A new class of exact solutions of the Klein–Gordon equation of a charged particle interacting with an electromagnetic plane wave in a medium

Analytic solutions of the wave equations of charged particles in the presence of strong radiation fields (such as those produced by high-intensity lasers) have played a crucial role in the description of multiphoton processes (Fedorov 1997). It is known that the Volkov states of an electron or other charged particle, being the exact solutions of the Dirac equation (Wolkow 1935), Klein–Gordon equation (Gordon 1927) or Schrödinger equation (Keldish 1964, Bunkin and Fedorov 1965), have long been an important theoretical tool in the study of these processes. Soon after the advent of the laser, these solutions were applied to describe various strong-field phenomena beyond the perturbative regime (see Brown and Kibble 1964, Eberly 1969, Neville and Rohrlich 1971, Gitman et al 1975, Ritus and Nikishov 1979 and Bergou and Varró 1980), and their study has nowadays received a renoved interest (see e.g. Panek et al 2002, Zakowicz 2005, Salamin et al 2006, Peatross et al 2007, Musakhayan 2008, Ehlotzky et al 2009, Boca 2011, and Harvey et al 2012). In the relativistic regime the closed analytic form of this set of states relies on the vacuum dispersion relation of the electromagnetic (EM) plane wave. If the EM field propagates in a medium (with a real index of refraction n_m ≠ 1), or the strong laser field is modelled by a time-periodic electric field, then the relativistic wave equations can be reduced to Mathieu or Hill equations (see Nikishov and Ritus 1967, Nikishov 1970, Narozhny and Nikishov 1974, Becker 1977, and Cronström and Noga 1977). This problem, in the context of pair creation processes in strong fields, has again received a considerable theoretical interest recently (see e.g. Narozhny et al 2004, Popov 2004, Dunne 2004, 2009, and Müller and Müller 2012). Except for some special cases, the solutions cannot be expressed in a closed finite form; they are related to higher transcendental functions. In this context it is interesting to note that if one describes a Schrödinger particle, interacting with an EM plane wave in vacuum, beyond the dipole approximation (by taking exactly into account the spatial variation of the field), then one also encounters Mathieu or Hill equations, with their associated stability charts and band structures (Bergou and Varró 1981). However, the occurrence of such band structures for the momenta has been shown to be a result of the inconsistent use of the nonrelativistic description for high-intensity fields, as has been pointed out by Drühl and McIver (1983).

In the present letter we show that there are exact closed form solutions of the Klein–Gordon equation of a charged particle moving in a monochromatic classical plane EM field in a medium of index of refraction n_m < 1. Recently we have solved the analogous problem for a Dirac particle (Varró 2013), and encountered (by now unknown) finite complex Fourier sums. The solutions to be derived in the forthcoming section are expressed in terms of known trigonometric polynomials; the Ince polynomials. Their arguments are integer or half-integer multiples of the EM wave's phase, and they form a doubly infinite discrete set of solutions labelled by two integer quantum numbers. This latter property is a completely new feature in comparison with both the Gordon–Volkov states (in vacuum) parametrized by continuous four momenta, and the Mathieu-type solutions characterized by a band structure of the allowed regions of the disposable parameters. We note that in mathematical physics Ince polynomials also appear in the description of various other systems. For instance, if one separates the two-dimensional harmonic oscillator wavefunction by using elliptic coordinates (Boyer et al 1975) then the eigenfunctions are products of Ince polynomials.

In section 2, during the course of the derivation of Ince's equation from the Klein–Gordon equation, we also make a brief comparison with the method leading to the usual Volkov states or to the Mathieu-type wavefunctions. Section 3 is devoted to the determination of the real polynomial solutions, which are associated with the eigenvalue problem of special finite tri-diagonal matrices. The solutions coincide with the Ince polynomials, labelled by two integer quantum numbers, which represent a quantized spectrum of the electron's momentum components along the (linear) polarization direction and along the propagation direction of the laser field. In section 4 we shall summarize the main mathematical properties of the solutions obtained, and few physical implications associated with them will be mentioned. In general, more emphasis shall be placed on the existence and analytic form of these solutions than on physical and numerical results. However, in section 4 some numerical examples will also be presented, just for illustration purposes. In section 5 a brief summary with conclusions closes our letter.

In an external electromagnetic field characterized by the four-vector potential A(x), the Klein–Gordon equation of a particle of charge e and of mass m has the form¹

$$\begin{eqnarray} &&\fl [(\mathrm{i}\partial -\varepsilon A)^{2}-{\kappa }^{2}]\Phi =0,\tqs \varepsilon =e/\hbar c,\quad \kappa =m c/\hbar , \end{eqnarray} \tag{ 1 }$$

where c is the velocity of light in vacuum, and ħ is Planck's constant divided by 2π. In a medium of index of refraction n_m, a general transverse electromagnetic plane wave of wavevector k can be represented by a vector potential

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle A(x)={e}_{1}{A}_{1}(\xi )+{e}_{2}{A}_{2}(\xi ),\tqs \xi =k\cdot x,\\ \displaystyle k\cdot A=0,\tqs k=\{ {k}^{\mu }\} ={k}^{0}(1,{n}_{\mathrm{m}}{\boldsymbol{e}}_{3}),\\ \displaystyle \{ {e}_{1,2}^{\mu }\} =(0,{\boldsymbol{e}}_{1,2}), \end{array} \end{eqnarray} \tag{ 2 }$$

where {e₁,e₂,e₃} form a right system of mutually orthogonal unit vectors. In principle, the scalar functions A_1,2(ξ) in (2) may have arbitrary form (satisfying, of course, certain regularity conditions). For a purely monochromatic field A_1,2(ξ) are simple sine and cosine functions and k⁰ = ω₀/c, where ω₀ = 2πν₀ is the circular frequency. In the case of a general plane EM field, the solutions can be expressed in a simple closed form Gordon (1927). By taking a modified de Broglie plane wave, associated with a four-momentum parameter p_μ, we have

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl \Phi ={\Phi }_{\mathrm{p}}(\xi )\exp (-\mathrm{i}p\cdot x), \\ \displaystyle \fl -{k}^{2}\hspace{0.167em} \frac{{\mathrm{d}}^{2}{\Phi }_{\mathrm{ p}}}{\mathrm{d}{\xi }^{2}}+2\mathrm{i}k\cdot p \hspace{0.167em} \frac{\mathrm{d}{\Phi }_{\mathrm{p}}}{\mathrm{d}\xi }+({p}^{2}-{\kappa }^{2}-2 \varepsilon p\cdot A \\ \displaystyle +~{\varepsilon }^{2}{A}^{2}){\Phi }_{\mathrm{ p}}=0,\\ \displaystyle \fl \xi =k\cdot x. \end{array} \end{eqnarray} \tag{ 3 }$$

In vacuum (n_m = 1, k = k₀(1,e₃), k² = 0) the factor of the second derivative in (3) is zero, and the remaining first-order equation can be directly integrated, yielding the Gordon–Volkov states,

$$\begin{eqnarray} \begin{array}{@{}rcl@{}} \displaystyle \fl {\Phi }_{\mathrm{p}}&\displaystyle =&\displaystyle (2 \pi )^{-3/2}(2{p}_{0})^{-\frac{1}{2}}\exp (-\mathrm{i}{S}_{\mathrm{p}}),\\ \displaystyle \fl {S}_{\mathrm{p}}&\displaystyle =&\displaystyle p\cdot x+\frac{1}{2 k\cdot p}\int [-2 \varepsilon p\cdot A+{\varepsilon }^{2}{A}^{2}]\hspace{0.167em} \mathrm{d}\xi ,\quad \quad {k}^{2}=0, \end{array} \end{eqnarray} \tag{ 4 }$$

where S_p is the classical action divided by ħ. Equation (4) displays the generic structure of the Gordon–Volkov type states, the exact solutions of the Klein–Gordon equation of a charged particle interacting with a plane electromagnetic wave in vacuum. Concerning the basic mathematical properties of these solutions, see the early papers by Brown and Kibble (1964), Eberly (1969), Bergou and Varró (1980), and the recent work by Boca (2011).

The analysis becomes more complicated if one considers the interaction with a plane wave propagating in a medium of index of refraction n_m ≠ 1. This is due to the presence of the non-vanishing second derivative in (3), since now ${k}^{2}={k}_{0}^{2}(1-{n}_{\mathrm{m}}^{2})\not = 0$. In this case we can avoid the appearance of the first derivative in (3), by using the following Ansatz,

$$\begin{eqnarray} &&\fl \Phi ={\Phi }_{\mathrm{p}}^{(\pm )}(\xi )\exp \left \{ \mp \mathrm{i}\left [p-\frac{(k\cdot p)}{{k}^{2}}k\right ]\cdot x\right \} \nonumber\\ &&\equiv {\Phi }_{\mathrm{p}}^{(\pm )}(\xi )\exp [\mp \mathrm{i}(-\hat {p}\hat {x}-{p}_{1}{x}_{1}-{p}_{2}{x}_{2})],\nonumber\\ &&\fl {k}^{2}\gt 0, \end{eqnarray} \tag{ 5a }$$

$$\begin{eqnarray} &&\fl \hat {p}\equiv \tilde {k}\cdot p/({k}^{2})^{\frac{1}{2}},\tqs \hat {x}\equiv \tilde {k}\cdot x/({k}^{2})^{\frac{1}{2}},\nonumber\\ &&\fl k=\{ {k}^{\mu }\} ={k}^{0}(1,{n}_{\mathrm{m}}{\boldsymbol{e}}_{3}),\tqs \tilde {k}=\{ {\tilde {k}}^{\mu }\} =-{k}^{0}({n}_{\mathrm{m}},{\boldsymbol{e}}_{3}),\nonumber\\ \end{eqnarray} \tag{ 5b }$$

$$\begin{eqnarray} &&\fl \frac{{\mathrm{d}}^{2}{\Phi }_{\mathrm{p}}^{(\pm )}}{\mathrm{d}{\xi }^{2}}+\frac{1}{-{k}^{2}}[-{p}_{\xi }^{2}/{k}^{2}+{p}^{2}-{\kappa }^{2}\mp 2 \varepsilon p\cdot A\nonumber\\ &&+~{\varepsilon }^{2}{A}^{2}]{\Phi }_{\mathrm{ p}}^{(\pm )}=0,\tqs {p}_{\xi }\equiv (k\cdot p), \end{eqnarray} \tag{ 5c }$$

where the ambient sign ∓ refers to the 'positive and negative energy solutions'. The plane wave factor in the first equation in (5a) has also been written as $\exp [\mp \mathrm{i}(-\hat {p}\hat {x}-{p}_{1}{x}_{1}-{p}_{2}{x}_{2})]$, where p₁ = e₁⋅p and p₂ = e₂⋅p are the transverse momentum components. We have introduced the 'complementary wavevector' $\tilde {k}=-{k}^{0}({n}_{\mathrm{m}},{\boldsymbol{e}}_{3})$ (${\tilde {k}}^{2}=-{k}^{2}$ and $\tilde {k}\cdot k=0$), with the help of which one can derive

