Eulerian Pantograph Equations, Half-Line Spectra, and Hermite Representations

1. Introduction

Functional-differential equations with proportional delay occur in self-similar and Mellin-type reductions of evolution and integral problems. The scalar Eulerian pantograph equation

$$At{q}^{\prime }\left(t\right)+Bq\left(t\right)+Cq\left(\alpha t\right)=0,0<\alpha <1,$$

(1)

has a singular endpoint at $t=0$, because the delay length $(1-\alpha )t$ collapses there. Setting

$$t={\mathrm{e}}^{-x},Q\left(x\right)=q\left({\mathrm{e}}^{-x}\right),L=log(1/\alpha )>0,$$

(2)

transforms (1) into

$$-A{Q}^{\prime }\left(x\right)+BQ\left(x\right)+CQ(x+L)=0,x\ge 0.$$

(3)

Thus the proportional delay becomes a constant advanced shift on the half-line.

The paper uses four related, but distinct, objects. The original unknown is $q\left(t\right)$ on $t>0$. After the logarithmic change $t={\mathrm{e}}^{-x}$, the new unknown is $Q\left(x\right)$, with $x\in \mathbb{R}$ for the full-line comparison model and $x\in [0,\infty )$ for the half-line problem associated with the singular endpoint $t=0$. The full-line operator is used only as a reference Fourier multiplier. The main operator studied below is the half-line operator with maximal domain.

More precisely, for the left-transport sign

$$v:=-A>0,$$

(4)

we analyze the maximal half-line operator

$${\mathcal{L}}_{+}=vD+BI+C{S}_L,D={\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}},Dom\left(D\right)=W^{1,p}(0,\infty ),$$

(5)

where the shift operator is

$$\left(S_{L}f\right)\left(x\right)=f(x+L),x\ge 0.$$

(6)

No boundary trace is imposed at $x=0$.

The results should be separated as follows. The logarithmic conjugacy, the full-line Fourier multiplier picture, the step-method interpretation of histories, and the Hermite translation matrix are standard or representational tools recalled for context. The new contribution of the paper is the explicit assembly, for the maximal half-line realization (5), of the point spectrum, boundary approximate spectrum, full spectral image, and sharp semigroup norm identity. The comparison with Mellin modes, Markov heuristics, and FNO-type parametrizations is interpretive; it is included to clarify how these familiar viewpoints relate to the half-line operator, not to claim that they are new spectral methods. Informally, the main distinction is that on the full line the Fourier transform sees only oscillatory modes, whereas on the half-line exponentially decaying modes are also admissible and become genuine eigenfunctions.

The main theorem identifies the spectrum and growth bound for (5). With

$$m\left(\mu \right)=v\mu +B+C{\mathrm{e}}^{L\mu },{\mathbb{H}}_{-}^{\circ }=\{\mu \in \mathbb{C}:\Re \mu <0\},$$

(7)

we prove

$$Spec\left({\mathcal{L}}_{+}\right)=\overline{m\left({\mathbb{H}}_{-}^{\circ }\right)},{\omega }_0\left({\mathcal{L}}_{+}\right)=\Re B+\left|C\right|.$$

(8)

Here $Spec\left({\mathcal{L}}_{+}\right)$ is the full spectrum of the unbounded operator ${\mathcal{L}}_{+}$, and ${\omega }_0\left({\mathcal{L}}_{+}\right)$ is the exponential growth rate of the corresponding time-evolution semigroup. This is a half-line result: the interior image consists of genuine $L^p$ eigenfunctions, the boundary image consists of approximate eigenvalues produced by wave packets escaping to infinity, and the closure gives the full spectrum.

The pantograph equation and proportional-delay models go back to classical work such as [8,9,10]; general delay-equation background and characteristic-root methods are treated in [5,6], while stability questions for time-delay systems are discussed in [7] and recent numerical schemes for pantograph-type equations in [16]. The present paper does not attempt to replace these theories. Its narrower aim is to record a precise operator realization for the Eulerian half-line case and to keep it separate from the full-line Fourier multiplier picture. Hermite functions and their Laguerre-polynomial translation matrix are included because they give a concrete finite-dimensional representation of histories, which is useful in computations and in operator-learning parameterizations.

