely large and we could not find apractical way to compress it to a manageable length. Themain difficulty in simplifying the square-well result is thatthe eigenvalue equation is muchmore complicated (cf. (14) to(15)). Nevertheless, having explicit formulas in terms of Airyfunctions, we verified numerically that the integrated resultdoes relate to the dipole matrix element as suggested by (8):

⟨𝜕𝐹𝜓𝑛| 𝜓𝑘⟩𝑆

=−𝑉0

(𝐸𝑛− 𝐸𝑘)3[𝜓𝑛(−𝑑) 𝜓

𝑘(−𝑑) − 𝜓

𝑛(𝑑) 𝜓𝑘(𝑑)] .

(33)

To conclude this section, we have shown that the couplingterms can be directly calculated using a new Airy integraltechnique detailed in the appendix. While our explicit calcu-lations do not justify the formal steps taken to obtain (8), theydo corroborate that the relation between the coupling termsand the dipole moment holds. Pragmatically, one shouldchoose to use the dipolematrix elements, since they are easierto calculate numerically.

5. Generalization for Arbitrary PiecewiseConstant Potentials

Comparing results (18) and (27) we see that they havea similar form, with the dipole moment matrix elementsexpressed in terms of wave function values evaluated atdiscontinuities of the potential 𝑉. While the same result canbe easily written explicitly with Airy functions, this particularform indicates that the expressions are in fact sums overatomic potential discontinuities, with weights correspondingto the potential-value jumps. This suggests the followinggeneralization of the dipole matrix element formulas for asystem with arbitrary piecewise constant potential:

𝑁2= ∑

𝑖

Δ𝑉𝑖

𝐹𝜓 (𝑥𝑖)2

, (34)


𝑘⟩𝑆=

1

(𝐸𝑛− 𝐸𝑘)2∑

𝑖

Δ𝑉𝑖𝜓𝑛(𝑥𝑖) 𝜓𝑘(𝑥𝑖) , (35)

where the sums run over all potential discontinuities. It is infact not too difficult to realize that the procedures utilized

above can be modified for a more general case of piecewiseconstant potential. From there, one can take a continuumlimit, approximating an arbitrary potential as a limit ofpiecewise constant functions, and arrive at

𝑁2=

1

𝐹∫𝑑𝑉

𝑑𝑥𝜓 (𝑥𝑖)2 d𝑥, (36)


𝑘⟩𝑆=

1

(𝐸𝑛− 𝐸𝑘)2∫𝑑𝑉

𝑑𝑥𝜓𝑛(𝑥) 𝜓𝑘(𝑥) d𝑥. (37)

We have assumed that the potential is short-ranged and theintegration in these formulas is along the real axis (i.e., thecontourC is not necessary for convergence).

This is an intriguing result, because an identical formulacan be derived for the discrete-energy eigenstates of a self-adjoint Hamiltonian by evaluating its double commutatorwith the position operator. However, here we have Stark res-onances represented by complex-valued functions living onthe contour C. So it seems that as long as the normalizationand “scalar product” are defined with the help of pairing (3),the Stark resonance states satisfy relations analogous to thoseobeyed by their self-adjoint counterparts.

We have used numerical simulations (not shown here)to verify that the relation between the Stark resonancepseudonorm and the generalized expectation value of the“atomic” potential gradient could also be valid for three-dimensional systems. It is tempting to speculate that the off-diagonal dipole element relation (37) could be generally validfor Stark resonant states in higher dimensions.

6. Conclusion

We have derived analytic expressions for a number of quanti-ties that characterize the Stark resonance states in two exactlysolvable systems. The first model studied in this work isthe one-dimensional particle in a Dirac-delta potential withadditional homogeneous field, and the second has the square-well potential.

We have studied these systems as open, non-Hermitianmodels, and identified a natural choice for the pairingconnecting the states in the domain of the Hamiltonian withthe states in the domain of its adjoint operator. With respectto this pairing, Stark resonances form an orthogonal system,and many of their properties can be evaluated analytically.

Despite the fact that both models have been studied foryears, explicit expressions for their (pseudo-) norms, dipolemoment expectation values, and their relations connectingthe resonant state wave functions at different field valuesare new. Our results thus further the understanding of themathematical properties that underline the Stark effect.