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle p=\frac{(k\cdot p)}{{k}^{2}}k-\frac{(\tilde {k}\cdot p)}{{k}^{2}}\tilde {k}-(p\cdot {e}_{1}){e}_{1}-(p\cdot {e}_{2}){e}_{2}, \\ \displaystyle \left [p-\frac{(k\cdot p)}{{k}^{2}}k\right ]\cdot x=-\hat {p}\hat {x}-{p}_{1}{x}_{1}-{p}_{2}{x}_{2}. \end{array} \end{eqnarray} \tag{ 5d }$$

If ${n}_{\mathrm{m}}^{2}\lt 1$, then $\tilde {k}$ is space-like, and $\hat {p}\equiv \tilde {k}\cdot p/({k}^{2})^{\frac{1}{2}}$ plays the role of a momentum component conjugated to the 'position variable' $\hat {x}\equiv \tilde {k}\cdot x/({k}^{2})^{\frac{1}{2}}$ (see e.g. Becker 1977). Thus, the Ansatz in (5a) means a separation of variables, where $\tilde {k}\cdot p$, p₁ and p₂ are 'three-momentum type parameters', and p_ξ = (k⋅p) may be said to be an 'energy-type parameter'. This latter parameter p_ξ does not explicitly show up in the solution, because (p_ξ/k²)k_μ is subtracted from p_μ (see (5d)).

In the case of a right-circularly polarized monochromatic plane EM wave, for instance, one has A_cir(x) = A₀(e_xcosξ + e_zsinξ), ${A}_{\mathrm{cir}}^{2}=-{A}_{0}^{2}=$ constant, and one immediately obtains a Mathieu equation for the modulation function ${\Phi }_{\mathrm{p}}^{(\pm )}$, which we write in the standard form (Meixner and Schäfke 1954)

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \frac{{\mathrm{d}}^{2}w}{\mathrm{d}{z}^{2}}+(\lambda -2{h}^{2}\cos 2 z)w=0,\tqs w\equiv {\Phi }_{\mathrm{ p}}^{(\pm )},\\\displaystyle \quad \lambda \equiv (2 k\cdot p)^{2}/({k}^{2})^{2},~{h}^{2}\equiv \mp 4{p}_{\perp }\varepsilon {A}_{0}/{k}^{2}, \end{array} \end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle z=(\xi -\alpha )/2,\tqs \xi ={\omega }_{0}(t-{n}_{\mathrm{m}}y/c),\sin \alpha ={p}_{z}/{p}_{\perp },\\\displaystyle \quad {p}_{\perp }=\sqrt{{p}_{x}^{2}+{p}_{z}^{2}},\tqs {p}^{2}={\kappa }^{2}+{\varepsilon }^{2}{A}_{0}^{2}. \end{array} \end{eqnarray} \tag{ 6b }$$

(See e.g. Becker 1977, and Cronström and Noga 1977.) We note that the two terms κ² and ${\varepsilon }^{2}{A}_{0}^{2}$ in the mass–shell relation are usually combined to ${\kappa }_{\ast }^{2}\equiv {\kappa }^{2}+{\varepsilon }^{2}{A}_{0}^{2}$, which is the intensity-dependent mass shift $\Delta m={m}_{\ast }-m=m\sqrt{1+{\mu }_{0}^{2}}-m$, where μ₀ = eF₀/mcω₀ is the well-known dimensionless intensity parameter (see e.g. Brown and Kibble 1964, and the recent works by Harvey et al 2012, and Müller and Müller 2012).

Henceforth, in equation (1) and in (5c) we specialize A = A(ξ) to represent a monochromatic linearly polarized plane wave,

$$\begin{eqnarray} \begin{array}{@{}rcl@{}} \displaystyle A(x)&\displaystyle =&\displaystyle {e}_{x}{A}_{0}\cos \xi ,\tqs \xi =k\cdot x,\\ \displaystyle \{ {k}^{\mu }\} &\displaystyle =&\displaystyle \frac{{\omega }_{0}}{c}(1,0,{n}_{\mathrm{m}},0),\\ \displaystyle \{ {e}_{x}^{\mu }\} &\displaystyle =&\displaystyle (0,1,0,0),\tqs {A}_{0}={F}_{0}/{k}_{0}, \end{array} \end{eqnarray} \tag{ 7 }$$

where ω₀ = 2πν₀ is the circular frequency and F₀ denotes the amplitude of the electric field strength. This is an x-polarized wave which propagates in the positive y-direction in the medium, i.e. the explicit form of the argument of the cosine is ξ = k⋅x = ω₀(t − n_my/c). The differential equation (5c) for the modulation function ${\Phi }_{\mathrm{p}}^{(\pm )}$ can be brought to the form,

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl \frac{{\mathrm{d}}^{2}{\Phi }_{\mathrm{ p}}^{(\pm )}}{\mathrm{d}{z}^{2}}+({\theta }_{0}+2{\theta }_{1}\cos 2 z+2{\theta }_{2}\cos 4 z){\Phi }_{\mathrm{p}}^{(\pm )}=0,\tqs \\ \displaystyle z=\xi /2, \end{array} \end{eqnarray} \tag{ 8a }$$

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl {\theta }_{0}\equiv \frac{4}{-{k}^{2}}[-(k\cdot p)^{2}/{k}^{2}+{p}^{2}-{\kappa }^{2}-{\varepsilon }^{2}{A}_{0}^{2}/2], \\ \displaystyle \fl 2{\theta }_{1}\equiv \frac{4}{-{k}^{2}}[\pm 2{p}_{x}\varepsilon {A}_{0}],\tqs 2{\theta }_{2}\equiv \frac{4}{-{k}^{2}}[-{\varepsilon }^{2}{A}_{0}^{2}/2]. \end{array} \end{eqnarray} \tag{ 8b }$$

In (8a) we have defined the independent variable z = ξ/2 (as is usual in the theory of differential equations with periodic coefficients), and followed the standard notations for the parameters θ_0,1,2. According to (8a), in the case of a linearly polarized monochromatic wave (7), the modulation function ${\Phi }_{\mathrm{p}}^{(\pm )}(\xi )$ satisfies the so-called Whittaker–Hill equation (or Hill's three-term equation), because now ${A}^{2}=-{A}_{0}^{2}(1+\cos 2 \xi )/2\not =$ constant. If one attempts to solve this equation in terms of a trigonometric series, that procedure leads to five-term recurrence relations between the coefficients, in contrast to the three-term expressions encountered with the Mathieu equation. Thus, the standard procedure (the technique of continued fractions) cannot be taken over from the theory of Mathieu equations. In looking for finite-term solutions, we overcome this difficulty by using the transformation, originally due to Ince (see Arscott 1964, and references therein). This is the basic new element in our present approach. In the spirit of this method, in order to solve (8a), we proceed by introducing the following Ansatz for ${\Phi }_{\mathrm{ps}}^{(\pm )}$

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle {\Phi }_{\mathrm{p}}^{(\pm )}=w\exp \left (-{\theta }_{2}^{\frac{1}{2}}\cos 2 z\right ),\\ \displaystyle \frac{{\mathrm{d}}^{2}w}{\mathrm{d}{z}^{2}}+4{\theta }_{2}^{\frac{1}{2}}\sin 2 z \hspace{0.167em} \frac{\mathrm{d}w}{\mathrm{d}z}+\left [\vphantom {\left (2\theta _1 + 4\theta _2^{\frac {1}{2}} \right )} {\theta }_{0}+2{\theta }_{2}\right .\\ \displaystyle \tqs +\left .\left (2{\theta }_{1}+4{\theta }_{2}^{\frac{1}{2}}\right )\cos 2 z\right ]w=0. \end{array} \end{eqnarray} \tag{ 9 }$$

By using a more condensed notation, from (9) we receive the standard form of Ince's equation,

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl \frac{{\mathrm{d}}^{2}w}{\mathrm{d}{z}^{2}}+a\sin 2 z \hspace{0.167em} \frac{\mathrm{d}w}{\mathrm{d}z}+(\eta -q a\cos 2 z)w=0,\tqs \\ \displaystyle z=\xi /2,\quad a\equiv 4{\theta }_{2}^{\frac{1}{2}}=4\vert \varepsilon \vert {A}_{0}/{k}_{\mathrm{p}}, \end{array} \end{eqnarray} \tag{ 10a }$$