For clarity, we use the following notation throughout the paper. The symbol $S_L$ always denotes translation by L, but its underlying space is indicated by context: on the full line it acts on functions on $\mathbb{R}$, whereas in the main half-line model it acts on functions on $[0,\infty )$. If A is a closed operator, then $Spec\left(A\right)$ denotes its spectrum, i.e., the set of complex numbers $\lambda$ for which $\lambda -A$ is not boundedly invertible. The point spectrum ${Spec}_p\left(A\right)$ is the set of eigenvalues. The approximate point spectrum ${Spec}_{\mathrm{ap}}\left(A\right)$ consists of those $\lambda$ for which there are unit vectors $u_n$ in the operator domain with $(A-\lambda )u_n\to 0$; such vectors are approximate eigenvectors. We write $Dom\left(A\right)$ for the domain of A, $Ran\left(A\right)$ for its range, $Ker\left(A\right)$ for its kernel, and ${\omega }_0\left(A\right)$ for the growth bound of the semigroup generated by A. The spectral bound is denoted by $s\left(A\right)=sup\{\Re \lambda :\lambda \in Spec(A\left)\right\}$. These conventions are standard in semigroup theory [4], but they are recalled here because the paper compares several related spectral notions.

2. Logarithmic Conjugacy and the Full-Line Model

Let

$$\mathcal{H}=L^2({\mathbb{R}}_{+},\mathrm{d}t/t).$$

(9)

The measure $\mathrm{d}t/t$ is the natural scale-invariant measure on the positive half-line: the dilation $t\mapsto \alpha t$ preserves this measure. This is why logarithmic coordinates turn dilations into translations without changing the $L^2$ norm. The map

$$\left(Uf\right)\left(x\right)=f\left({\mathrm{e}}^{-x}\right)$$

(10)

is unitary from $\mathcal{H}$ onto $L^2\left(\mathbb{R}\right)$. Under this map,

$$U\left(t{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\right)U^{-1}=-{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}.$$

(11)

Thus the Euler operator is the logarithmic form of the translation generator. This subsection recalls a standard full-line comparison model; it is not the main half-line spectral theorem of the paper.

On the full line, define

$$\mathcal{L}=-A{\partial }_x+BI+C{S}_L,Dom\left(\mathcal{L}\right)=H^1\left(\mathbb{R}\right)\subset L^2\left(\mathbb{R}\right).$$

(12)

Here ${\partial }_x$ denotes differentiation on the full real line and $H^1\left(\mathbb{R}\right)$ is the Sobolev space of square-integrable functions with square-integrable weak derivative. The Fourier transform diagonalizes this realization:

$$\mathcal{F}\mathcal{L}{\mathcal{F}}^{-1}=M_{\widehat{\mathcal{L}}},\widehat{\mathcal{L}}\left(k\right)=-\mathrm{i}Ak+B+C{\mathrm{e}}^{\mathrm{i}kL}.$$

(13)

Consequently, by the standard spectral theorem for multiplication operators,

$$Spec\left(\mathcal{L}\right)=\overline{\widehat{\mathcal{L}}\left(\mathbb{R}\right)}.$$

(14)

This full-line spectrum is a multiplier spectrum. It is recalled only as a comparison object and should not be confused with the half-line spectrum of the maximal operator (5), where exponentially decaying modes become actual eigenfunctions.

Remark 1. 

If A is real and $v=-A>0$, then (13) becomes $\widehat{\mathcal{L}}\left(k\right)=\mathrm{i}vk+B+C{\mathrm{e}}^{\mathrm{i}kL}=m\left(\mathrm{i}k\right)$. Hence the full-line spectrum is exactly the boundary curve appearing in the half-line theorem below.

2.1. Probabilistic Interpretation Under Markov Assumptions

The same logarithmic change of variables also gives a limited connection with Markov processes and their generators. This connection is not used in the spectral proof, but it is useful for placing the deterministic half-line operator in the broader context of functional differential equations and their stochastic variants. Suppose that the coefficients are real and that $C\ge 0$ is interpreted as the intensity of multiplicative jumps $t\mapsto \alpha t$. In this interpretation, a trajectory moves deterministically between jumps, and at random jump times it is multiplied by $\alpha$. The infinitesimal Markov generator records the first-order change of expected observables. On functions of $t>0$ it is then

$$\left(\mathcal{G}f\right)\left(t\right)=At{f}^{\prime }\left(t\right)+C\left(f\left(\alpha t\right)-f\left(t\right)\right).$$

(15)

In logarithmic variables $x=-logt$, this becomes

$$\left({\mathcal{G}}_{x}F\right)\left(x\right)=v{F}^{\prime }\left(x\right)+C\left(F(x+L)-F\left(x\right)\right),v=-A,L=log(1/\alpha ).$$