Moreover, we have shown that certain results naturallyextend to a wide class of one-dimensional models and wehave also identified relations that appear to be candidatesfor properties generally applicable to three-dimensional Starksystems. In particular, we have found that the generalizeddipole moment matrix elements between the nonphysicalresonant states can be related to the expectation values ofthe atomic potential gradient in a way that is completely


analogous to relations that hold for real-valued discrete-energy eigenfunctions in Hermitian systems. We speculatethat these relations apply generally in three dimensions andcould be used in numerical calculations to assess the fidelityof the resonant eigenfunctions.

An important by-product of this study is a new integra-tion technique applicable to combinations of Airy functionsthat represent Stark resonances in one-dimensional modelswith piecewise constant potentials.

Results presented in this work have also an immediatepractical impact on modeling of light-matter interactions instrong time-dependent optical fields in the framework of theMetastable Electronic State Approach.

Appendices

TheAiry integral technique outlined in this appendix is usedto evaluate integrals that contain linear combinations of Airyfunctions𝐴, 𝐵 and their derivatives𝐴, 𝐵, where the two setsof functions have shifted arguments. We use the shorthandnotation of 𝐴 = 𝐴[𝛼(𝑥 + 𝛽

𝑛)] and 𝐵 = 𝐵[𝛼(𝑥 + 𝛽

𝑘)].

The two unknown integrals that we are required to solveare

∫

∞

−∞

𝐴𝐵 d𝑥,

∫

∞

−∞

𝑥𝐴𝐵 d𝑥,

(∗)

where the integration is understood over 𝑥 along the contourC. Simply applying integration by parts leads nowhere sincethe solutions always involve integrals that are also not known.The technique described here circumvents that problem. Foran integral whose integrand has the form 𝑥𝑛𝐴𝐵, we performthe following steps:

(1) Multiply integrand 𝑥𝑛𝐴𝐵 by 𝑥.

(2) Differentiate with respect to 𝑥 and integrate.

(3) Repeat (1) and (2) with the “symmetric” integrand𝑥𝑛𝐴𝐵.

(4) Subtract the two equations.

And we find that we can write the unknown integral in termsof integrals that do not contain derivatives of 𝐴 or 𝐵 andare therefore found in published literature [35]. In the nextsectionswe demonstrate the procedure on the integrands𝐴𝐵and 𝑥𝐴𝐵.

The procedure relies on the fact that wave functions andtheir derivatives are continuous at jumps of 𝑉(𝑥). Moreover,𝐴 and 𝐵 must be resonant eigenstates that belong to theorthonormal system with respect to the pairing defined bythe contour integral along C (3). As such, the formulaswe derive below do not apply to arbitrary combinations ofAiry functions. On the other hand, the method does applyto a general case of Stark resonances in piecewise constantpotential and in this sense the result shown below is general.

A. Integrating 𝐴𝐵

To solve the integral of𝐴𝐵, wemultiply by𝑥 and differentiate

𝜕𝑥(𝑥𝐴𝐵) = 𝐴

𝐵 + 𝛼𝑥𝐴

𝐵 + 𝛼𝑥𝐴

𝐵

= 𝐴𝐵 + 𝛼2𝑥 (𝑥 + 𝛽

𝑛) 𝐴𝐵 + 𝛼𝑥𝐴

𝐵.

(A.1)

Next, we integrate both sides and find that there are knownand unknown Airy integrals:

𝑥𝐴𝐵

+∞

−∞⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=0

= ∫

∞

−∞

𝐴𝐵 d𝑥

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

want

+ 𝛼2∫

+∞

−∞

𝑥2𝐴𝐵 d𝑥 + 𝛼2𝛽

𝑛∫

∞

−∞

𝑥𝐴𝐵 d𝑥⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

known

+ 𝛼∫

∞

−∞

𝑥𝐴𝐵 d𝑥

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

unknown

.