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl 2{\theta }_{1}\equiv -(q+1)a\rightarrow 2{p}_{x}=(q+1){k}_{p},\\ \displaystyle \fl {k}_{p}\equiv \sqrt{{k}^{2}}={k}_{0}\sqrt{1-{n}_{\mathrm{m}}^{2}}\equiv {\omega }_{\mathrm{p}}/c, \end{array} \end{eqnarray} \tag{ 10b }$$

$$\begin{eqnarray} \begin{array}{@{}rcl@{}} \displaystyle \fl \eta \equiv {\theta }_{0}+2{\theta }_{2}&\displaystyle =&\displaystyle 4[{p}_{\xi }^{2}/{k}_{\mathrm{ p}}^{2}+{\varepsilon }^{2}{A}_{0}^{2}+{\kappa }^{2}-{p}^{2}]/{k}_{\mathrm{ p}}^{2},\\ \displaystyle \fl \eta &\displaystyle =&\displaystyle 4[{\hat {p}}^{2}+{\boldsymbol{p}}_{\perp }^{2}+{\kappa }^{2}+{\varepsilon }^{2}{A}_{0}^{2}]/{k}_{\mathrm{ p}}^{2}. \end{array} \end{eqnarray} \tag{ 10c }$$

The second equation of (10c) from the first one has been derived by using the relations

$$\begin{eqnarray} \begin{array}{@{}rcl@{}} \displaystyle {p}_{0}^{2}-{p}_{y}^{2}&\displaystyle =&\displaystyle {p}_{\xi }^{2}/{k}_{\mathrm{ p}}^{2}-{\hat {p}}^{2}, \\ \displaystyle {p}^{2}&\displaystyle =&\displaystyle {p}_{0}^{2}-{p}_{y}^{2}-{\boldsymbol{p}}_{\perp }^{2}={p}_{\xi }^{2}/{k}_{\mathrm{ p}}^{2}-{\hat {p}}^{2}-{\boldsymbol{p}}_{\perp }^{2}, \end{array} \end{eqnarray} \tag{ 10d }$$

which is valid for an arbitrary four vector p_μ, where $\hat {p}$ and p_ξ have been defined in (5b) and (5c), respectively. In obtaining (10a)–(10c) from (8a) and (8b) we have taken a negative test charge (electron; ε < 0), and considered 'positive solutions'. Note that equation (10a) is unchanged under the simultaneous substitutions z → z + π/2 and a → −a, thus we need not consider the case ε > 0 and ε < 0 separately. In (10b) we have introduced the new parameters q = (2p_x/k_p) − 1 and ${k}_{\mathrm{p}}\equiv {k}_{0}\sqrt{1-{n}_{\mathrm{m}}^{2}}\equiv {\omega }_{\mathrm{p}}/c$, where the subscript 'p' in the latter symbol refers to the word 'plasma', though at the present stage we do not need to specify the nature of the medium (in section 4 we shall deal with this interpretation).

In the rest of the present letter we shall study the general equation (10a); however, we shall not carry out a complete analysis of this equation, which has a quite unexplored infinite set of transcendental solutions. Our analysis is restricted to the special class of solutions, the (finite) Ince polynomials, which form a subset of all solutions. The exceptional feature of these solutions is that they form a doubly infinite countable set, corresponding to discrete values of the transverse and longitudinal momentum parameters.

In section 2 we have derived Ince's equation (10a) for the modulation function of a de Broglie plane wave solution of the original Klein–Gordon equation (1). In order to obtain periodic finite solutions we follow the procedure explained in chapter 7 of Arscott (1964). At this point we would like to note that we have already used an analogous method for obtaining a new class of exact solutions of the corresponding Dirac equation, where, in contrast to (10a), we have encountered a complex equation, whose solutions turned out to be finite complex Fourier sums (Varró 2013). Though the method itself used there is analogous to that we are now using, the complex wavefunctions obtained have quite different properties from those of the real Ince polynomials, which we are deriving in the present section.

Let us first expand the solution of Ince's equation (10a) as a cosine series,

$$\begin{eqnarray} &&w=\sum _{r=0}^{\infty }{A}_{r}\cos 2 r z. \end{eqnarray} \tag{ 11 }$$

After substitution (10a), we have the recurrence relations for the unknown coefficients,

$$\begin{eqnarray} &&\fl {-}\eta {A}_{0}+\left (\frac{1}{2}q+1\right )a{A}_{1}=0, \end{eqnarray} \tag{ 11a }$$

$$\begin{eqnarray} &&\fl q a{A}_{0}+(4-\eta ){A}_{1}+\left (\frac{1}{2}q+2\right )a{A}_{2}=0, \end{eqnarray} \tag{ 11b }$$

$$\begin{eqnarray} &&\fl \left (\frac{1}{2}q-r\right )a{A}_{r}+[4(r+1)^{2}-\eta ]{A}_{r+1}\nonumber\\ &&+~\left (\frac{1}{2}q+r+2\right )a{A}_{r+2}=0. \end{eqnarray} \tag{ 11c }$$

The parameter q has been defined in (10b), according to which q = (2p_x/k_p) − 1. If q is a positive even integer, q = 2n, and if moreover η is chosen so that A_n+1 = 0, then, according to (11c) with r = n we have A_n+2 = 0 also, thus, successively A_n+3 = A_n+4 = ⋯ = 0. This means that the series terminates with the term A_ncos2nz, and (11a)–(11c) becomes

$$\begin{eqnarray} &&\fl {-}\eta {A}_{0}+(n+1)a{A}_{1}=0, \end{eqnarray} \tag{ 12a }$$

$$\begin{eqnarray} &&\fl 2 n a{A}_{0}+(4-\eta ){A}_{1}+(n+2)a{A}_{2}=0, \end{eqnarray} \tag{ 12b }$$

$$\begin{eqnarray} &&\fl (n-r)a{A}_{r}+[4(r+1)^{2}-\eta ]{A}_{r+1}\nonumber\\ &&+~(n+r+2)a{A}_{r+2}=0. \end{eqnarray} \tag{ 12c }$$

The solution of this system corresponds to a standard algebraic eigenvalue problem of the following tri-diagonal (n + 1) × (n + 1) matrix.

$$\begin{eqnarray} \fl \left [\begin{array}{@{}ccccc@{}} \displaystyle 0&\displaystyle (n+1)a&\displaystyle &\displaystyle &\displaystyle \\ \displaystyle 2 n a&\displaystyle 4&\displaystyle (n+2)a&\displaystyle &\displaystyle \\ \displaystyle &\displaystyle (n-1)a&\displaystyle 4\cdot {2}^{2}&\displaystyle \cdots &\displaystyle \\ \displaystyle &\displaystyle &\displaystyle \cdots &\displaystyle \cdots &\displaystyle 2 n a\\ \displaystyle &\displaystyle &\displaystyle &\displaystyle (n-(n-1))a&\displaystyle 4{n}^{2} \end{array}\right ]\cdot ~\left [\begin{array}{@{}c@{}} \displaystyle {A}_{0}\\ \displaystyle {A}_{1}\\ \displaystyle \vdots \\ \displaystyle \vdots \\ \displaystyle {A}_{n} \end{array}\right ]\nonumber\\ &&=\eta \cdot \left [\begin{array}{@{}c@{}} \displaystyle {A}_{0}\\ \displaystyle {A}_{1}\\ \displaystyle \vdots \\ \displaystyle \vdots \\ \displaystyle {A}_{n} \end{array}\right ],\tqs {\boldsymbol{L}}_{n}(a)\boldsymbol{A}=\eta \boldsymbol{A}. \end{eqnarray} \tag{ 12d }$$

Following Arscott's notation, we use the symbol for the tri-diagonal matrix,

$$\begin{eqnarray} \fl {\boldsymbol{L}}_{n}&\equiv &\left [\begin{array}{@{}cccccc@{}} \displaystyle {\beta }_{0}&\displaystyle {\gamma }_{0}&\displaystyle &\displaystyle &\displaystyle &\displaystyle \\ \displaystyle {\alpha }_{1}&\displaystyle {\beta }_{1}&\displaystyle {\gamma }_{1}&\displaystyle &\displaystyle &\displaystyle \\ \displaystyle &\displaystyle {\alpha }_{2}&\displaystyle {\beta }_{2}&\displaystyle {\gamma }_{2}&\displaystyle &\displaystyle \\ \displaystyle &\displaystyle &\displaystyle \ddots &\displaystyle \ddots &\displaystyle \ddots &\displaystyle \\ \displaystyle &\displaystyle &\displaystyle &\displaystyle \ddots &\displaystyle \ddots &\displaystyle {\gamma }_{n-1}\\ \displaystyle &\displaystyle &\displaystyle &\displaystyle &\displaystyle {\alpha }_{n}&\displaystyle {\beta }_{n} \end{array}\right ]=\left \langle \begin{array}{@{}ccccc@{}} \displaystyle &\displaystyle {\gamma }_{0}&\displaystyle \ldots &\displaystyle \ldots &\displaystyle {\gamma }_{n-1}\\ \displaystyle {\beta }_{0}&\displaystyle {\beta }_{1}&\displaystyle \ldots &\displaystyle \ldots &\displaystyle {\beta }_{n}\\ \displaystyle {\alpha }_{1}&\displaystyle {\alpha }_{2}&\displaystyle \ldots &\displaystyle {\alpha }_{n}&\displaystyle \end{array}\right \rangle .\nonumber\\ \end{eqnarray} \tag{ 13 }$$

With this notation the tri-diagonal matrix, L_n(a) in (12d) is represented by the following symbol

$$\begin{eqnarray} \begin{array}{@{}c@{}} \displaystyle \left \langle \begin{array}{@{}ccccc@{}} \displaystyle &\displaystyle (n+1)a&\displaystyle (n+2)a&\displaystyle \ldots &\displaystyle 2 n a\\ \displaystyle 0&\displaystyle 4&\displaystyle 1 6&\displaystyle \ldots &\displaystyle 4{n}^{2}\\ \displaystyle 2 n a&\displaystyle (n-1)a&\displaystyle (n-2)a&\displaystyle \ldots a&\displaystyle \end{array}\right \rangle ={\boldsymbol{L}}_{n}(a),\\ \displaystyle {\lambda }_{n}(\eta ,a)=\det \hspace{0.167em} [{\boldsymbol{L}}_{n}(a)-\eta \boldsymbol{I}]. \end{array} \end{eqnarray} \tag{ 14 }$$

In (14) we have also introduced the characteristic polynomial λ_n(η,a) associated with the eigenvalue problem, where I is the (n + 1) × (n + 1) unit matrix. The eigenvalues are given by the zeros of the characteristic polynomial ${\lambda }_{n}(\eta ,a)={\mathop{\prod }\nolimits }_{k=0}^{n}(\eta -{\eta }_{n}^{(k)})$, in our case they are all real, and they are all simple roots of the equation λ_n(η,a) = 0. This is a consequence of the following lemma quoted on page 21 by Arscott (1964).

Lemma 1 (a) Let the numbers α_i,β_i,γ_i be all real, each product α_i+1γ_i positive, and let L_n denote the tri-diagonal matrix above (13), all other elements being zero. Then the latent roots of L_n are all real and different.

(We have inserted '(13)' into the quotation, in order to refer our equation (13) above.) There are (n + 1) real linearly independent eigenvectors $\{ {A}_{r}^{(k)}\}$, associated with the eigenvalues ${\eta }_{n}^{(k)}~(0\leq k\leq n)$. According to the above considerations, if q = 2n, i.e. if ${p}_{x}={k}_{\mathrm{p}}(n+\frac{1}{2})$, then the potentially infinite cosine series in (11) reduces to the Ince polynomial

$$\begin{eqnarray} &&\fl w={\varphi }_{n}^{k}(\xi \vert a,\mathrm{c})=\sum _{r=0}^{n}{A}_{r}^{(k)}(a\vert n+1)\cos 2 r z,\tqs z=\xi /2,\nonumber\\ &&(n=1,2,\ldots ),\quad (0\leq k\leq n). \end{eqnarray} \tag{ 15 }$$

These solutions may be called 'even solutions' of (10a). In the notation ${\varphi }_{n}^{k}(\xi \vert a,\mathrm{c})$ the letter 'c' refers to the word 'cosine', and in ${A}_{r}^{(k)}(a~\vert ~n+1)$ the second argument refers to there being n + 1 sets of the coefficients (eigenvectors), and the superscript labels the kth such set.

We note that Ince introduced the following notations for the ordered roots of λ_n(η,a) = 0 and for the associated polynomials (Arscott 1964)

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl \eta ={\eta }_{n}^{(m)}\equiv {a}_{2 n}^{2 m}(a)\tqs (m=0,1,\ldots ,n),\\\displaystyle \quad {a}_{2 n}^{0}\lt {a}_{2 n}^{2}\lt \cdots \lt {a}_{2 n}^{2 n},\\\displaystyle \quad \fl w={C}_{2 n}^{2 m}(z,a)=\sum _{r=0}^{n}{A}_{r}\cos 2 k z. \end{array} \end{eqnarray} \tag{ 16a }$$

These functions are normalized and they are orthogonal to each other,

$$\begin{eqnarray} &&\fl \int \nolimits _{-\pi }^{\pi }[{C}_{2 n}^{2 m}(z,a)]^{2}\hspace{0.167em} \mathrm{d}z=\pi , \end{eqnarray} \tag{ 16b }$$

$$\begin{eqnarray} &&\fl \int \nolimits _{-\pi }^{\pi }{\mathrm{e}}^{-(a/2)\cos 2 z}{C}_{2 n}^{2 l}(z,a){C}_{2 n}^{2 m}(z,a)\hspace{0.167em} \mathrm{d}z=0,\tqs l\not = m.\nonumber\\ \end{eqnarray} \tag{ 16c }$$

The latter orthogonality relation can be derived in the usual way, by using (8a), (9) and (10a). The wavefunction in equation (15) is the Ince polynomial ${C}_{2 n}^{2 k}(z,a)$ displayed in (16a),

$$\begin{eqnarray} &&\fl w={\varphi }_{n}^{k}(\xi \vert a,\mathrm{c})=\sum _{r=0}^{n}{A}_{r}^{(k)}(a\vert n+1)\cos 2 r z={C}_{2 n}^{2 k}(z,a).\nonumber\\ \end{eqnarray} \tag{ 16d }$$

Still for q an even positive number, q = 2n, there are also odd solutions of Ince's equation (10a). Let us now expand the wavefunction as a series of sines,

$$\begin{eqnarray} &&w=\sum _{r=0}^{\infty }{B}_{r}\sin 2 r z. \end{eqnarray} \tag{ 17 }$$

The recurrence relations for the unknowns have the same structure as equations (12a)–(12c), except that the first is omitted, and we take B₀ = 0,

$$\begin{eqnarray} &&\fl (4-\eta ){B}_{1}+\left (\frac{1}{2}q+2\right )a{B}_{2}=0, \end{eqnarray} \tag{ 17a }$$

$$\begin{eqnarray} &&\fl \left (\frac{1}{2}q-r\right )a{B}_{r}+[4(r+1)^{2}-\eta ]{B}_{r+1}\nonumber\\ &&+~\left (\frac{1}{2}q+r+2\right )a{B}_{r+2}=0. \end{eqnarray} \tag{ 17b }$$

If q is a positive even integer, q = 2n, and if moreover η is chosen such that B_n+1 = 0, then, according to equation (17b), with r = n we have B_n+2 = 0 also, thus, successively B_n+3 = B_n+4 = ⋯ = 0, which means that the series terminates with the term B_nsin2nz,

$$\begin{eqnarray} &&\fl (4-\eta ){B}_{1}+(n+2)a{B}_{2}=0, \end{eqnarray} \tag{ 18a }$$

$$\begin{eqnarray} &&\fl (n-r)a{B}_{r}+[4(r+1)^{2}-\eta ]{B}_{r+1}\nonumber\\ &&+~(n+r+2)a{B}_{r+2}=0. \end{eqnarray} \tag{ 18b }$$

The solutions of this system are determined by solving the eigenvalue problem of the n × n tri-diagonal matrix

$$\begin{eqnarray} \fl \left [\begin{array}{@{}ccccc@{}} \displaystyle 4&\displaystyle (n+2)a&\displaystyle &\displaystyle &\displaystyle \\ \displaystyle (n-1)a&\displaystyle 4\cdot {2}^{2}&\displaystyle (n+3)a&\displaystyle &\displaystyle \\ \displaystyle &\displaystyle (n-2)a&\displaystyle 4\cdot {3}^{2}&\displaystyle \cdots &\displaystyle \\ \displaystyle &\displaystyle &\displaystyle \cdots &\displaystyle \cdots &\displaystyle (n+n)a\\ \displaystyle &\displaystyle &\displaystyle &\displaystyle (n-(n-1))a&\displaystyle 4{n}^{2} \end{array}\right ]\nonumber\\ &&\cdot \left [\begin{array}{@{}c@{}} \displaystyle {B}_{1}\\ \displaystyle {B}_{2}\\ \displaystyle \vdots \\ \displaystyle \vdots \\ \displaystyle {B}_{n} \end{array}\right ]=\eta \cdot \left [\begin{array}{@{}c@{}} \displaystyle {B}_{1}\\ \displaystyle {B}_{2}\\ \displaystyle \vdots \\ \displaystyle \vdots \\ \displaystyle {B}_{n} \end{array}\right ],\tqs {\boldsymbol{M}}_{n}(a)\boldsymbol{B}=\eta \boldsymbol{B}. \end{eqnarray} \tag{ 18c }$$

By using the condensed notation, as in (14), we write

$$\begin{eqnarray} \begin{array}{@{}c@{}} \displaystyle \left \langle \begin{array}{@{}cccc@{}} \displaystyle &\displaystyle (n+2)a&\displaystyle \cdots &\displaystyle 2 n a\\ \displaystyle 4&\displaystyle 4\cdot {2}^{2}&\displaystyle \cdots &\displaystyle 4{n}^{2}\\ \displaystyle (n-1)a&\displaystyle (n-2)a&\displaystyle \cdots a&\displaystyle \end{array}\right \rangle ={\boldsymbol{M}}_{n}(a),\\ \displaystyle {\mu }_{n}(\eta ,a)=\det [{\boldsymbol{M}}_{n}(a)-\eta \boldsymbol{I}], \end{array} \end{eqnarray} \tag{ 19 }$$

where I means here the n × n unit matrix. The eigenvalues are the roots of the characteristic polynomial, i.e. ${\mu }_{n}({\eta }_{n}^{(k)},a)=0$. The matrix M_n(a) in (19) also satisfies the conditions of the above lemma, thus the eigenvalues are all real and simple. In this way, we have found that, if q = 2n, i.e. if ${p}_{x}={k}_{\mathrm{p}}(n+\frac{1}{2})$, then the potentially infinite sine series in (17) reduces to the following Ince polynomial

$$\begin{eqnarray} &&\fl w={\varphi }_{n}^{k}(\xi \vert a,\mathrm{s})=\sum _{r=1}^{n}{B}_{r}^{(k)}(a\vert n)\sin 2 r z,\tqs \nonumber\\ &&z=\xi /2,\quad (n=1,2,\ldots ),\quad (1\leq k\leq n). \end{eqnarray} \tag{ 20 }$$