(A.2)

To remove the unknown integrals that contain both deriva-tive functions 𝐴 and 𝐵, we differentiate the “symmetric”integrand

𝜕𝑥(𝑥𝐴𝐵

) = 𝐴𝐵

+ 𝛼𝑥𝐴

𝐵+ 𝛼𝑥𝐴𝐵

= 𝐴𝐵+ 𝛼𝑥𝐴

𝐵+ 𝛼2𝑥 (𝑥 + 𝛽

𝑘) 𝐴𝐵

(A.3)

and then integrate to find the complementary equation to(A.2):

𝑥𝐴𝐵

+∞

−∞⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=0

= ∫

∞

−∞

𝐴𝐵 d𝑥

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

unknown

+ 𝛼2∫

∞

−∞

𝑥2𝐴𝐵 d𝑥 + 𝛼2𝛽

𝑘∫

∞

−∞

𝑥𝐴𝐵 d𝑥⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

known

+ 𝛼∫

∞

−∞

𝑥𝐴𝐵 d𝑥

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

unknown

.

(A.4)

We subtract (A.2) and (A.4)

∫

∞

−∞

𝐴𝐵 d𝑥 − ∫

∞

−∞

𝐴𝐵 d𝑥

= −𝛼2(𝛽𝑛− 𝛽𝑘) ∫

∞

−∞

𝑥𝐴𝐵 d𝑥.(A.5)

And then we relate the two terms on the LHS using thefact that 𝐴𝐵 vanishes at the ends of the integration contour:∫𝐴𝐵 d𝑥 + ∫𝐴𝐵 d𝑥 = 0. We arrive at the unknown integral,

in terms of a known one (VS 3.54):

∫

∞

−∞

𝐴𝐵 d𝑥 = −𝛼

2

2(𝛽𝑛− 𝛽𝑘) ∫

∞

−∞

𝑥𝐴𝐵 d𝑥⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

3.54

. (A.6)


Here we have used the integration along the entire contour,but it is possible to apply the same technique to integrals overa finite range (𝑎, 𝑏).The only changewould be the inclusion ofthe LHS terms of (A.2) and (A.4) evaluated at the points 𝑎 and𝑏. Such a finite-range integral over (−𝑑, 𝑑) appears during thecalculation of the coupling term for the square-well potentialin (30).

B. Integrating 𝑥𝐴𝐵

To solve this integral, we use the same procedure: multiplyintegrand by 𝑥 and differentiate and subtract “symmetric”integrand. Consider

𝜕𝑥(𝑥2𝐴𝐵) = 2𝑥𝐴

𝐵 + 𝛼𝑥

2𝐴𝐵 + 𝛼𝑥

2𝐴𝐵

= 2𝑥𝐴𝐵 + 𝛼2𝑥2(𝑥 + 𝛽

𝑛) 𝐴𝐵

+ 𝛼𝑥2𝐴𝐵,

(B.1)

𝜕𝑥(𝑥2𝐴𝐵) = 2𝑥𝐴𝐵

+ 𝛼𝑥2𝐴𝐵+ 𝛼𝑥2𝐴𝐵

= 2𝑥𝐴𝐵+ 𝛼2𝑥2(𝑥 + 𝛽

𝑘) 𝐴𝐵

+ 𝛼𝑥2𝐴𝐵.

(B.2)

Subtract (B.1) and (B.2) and then integrate

0 = 2[[

[

∫

∞

−∞

𝑥𝐴𝐵 d𝑥

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

want

− ∫

∞

−∞

𝑥𝐴𝐵 d𝑥

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

unknown

]]

]

+ 𝛼2(𝛽𝑛− 𝛽𝑘) ∫

∞

−∞

𝑥2𝐴𝐵 d𝑥

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

known

.

(B.3)

Again the two terms above can be related using integrationper parts plus an orthogonality argument. As a result, we canwrite the sought integral in terms of (VS 3.55). Consider

∫

∞

−∞

𝑥𝐴𝐵 d𝑥 = −

𝛼2(𝛽𝑛− 𝛽𝑘)

4∫

∞

−∞

𝑥2𝐴𝐵 d𝑥

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

3.55

. (B.4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

Thisworkwas supported by theUnited States Air ForceOfficefor Scientific Research under Grants nos. FA9550-13-1-0228and FA9550-10-1-0561.

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