These solutions may be called 'odd solutions' of (10a). In the notation ${\varphi }_{n}^{k}(\xi \vert a,\mathrm{s})$ the letter 's' refers to the word 'sine', and in the coefficients ${B}_{r}^{(k)}(a\vert n)$ the second argument shows that there are n sets of the coefficients (eigenvectors), where the superscript labels the kth such set. The properties of the 'even sine-type solutions' discovered have been summarized by Arscott (1964) in complete analogy with (16a)–(16c),

$$\begin{eqnarray} &&\fl \eta ={\eta }_{n}^{(m)}\equiv {b}_{2 n}^{2 m}(a)\tqs (m=1,2,\ldots ,n),\tqs \nonumber\\ &&{b}_{2 n}^{2}\lt {b}_{2 n}^{4}\lt \cdots \lt {b}_{2 n}^{2 n},\nonumber\\ &&\fl w={S}_{2 n}^{2 m}(z,a)=\sum _{r=1}^{n}{B}_{r}\sin 2 r z, \end{eqnarray} \tag{ 21a }$$

$$\begin{eqnarray} &&\fl \int \nolimits _{-\pi }^{\pi }[{S}_{2 n}^{2 m}(z,a)]^{2}\hspace{0.167em} \mathrm{d}z=\pi , \end{eqnarray} \tag{ 21b }$$

$$\begin{eqnarray} &&\fl \int \nolimits _{-\pi }^{\pi }{\mathrm{e}}^{-(a/2)\cos 2 z}{S}_{2 n}^{2 l}(z,a){S}_{2 n}^{2 m}(z,a)\hspace{0.167em} \mathrm{d}z=0,l\not = m.\nonumber\\ \end{eqnarray} \tag{ 21c }$$

The 'even sine-type solutions' in (20) thus coincide with the corresponding Ince polynomials in (21a),

$$\begin{eqnarray} &&\fl w={\varphi }_{n}^{k}(\xi \vert a,\mathrm{s})=\sum _{r=0}^{n}{B}_{r}^{(k)}(a\vert n)\sin 2 r z={S}_{2 n}^{2 k}(z,a). \end{eqnarray} \tag{ 21d }$$

There are finite solutions with q being an odd number. Let us try to find these in the form

$$\begin{eqnarray} &&w=\sum _{r=0}^{\infty }{A}_{r}^{\prime}\cos (2 r+1)z. \end{eqnarray} \tag{ 22 }$$

The coefficients satisfy the recurrence relations

$$\begin{eqnarray} &&\fl \left [\frac{1}{2}(q+1)a-(\eta -1)\right ]{A}_{0}^{\prime}+\frac{1}{2}(q+3)a{A}_{1}^{\prime}=0, \end{eqnarray} \tag{ 22a }$$

$$\begin{eqnarray} &&\fl \frac{1}{2}[q-(2 r+1)]a{A}_{r}^{\prime}+[(2(r+1)+1)^{2}-\eta ]{A}_{r+1}^{\prime}\nonumber\\ &&+~\frac{1}{2}[q+(2(r+2)+1)]a{A}_{r+2}^{\prime}=0. \end{eqnarray} \tag{ 22b }$$

Let us assume that q is a positive odd integer, q = 2n + 1, and require that η is such that at r = n + 1 the coefficient ${A}_{n+1}^{\prime}=0$ vanishes. Then, by putting r = n in (22b) we see that the first two terms on the left-hand side are zero, consequently ${A}_{n+2}^{\prime}=0$ vanishes, too, and all the higher index coefficient vanish—thus we have again a finite-term expression for the wavefunction. Accordingly, (22a) and (22b) are rewritten as

$$\begin{eqnarray} &&\fl [-\eta +1+(n+1)a]{A}_{0}^{\prime}+(n+2)a{A}_{1}^{\prime}=0, \end{eqnarray} \tag{ 23a }$$

$$\begin{eqnarray} &&\fl (n-r)a{A}_{r}^{\prime}+[(2(r+1)+1)^{2}-\eta ]{A}_{r+1}^{\prime}\nonumber\\ &&+~(n+1+r+2)a{A}_{r+2}^{\prime}=0. \end{eqnarray} \tag{ 23b }$$

The characteristic numbers are solutions of the eigenvalue problem of the (n + 1) × (n + 1) tri-diagonal matrix

$$\begin{eqnarray} \fl \left [\begin{array}{@{}ccccc@{}} \displaystyle 1+(n+1)a&\displaystyle (n+2)a&\displaystyle &\displaystyle &\displaystyle \\ \displaystyle n a&\displaystyle {3}^{2}&\displaystyle (n+3)a&\displaystyle &\displaystyle \\ \displaystyle &\displaystyle (n-1)a&\displaystyle {5}^{2}&\displaystyle \cdots &\displaystyle \\ \displaystyle &\displaystyle &\displaystyle \cdots &\displaystyle \cdots &\displaystyle (2 n+1)a\\ \displaystyle &\displaystyle &\displaystyle &\displaystyle a&\displaystyle (2 n+1)^{2} \end{array}\right ]\cdot ~\left [\begin{array}{@{}c@{}} \displaystyle {A}_{0}^{\prime}\\ \displaystyle {A}_{1}^{\prime}\\ \displaystyle \vdots \\ \displaystyle \vdots \\ \displaystyle {A}_{n}^{\prime} \end{array}\right ]\nonumber\\ &&=\eta \cdot \left [\begin{array}{@{}c@{}} \displaystyle {A}_{0}^{\prime}\\ \displaystyle {A}_{1}^{\prime}\\ \displaystyle \vdots \\ \displaystyle \vdots \\ \displaystyle {A}_{n}^{\prime} \end{array}\right ],\tqs {\boldsymbol{K}}_{n}(a){\boldsymbol{A}}^{\prime}=\eta {\boldsymbol{A}}^{\prime}. \end{eqnarray} \tag{ 23c }$$

By using the condensed notation introduced in (13), we have

$$\begin{eqnarray} \fl \left \langle \begin{array}{@{}ccccc@{}} \displaystyle &\displaystyle (n+2)a&\displaystyle (n+3)a&\displaystyle \cdots &\displaystyle (2 n+1)a\\ \displaystyle 1+(n+1)a&\displaystyle {3}^{2}&\displaystyle {5}^{2}&\displaystyle \cdots &\displaystyle (2 n+1)^{2}\\ \displaystyle n a&\displaystyle (n-1)a&\displaystyle (n-2)a&\displaystyle \cdots a&\displaystyle \end{array}\right \rangle \nonumber\\ &&={\boldsymbol{K}}_{n}(a),\quad {\kappa }_{n}(\eta ,a)=\det [{\boldsymbol{K}}_{n}(a)-\eta \boldsymbol{I}], \end{eqnarray} \tag{ 24 }$$

where I means the (n + 1) × (n + 1) unit matrix. According to the above lemma, the eigenvalues (the roots of the characteristic equation κ_n(η,a) = 0) are all real and different. We have shown that if q = 2n + 1, i.e. if p_x = k_p(n + 1), then the potentially infinite cosine series in (22) reduces to the Ince polynomial

$$\begin{eqnarray} &&\fl w={\chi }_{n}^{k}(\xi \vert a,\mathrm{c})=\sum _{r=0}^{n}{A}_{r}^{{^{\prime}}~(k)}(a\vert n+1)\cos (2 r+1)z,\nonumber\\ &&z=\xi /2,\quad (n=0,1,\ldots ),\quad (0\leq k\leq n). \end{eqnarray} \tag{ 25 }$$

By using Ince's notation, the ordered roots of κ_n(η,a) = 0 and the associated polynomials are

$$\begin{eqnarray} &&\fl \eta ={\eta }_{n}^{(m)}\equiv {a}_{2 n+1}^{2 m+1}(a)\tqs (m=0,1,\ldots ,n),\nonumber\\ &&{a}_{2 n+1}^{1}\lt {a}_{2 n+1}^{2+1}\lt \cdots \lt {a}_{2 n+1}^{2 n+1},\nonumber\\ &&\fl w={C}_{2 n+1}^{2 m+1}(z,a)=\sum _{r=0}^{n}{A}_{r}^{\prime}\cos (2 r+1)z. \end{eqnarray} \tag{ 26a }$$

The normalization condition and the orthogonality relation of ${C}_{2 n+1}^{2 k+1}(z,a)$ are the same as in (16b) and (16c),

$$\begin{eqnarray} &&\fl \int \nolimits _{-\pi }^{\pi }[{C}_{2 n+1}^{2 m+1}(z,a)]^{2}\hspace{0.167em} \mathrm{d}z=\pi , \end{eqnarray} \tag{ 26b }$$

$$\begin{eqnarray} &&\fl \int \nolimits _{-\pi }^{\pi }{\mathrm{e}}^{-(a/2)\cos 2 z}{C}_{2 n+1}^{2 l+1}(z,a){C}_{2 n+1}^{2 m+1}(z,a)\hspace{0.167em} \mathrm{d}z=0,\tqs \nonumber\\ &&l\not = m. \end{eqnarray} \tag{ 26c }$$

The wavefunction in equation (25) can be expressed with the Ince polynomial ${C}_{2 n+1}^{2 k+1}(z,a)$ in (26a),

$$\begin{eqnarray} w&=&{\chi }_{n}^{k}(\xi \vert a,\mathrm{c})=\sum _{r=0}^{n}{A}_{r}^{{^{\prime}}~(k)}(a\vert n+1)\cos (2 r+1)z\nonumber\\ &=&{C}_{2 n+1}^{2 k+1}(z,a). \end{eqnarray} \tag{ 26d }$$

Finally, we study the 'odd sine-type solutions'

$$\begin{eqnarray} &&w=\sum _{r=0}^{\infty }{B}_{r}^{\prime}\sin (2 r+1)z. \end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray} &&\fl \left [-\eta +1-\frac{1}{2}(q+1)a\right ]{B}_{0}^{\prime}+\frac{1}{2}(q+3)a{B}_{1}^{\prime}=0, \end{eqnarray} \tag{ 27a }$$

$$\begin{eqnarray} &&\fl \frac{1}{2}[q-(2 r+1)]a{B}_{r}^{\prime}+[(2(r+1)+1)^{2}-\eta ]{B}_{r+1}^{\prime}\nonumber\\ &&+~\frac{1}{2}[q+(2(r+2)+1)]a{B}_{r+2}^{\prime}=0. \end{eqnarray} \tag{ 27b }$$

By using the same reasoning as above for an odd q = 2n + 1, from (27a) and (27b) we have

$$\begin{eqnarray} &&\fl [-\eta +1-(n+1)a]{B}_{0}^{\prime}-(n+2)a{B}_{1}^{\prime}=0, \end{eqnarray} \tag{ 28a }$$

$$\begin{eqnarray} &&\fl (n-r)a{B}_{r}^{\prime}+[(2(r+1)+1)^{2}-\eta ]{B}_{r+1}^{\prime}\nonumber\\ &&+~(n+1+r+2)a{B}_{r+2}^{\prime}=0, \end{eqnarray} \tag{ 28b }$$

$$\begin{eqnarray} \fl \left [\begin{array}{@{}ccccc@{}} \displaystyle 1-(n+1)a&\displaystyle (n+2)a&\displaystyle &\displaystyle &\displaystyle \\ \displaystyle n a&\displaystyle {3}^{2}&\displaystyle (n+3)a&\displaystyle &\displaystyle \\ \displaystyle &\displaystyle (n-1)a&\displaystyle {5}^{2}&\displaystyle \cdots &\displaystyle \\ \displaystyle &\displaystyle &\displaystyle \cdots &\displaystyle \cdots &\displaystyle (2 n+1)a\\ \displaystyle &\displaystyle &\displaystyle &\displaystyle a&\displaystyle (2 n+1)^{2} \end{array}\right ]\nonumber\\ &&\cdot ~\left [\begin{array}{@{}c@{}} \displaystyle {B}_{0}^{\prime}\\ \displaystyle {B}_{1}^{\prime}\\ \displaystyle \vdots \\ \displaystyle \vdots \\ \displaystyle {B}_{n}^{\prime} \end{array}\right ]=\eta \cdot \left [\begin{array}{@{}c@{}} \displaystyle {B}_{0}^{\prime}\\ \displaystyle {B}_{1}^{\prime}\\ \displaystyle \vdots \\ \displaystyle \vdots \\ \displaystyle {B}_{n}^{\prime} \end{array}\right ],\tqs {\tilde {\boldsymbol{K}}}_{n}(a){\boldsymbol{B}}^{\prime}=\eta {\boldsymbol{B}}^{\prime}. \end{eqnarray} \tag{ 28c }$$

With the condensed notation introduced in (13), we have

$$\begin{eqnarray} \fl \left \langle \begin{array}{@{}ccccc@{}} \displaystyle &\displaystyle (n+2)a&\displaystyle (n+3)a&\displaystyle \cdots &\displaystyle (2 n+1)a\\ \displaystyle 1-(n+1)a&\displaystyle {3}^{2}&\displaystyle {5}^{2}&\displaystyle \ldots &\displaystyle (2 n+1)^{2}\\ \displaystyle n a&\displaystyle (n-1)a&\displaystyle (n-2)a&\displaystyle \cdots a&\displaystyle \end{array}\right \rangle \nonumber\\ &&={\tilde {\boldsymbol{K}}}_{n}(a),\quad {\tilde {\kappa }}_{n}(\eta ,a)=\det [{\tilde {\boldsymbol{K}}}_{n}(a)-\eta \boldsymbol{I}], \end{eqnarray} \tag{ 29 }$$

where I is the (n + 1) × (n + 1) unit matrix. As has been pointed out by Arscott (1964), the characteristic equation ${\tilde {\kappa }}_{n}(\eta ,a)=0$ is equivalent to κ_n(η, − a) = 0, where κ_n(η,a) has been defined in (24). To summarize this part of the section, we have found that if q = 2n + 1, i.e. if p_x = k_p(n + 1), then the cosine series in (22) reduces to the Ince polynomial

$$\begin{eqnarray} &&\fl w={\chi }_{n}^{k}(\xi \vert a,\mathrm{s})=\sum _{r=0}^{n}{B}_{r}^{{^{\prime}}~(k)}(a\vert n+1)\sin (2 r+1)z,\tqs \nonumber\\ &&z=\xi /2,\quad (n=0,1,\ldots ),\quad (0\leq k\leq n). \end{eqnarray} \tag{ 30 }$$

By using Ince's notation, the ordered roots of ${\tilde {\kappa }}_{n}(\eta ,a)=0$ and the associated polynomials are

$$\begin{eqnarray} &&\fl \eta ={\eta }_{n}^{(m)}\equiv {b}_{2 n+1}^{2 m+1}(a)\tqs (m=0,1,\ldots ,n),\nonumber\\ &&{b}_{2 n+1}^{1}\lt {b}_{2 n+1}^{2+1}\lt \cdots \lt {b}_{2 n+1}^{2 n+1},\nonumber\\ &&\fl w={S}_{2 n+1}^{2 m+1}(z,a)=\sum _{r=0}^{n}{B}_{r}^{{^{\prime}}~(m)}\sin (2 r+1)z. \end{eqnarray} \tag{ 31a }$$

The normalization condition and the orthogonality relation of ${S}_{2 n+1}^{2 m+1}(z,a)$ are the same as in (26b) and (26c),

$$\begin{eqnarray} &&\fl \int \nolimits _{-\pi }^{\pi }\left [{S}_{2 n+1}^{2 m+1}(z,a)\right ]^{2}\hspace{0.167em} \mathrm{d}z=\pi , \end{eqnarray} \tag{ 31b }$$

$$\begin{eqnarray} &&\fl \int \nolimits _{-\pi }^{\pi }{\mathrm{e}}^{-(a/2)\cos 2 z}{S}_{2 n+1}^{2 l+1}(z,a){S}_{2 n+1}^{2 m+1}(z,a)\hspace{0.167em} \mathrm{d}z=0,\nonumber\\ &&l\not = m. \end{eqnarray} \tag{ 31c }$$

The wavefunction in equation (30) can be expressed with the Ince polynomial ${S}_{2 n+1}^{2 k+1}(z,a)$ in (31a),

$$\begin{eqnarray} w&=&{\chi }_{n}^{k}(\xi \vert a,\mathrm{s})=\sum _{r=0}^{n}{B}_{r}^{{^{\prime}}~(k)}(a\vert n+1)\sin (2 r+1)z\nonumber\\ &=&{S}_{2 n+1}^{2 k+1}(z,a). \end{eqnarray} \tag{ 31d }$$

According to (5a), (9), (10a), (15), (20), (25) and (30) the exact polynomial solutions of the Klein–Gordon equation (1) can be written as

$$\begin{eqnarray} {\Phi }_{\mathrm{p}}&=&\exp [+\mathrm{i}(\hat {p}\hat {x}+{p}_{x}x+{p}_{z}z)]\times \exp [-(a/4)\cos \xi ]\nonumber\\ &&\times ~I{P}_{n}^{k}(\xi \vert a), \end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray} &&I{P}_{n}^{k}(\xi \vert a)\equiv {\varphi }_{n}^{k}(\xi \vert a,\mathrm{c}),\tqs {\varphi }_{n}^{k}(\xi \vert a,\mathrm{s}),\nonumber\\ &&{\chi }_{n}^{k}(\xi \vert a,\mathrm{c})\tqs \text{or}\tqs {\chi }_{n}^{k}(\xi \vert a,\mathrm{s}). \end{eqnarray} \tag{ 32a }$$

For short, we have denoted the (Ince) polynomials (15), (20), (25) and (30), by the general symbol $I{P}_{n}^{k}(\xi \vert a)$. For each n-values (corresponding to a discrete set of transverse momenta 2p_x = (q + 1)k_p), the coefficients of the polynomials given by (15), (20), (25) and (30), satisfy the inhomogeneous equations (12d), (18c), (23c) and (28c), respectively. There are n (or n + 1) real, linearly independent eigenvectors A^(k), B^(k), A'^(k) and B'^(k). The eigenfunctions associated with the eigenvalues ${\eta }_{n}^{(k)}$ form a doubly infinite set labelled by the integer numbers n and k. The eigenvalues are all real, and the eigenfunctions are real trigonometric polynomials, which coincide with the Ince polynomials, as has been explicitly displayed by (16d), (21d), (26d) and (31d). The possible values of the momentum parameter of the solutions are connected to eigenvalues ${\eta }_{n}^{(k)}$ by (10c). Since p_x is given (it is either ${p}_{x}=(n+\frac{1}{2}){k}_{\mathrm{p}}$ or p_x = (n' + 1)k_p, where n = 1,2,... and n' = 0,1,...), the relation (10c) determines n (or n') possible values for $\hat {p}$, which may be labelled as ${\hat {p}}_{n}^{(k)}$, where 0,1 ≤ k ≤ n. We note that for any value of the transverse momentum p_x = (q + 1)k_p/2 (regardless of q is an integer or not) it is always possible to find a denumerably infinite set of basically periodic even solutions ${C}_{q}^{m}(z,a)$ and odd solutions ${S}_{q}^{m}(z,a)$, associated with the eigenvalues ${a}_{q}^{m}(a)$ and ${b}_{q}^{m}(a)$, respectively (Arscott 1964). In the limit of small intensities (a → 0) these transcendental Ince functions go over to cosmz and sinmz. Research on the asymptotic behaviour of the so-called instability zones of Whittaker–Hill type equations, such as (8a), has recently led to new significant results (Djakov and Mityagin 2007), which may have importance in applying our solutions; however, the discussion of these is beyond the scope of the present letter.

We also note that an analogous new set of exact solutions of the corresponding Dirac equation have been recently found by us (Varró 2013), which are also expressed by finite trigonometric polynomials. However, these new polynomials are complex Fourier sums, whose interrelation with the Ince polynomials (which have an already well-developed theory) has not been explored.

The investigation of various possible physical applications of the exact solutions (32) is out of the scope of the present letter. The emphasis has been on proving the existence and on constructing the general polynomial solutions. In the present section we shall only briefly discuss a possible physical background of these solutions, and give a few numerical illustrations.

Each of the solutions (32) contains the exponential factor exp[ − (a/4)cosξ], which can be expanded into an infinite Fourier series (Gradshteyn and Ryzhik 1980 or Abramowitz and Stegun 1965),

$$\begin{eqnarray} \fl \exp [-(a/4)\cos \xi ]&=&\sum _{l=-\infty }^{\infty }{I}_{l}(a/4)\exp [\mathrm{i}l(\xi -\pi )]={I}_{0}(a/4)\nonumber\\ &&+~2\sum _{l=1}^{\infty }{I}_{l}(a/4)\cos [l(\xi -\pi )], \end{eqnarray} \tag{ 33 }$$

where I_l(z) denotes the modified Bessel functions of first kind of order l. This means that, though the polynomials $I{P}_{n}^{k}(\xi \vert a)$ are finite-term expressions, the complete wavefunction (32) contains all the higher harmonics of the fundamental frequency. If a is large, then this function is peaked at the points ξ = π − 2kπ (k being an arbitrary integer) with an exponentially large contrast, as is shown in figure 1.

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Figure 1 illustrates the simple fact that the exponential factor exp[ − (a/2)cosξ] behaves as an envelope on the base interval, rather than showing some structured modulations, which might be expected on the basis of its high harmonic content, according to (33). For large a, the spectrum, governed by the modified Bessel function of first kind, is practically independent of the index l, owing to the asymptotic behaviour ${I}_{l}(z)=({\mathrm{e}}^{z}/\sqrt{\pi z})[1+\mathrm{O}(1/z)]$ (Abramowitz and Stegun 1965).

The differential equation (10a) has led to a two-parameter eigenvalue problem, where we have considered a = 4|ε|A₀/k_p as a 'fundamental parameter', and 2p_x = (q + 1)k_p and η were 'disposable parameters'. We have seen that the condition q + 1 = 2n leads to even cosine-type and even sine-type solutions, and q + 1 = 2n + 1 to odd cosine-type and odd sine-type solutions. On the basis of the two equivalent forms of the eigenvalues η in (10c), the momentum parameters can be expressed as

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl {\hat {p}}_{n}^{(k)}=\pm ({k}_{\mathrm{p}}/2)\\ \displaystyle \fl \times ~\sqrt{{\eta }_{n}^{(k)}-(q+1)^{2}-(2{p}_{z}/{k}_{\mathrm{p}})^{2}-(2 \kappa /{k}_{\mathrm{p}})^{2}-(a/2)^{2}},\\ \displaystyle \fl 2{p}_{\xi }/{k}_{\mathrm{p}}^{2}=\pm \sqrt{{\eta }_{n}^{(k)}-(a/2)^{2}}. \end{array} \end{eqnarray} \tag{ 34 }$$

The second equation of a simpler form in (34) has been obtained by imposing (in addition) the free mass–shell condition (p² = κ²) for the momentum parameter p_μ, where the definition of p_ξ in (10c) has also been used. The eigenvalues ${\eta }_{n}^{(k)}$ are the roots of the characteristic equations λ_n(η,a) = 0, μ_n(η,a) = 0, κ_n(η,a) = 0 and ${\tilde {\kappa }}_{n}(\eta ,a)=0$, where the characteristic polynomials have been defined in (14), (19), (24) and (29). In general, p₀ and p_y can be expressed as

$$\begin{eqnarray} \begin{array}{@{}rcl@{}} \displaystyle {p}_{0}&\displaystyle =&\displaystyle {k}_{0}({p}_{\xi }/{k}_{\mathrm{p}}^{2}+{n}_{\mathrm{ m}}\hat {p}/{k}_{\mathrm{p}}),\tqs \\ \displaystyle {p}_{y}&\displaystyle =&\displaystyle {k}_{0}({n}_{\mathrm{m}}{p}_{\xi }/{k}_{\mathrm{p}}^{2}+\hat {p}/{k}_{\mathrm{ p}}), \end{array} \end{eqnarray} \tag{ 34a }$$

regardless of any constraints. If we impose the free mass–shell condition p² = κ², then, from (34), by choosing the positive sign in the second equation, we have

$$\begin{eqnarray} &&\fl {p}_{0}=\frac{{k}_{0}}{2}\left [\sqrt{{\eta }_{n}^{(k)}-(a/2)^{2}}\pm {n}_{\mathrm{m}}\{ {\eta }_{n}^{(k)}-(a/2)^{2}-(q+1)^{2}\right .\nonumber\\ &&-~\left .\vphantom {\frac {{k}_{0}}{2}\left [\sqrt {{\eta }_{n}^{(k)}-(a/2)^{2}}\pm {n}_{\mathrm {m}}\{ {\eta }_{n}^{(k)}-(a/2)^{2}-(q+1)^{2}\right ]} (2 \kappa /{k}_{\mathrm{ p}})^{2}-(2{p}_{z}/{k}_{\mathrm{p}})\} ^{1/2}\right ], \end{eqnarray} \tag{ 34b }$$

$$\begin{eqnarray} &&\fl {p}_{y}=\frac{{k}_{0}}{2}\left [{n}_{\mathrm{m}}\sqrt{{\eta }_{n}^{(k)}-(a/2)^{2}}\pm \{ {\eta }_{n}^{(k)}-(a/2)^{2}\right .\nonumber\\ &&-~\left .\vphantom {\frac {{k}_{0}}{2}\left [{n}_{\mathrm {m}}\sqrt {{\eta }_{n}^{(k)}-(a/2)^{2}}\pm \{ {\eta }_{n}^{(k)}-(a/2)^{2}\right ]} (q+1)^{2}-(2 \kappa /{k}_{\mathrm{ p}})^{2}-(2{p}_{z}/{k}_{\mathrm{p}})\} ^{1/2}\right ]. \end{eqnarray} \tag{ 34c }$$

According to (34), the parameter p_ξ (which reduces to the energy p_ξ/k₀ → p₀ in the limit n_m → 0 of the pure electric field case) can only be real if η ≥ a²/4. For these eigenvalues, ${p}_{\xi }/{k}_{0}={p}_{0}-{n}_{\mathrm{m}}{p}_{y}=\pm ({k}_{\mathrm{p}}^{2}/2{k}_{0})\sqrt{{\eta }_{n}^{(k)}-{a}^{2}/4}$ would correspond to a sort of 'gap state', because it can happen that −mc² < ħcp₀ <+ mc² in the pure electric case. On the other hand, one has to keep in mind that, in general, the parameters p₀ and p_y are not 'good quantum numbers' in the case of the interaction we are considering. Besides the transverse momenta p_x = k_p(q + 1)/2 and p_z, only the 'longitudinal combination' $\hat {p}=({k}_{0}/{k}_{\mathrm{p}})({p}_{y}-{n}_{\mathrm{m}}{p}_{0})$ is a constant of motion, and p₀ and p_y separately have no immediate physical meaning. Moreover, not all the eigenvalues give physically acceptable solutions of the Klein–Gordon equation. If $\hat {p}$ becomes purely imaginary, then the wavefunctions necessarily contain the exponentially growing factor $\exp [\pm \vert \hat {p}\vert ({k}_{0}^{2}/{k}_{\mathrm{p}}^{2})(y-{n}_{\mathrm{m}}{x}_{0})]$ in the space-like direction $\hat {k}$. This would not be an acceptable solution, except for the case when the interaction is limited in a finite space–time region. In the optical range (2κ/k_p)² is on the order of (10⁶)², thus η must be as large, in order to have a real square root and propagation constant in (34). We note that, alternatively, the laser-modified mass–shell relation ${p}^{2}={\kappa }_{\ast }^{2}$ may also be prescribed, where ${\kappa }_{\ast }^{2}={\kappa }^{2}+{\varepsilon }^{2}{A}_{0}^{2}$ contains the intensity-dependent mass shift (see e.g. Harvey et al 2012 or Müller and Müller 2012). In this case the first equation of (34) remains the same (because it does not depend on any restriction for the momentum parameter). The right-hand side of the second equation of (34) becomes different, it is then $2{p}_{\xi }/{k}_{\mathrm{p}}^{2}=\pm \sqrt{{\eta }_{n}^{(k)}}$.

Concerning possible physical applications of the exact solutions embodied in (32), one has to keep in mind that, in general, the index of refraction depends on the frequencies of the spectral components of the radiation field, i.e. n_m = n_m(ω). The assumed dependence of the field amplitudes merely through the combination ξ = k⋅x = k⁰c(t − n_me₃ ⋅r/c) with some constant n_m, is, of course, an idealization. The monochromatic field is usually meant to model a real quasi-monochromatic field having a very small spectral spread Δω ≪ ω₀ around some central frequency ω₀ = ck₀, and in the present letter we assume that the associated variation of the index of refraction is negligible in this spectral range, i.e. Δn_m ≪ n_m. For example, our analysis may be relevant for the description of the interaction of a relativistic electron with a strong laser field propagating in an underdense plasma. In the case when the Drude free electron model applies, the index of refraction may well be approximated by the formula ${n}_{\mathrm{m}}^{2}(\omega )=1-{\omega }_{\mathrm{p}}^{2}/{\omega }^{2}$, with ${\omega }_{\mathrm{p}}^{2}=4 \pi {n}_{\mathrm{e}}{e}^{2}/m$, where ω_p is a real plasma frequency, where n_e is the density of the plasma-electrons. If $\omega \gt {2}^{\frac{1}{2}}{\omega }_{\mathrm{p}}$, then Δn_m/n_m < Δω/ω is satisfied for all such frequencies, thus in case of Δω/ω ≪ 1 the relative spectral variation of the index of refraction is also negligible, Δn_m/n_m ≪ 1. Under these conditions, the vector potential displayed in (2) can be a faithful representation of a quasi-monochromatic radiation field in a homogeneous plasma medium. The invariant square (effective mass of the EM field) of the propagation vector ${k}_{\mu }{k}^{\mu }={k}_{0}^{2}(1-{n}_{\mathrm{m}}^{2}(\omega ))={\omega }_{\mathrm{p}}^{2}/{c}^{2}$ does not depend on the circular frequency ω of the laser field.

The fundamental parameter a can be expressed in terms of various combinations of parameters characterizing the applied monochromatic field and the medium. Taking the free electron model of a plasma medium, with a dielectric permittivity ${\varepsilon }_{\mathrm{m}}(\omega )=1-{\omega }_{\mathrm{p}}^{2}/{\omega }^{2}={n}_{\mathrm{m}}^{2}(\omega )$, the electromagnetic plane wave under discussion has the dispersion relation $\omega ({k}_{y})=\sqrt{{\omega }_{\mathrm{p}}^{2}+(c{k}_{y})^{2}}$, with k_y = n_mω/c, corresponding to the phase ξ = ω(t − n_my/c) of the laser field. In this case the parameter a can be written as the work done by the electric force along the plasma wavelength divided by the photon energy. The ratio of the photon density and the electron density also naturally appears,

$$\begin{eqnarray} &&\fl a=4 \varepsilon {A}_{0}/{k}_{\mathrm{p}}=4\frac{e{F{}_{0}\bar{\lambda}}_{\mathrm{p}}}{\hbar {\omega }_{0}}=4\sqrt{(2 m{c}^{2}/\hbar {\omega }_{0})({n}_{\mathrm{ph}}/{n}_{\mathrm{e}})}\nonumber\\ &&=2{\mu }_{0}(2 m{c}^{2}/\hbar {\omega }_{\mathrm{ p}}), \end{eqnarray} \tag{ 35a }$$

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl {n}_{\mathrm{ph}}\equiv \frac{{I}_{0}}{c\hbar {\omega }_{0}}=2.0 8\times {1 0}^{8}(S/{E}_{\mathrm{ph}})~({\mathrm{cm}}^{-3}),\\ \displaystyle {\mu }_{0}\equiv \frac{e{F}_{0}}{m c{\omega }_{0}}=1.0 6\times 1{0}^{-9}{S}^{\frac{1}{2}}/{E}_{\mathrm{ph}}, \end{array} \end{eqnarray} \tag{ 35b }$$

$$\begin{eqnarray} &&\fl {\omega }_{\mathrm{p}}=\sqrt{4 \pi {n}_{\mathrm{e}}{e}^{2}/m},\tqs {k}_{\mathrm{p}}={\omega }_{\mathrm{p}}/c=1{/\bar{\lambda}}_{\mathrm{p}}=2 \pi /{\lambda }_{\mathrm{p}}.\nonumber\\ \end{eqnarray} \tag{ 35c }$$

We have used the definition ${k}_{\mathrm{p}}={k}_{0}\sqrt{1-{n}_{\mathrm{m}}^{2}}$ of the 'plasma wavenumber' in (10b), and introduced the true plasma frequency ω_p. For example, the electron density n_e = 7.242 × 10²⁰ cm⁻³ corresponds to the plasmon energy ħω_p = 1 eV (in which case the plasma wavelength is λ_p = 1240 nm). In the numerical expressions of the photon number density n_ph and of the well-known 'dimensionless intensity parameter' μ₀ (which nowadays is also called 'scaled vector potential'), the quantities S = I₀ (W cm⁻²)⁻¹ and E_ph = ħω₀ (eV)⁻¹ measure the intensity in watts per square centimetre and the photon energy in electron volts, respectively. For a Ti:Sa laser with λ₀ = 795 nm we have ħω₀ = 1.563 eV > ħω_p, which means that this radiation freely propagates in the underdense plasma, with the phase velocity c/n_m and group velocity cn_m. In this case a = 2 × 10⁶μ₀ and μ₀ = 6.782 × 10⁻¹⁰S^1/2, showing that for optical frequencies the parameter a is by six orders of magnitude larger than the usual intensity parameter μ₀.

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Even for a relatively moderate intensity of 100 MW cm⁻² (S = 10⁸), the value of a = 13.56 ≫ 1 is already quite large. In figure 1 we present illustrations for the eigenvalue spectrum in the case of such a moderate intensity, for which we assumed a = 14, and have taken n = 20. In a recent experiment Kiefer et al (2013) have used a Ti:Sa laser of intensity I₀ = 6 × 10²⁰ W cm⁻², in which case μ₀ = 16.61 is well beyond the relativistic limit (μ₀ = 1), and consequently a = 3.32 × 10⁷ is very large. We note that in such extreme cases there exist approximate analytic formulae for both the eigenvalues and for the Ince polynomials; however, in the present letter we do not have space to enter into these details.

In figure 3 we show the distribution of the harmonic strengths (such as $[{A}_{r}^{(k)}(a\vert n+1)]^{2}$ from (15)) in the expansion of the polynomial factors, in the case of the interaction with a Ti:Sa laser field of photon energy ħω₀ = 1.563 eV and peak intensity I₀ = 100 MW cm⁻² in a plasma medium of electron density n_e = 7.242 × 10²⁰ cm⁻³ (ħω_p = 1 eV). Of course, the complete physical spectrum may in principle be influenced by the exponential prefactor leading to the modified Bessel function series in (33), but in the present letter we would like to put the emphasis on the properties of the less-known polynomial factors. According to figure 3, these spectra are in general qualitatively different from each other. As the index k of the eigenvalue increases, the single-peaked structure gradually shifts from the higher harmonics to the lower harmonics. When the first negative eigenvalue is reached, then the harmonic distribution becomes oscillatory, and beyond this border oscillations are also present in the solutions belonging to positive eigenvalues. In figure 4 we give an overview of the harmonic spectra in figure 3.

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In figure 5 we illustrate the detailed temporal evolution of the even cosine-type wavefunctions, whose harmonic spectra have been shown in figure 3. Finally, figure 6 illustrates the temporal evolution of the even sine-type wavefunctions. The further numerical study of the exact solutions (32) is out of the scope of the present letter.

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We have presented exact closed form solutions of the Klein–Gordon equation of a charged particle (electron or proton) moving in a monochromatic classical plane electromagnetic field in a medium of index of refraction n_m < 1. These solutions have been expressed in terms of Ince polynomials, which form a doubly infinite set, belonging to discrete momentum components of the particle's de Broglie wave along the polarization vector and along the propagation direction of the EM field. The odd Ince polynomials are finite sums of cosines or sines of the half-integer multiples of the phase of the incoming field, thus they describe half-integer order higher harmonics.

We have considered the medium as an underdense plasma, in which case the transverse momentum of the test charge is an integer or half-integer of the plasma momentum k_p = ω_p/c, i.e. p_x = nk_p or ${p}_{x}=(n+\frac{1}{2}){k}_{\mathrm{p}}$, where ω_p is the plasma frequency. The other quantum number determines the possible eigenvalues of the longitudinal energy–momentum parameter. The fundamental (input) parameter a (see (35a)), characterizing the strength of the interaction, is the work done on the charge by the electric force of the EM field along a plasma wavelength divided by the photon energy. In contrast to the well-known intensity parameter μ₀ (see (35b)), which often appears in the theory of strong laser field matter interactions, this a is a quantum parameter, and it is typically by many orders of magnitudes larger than the former one. We have found 'void regions' (or 'bubbles') in the particle density, centred in the middle of the plasma wave cycle with an exponentially large contrast (e^a/2). This behaviour of the density corresponds to the formation of a regular and periodic structure on the plasma length scale, and it may have relevance in the study of novel acceleration mechanisms (see e.g. Xia et al 2012, Mourou et al 2013, Pukhov et al 2012).

In section 2 we have derived the relevant Ince equation from the original Klein–Gordon equation, and made a brief comparison with the derivation of the usual Volkov states, and of the Mathieu-type wavefunctions in a medium. In section 3 we have derived the two even and two odd periodic, finite solutions, which coincide with the corresponding Ince polynomials. These solutions are associated with the eigenvalue problem of finite special tri-diagonal matrices equation (13) and (16a) and (16b). We have restricted our analysis only to these special class of solutions, proportional with polynomial expressions. They form a subset of all solutions of the Ince equation, including the transcendental Ince functions. In section 4 we have given some numerical examples for the relevant parameters appearing in the analysis, and presented numerical illustrations for the discrete eigenvalues, the temporal shape of the wavefunctions, and for the harmonic spectra which are associated with them.

This work has been supported by the Hungarian Scientific Research Foundation OTKA, Grant No. K 104260. The author thanks the referees for many valuable questions, comments and criticism, which have been answered, and taken into account in compiling the final version of the present letter.