(16)

Consequently, under these additional positivity assumptions,

$${\mathcal{L}}_{+}={\mathcal{G}}_x+(B+C)I.$$

(17)

The scalar shift in (17) may or may not correspond to a genuine killing rate, depending on its sign. Thus the Markov interpretation is conditional, whereas the operator-theoretic spectrum below is formulated for complex B and C and does not rely on positivity. On Mellin functions $t^z$, one has

$$\mathcal{G}{t}^z=\left(Az+C({\alpha }^z-1)\right)t^z,$$

(18)

so the characteristic equation of the pantograph mode is the shifted generator symbol $Az+B+C{\alpha }^z=0$. This is the probabilistic counterpart of the logarithmic translation symbol used below; standard references for Markov generators and semigroups include [11,12].

3. The Maximal Half-Line Operator

Throughout this section $1\le p<\infty$ and $X_p=L^p(0,\infty )$. We write $D=\mathrm{d}/\mathrm{d}x$ with maximal domain $W^{1,p}(0,\infty )$. Let

$$\left(T_0\left(t\right)f\right)\left(x\right)=f(x+vt),t\ge 0.$$

(19)

Then ${\left(T_0\left(t\right)\right)}_{t\ge 0}$ is a strongly continuous contraction semigroup on $X_p$, and its generator is $vD$. Moreover

$$S_L=T_0(L/v)={\mathrm{e}}^{LD}.$$

(20)

Therefore (5) is a bounded perturbation of $vD$. The word “maximal” means that every $W^{1,p}$ function is allowed in the domain; in particular, no boundary value such as $Q\left(0\right)=0$ is imposed. This choice is essential for the spectrum below. The semigroup $T_0\left(t\right)$ simply transports the graph to the left: the value observed at x after time t is the old value at $x+vt$.

Proposition 1 

(Transport spectrum). For $1\le p<\infty$,

$$Spec\left(vD\right)=\{\lambda \in \mathbb{C}:\Re \lambda \le 0\}.$$

(21)

The point spectrum consists precisely of the open half-plane $\Re \lambda <0$, with eigenfunction $x\mapsto exp(\lambda x/v)$. For $\Re \lambda >0$,

$$\left({(\lambda -vD)}^{-1}f\right)\left(x\right)={\displaystyle \frac{1}{v}}{\int }_x^{\infty }exp\left(-{\displaystyle \frac{\lambda }{v}}(y-x)\right)f\left(y\right)\mathrm{d}y.$$

(22)

Proof. 

Solving $v{f}^{\prime }=\lambda f$ gives the eigenvalue statement. Formula (22) follows by variation of constants and gives a bounded inverse for $\Re \lambda >0$. The line $\Re \lambda =0$ belongs to approximate spectrum by the usual cutoff plane waves translated to infinity. These approximate eigenvectors are not true eigenfunctions, because pure oscillations do not belong to $L^p(0,\infty )$, but they look like eigenfunctions on longer and longer intervals far from the endpoint. □

4. Point Spectrum and Characteristic Roots

For $f_{\mu }\left(x\right)={\mathrm{e}}^{\mu x}$ with $\Re \mu <0$,

$${\mathcal{L}}_{+}{f}_{\mu }=m\left(\mu \right)f_{\mu }.$$

(23)

Thus $m\left({\mathbb{H}}_{-}^{\circ }\right)$ is contained in the point spectrum ${Spec}_p\left({\mathcal{L}}_{+}\right)$, the set of eigenvalues of ${\mathcal{L}}_{+}$. For the converse we use, as a known input, the Carleman-transform representation theorem for autonomous half-line functional-differential equations. The role of this theorem is to rule out hidden $L^p$ eigenfunctions: every such eigenfunction must be built from exponential modes associated with zeros of the characteristic determinant.

Lemma 1 

(Carleman representation). Let $u\in W^{1,p}(0,\infty )$ solve

$$v{u}^{\prime }\left(x\right)+(B-\lambda )u\left(x\right)+Cu(x+L)=0,x>0.$$

(24)

If $u\in L^p(0,\infty )$, then u is a finite sum

$$u\left(x\right)=\sum _j{P}_j\left(x\right){\mathrm{e}}^{{\mu }_{j}x},$$

(25)

where $P_j$ are polynomials and ${\mu }_j$ are zeros in $\Re \mu <0$ of

$${\Delta }_{\lambda }\left(\mu \right)=\lambda -B-v\mu -C{\mathrm{e}}^{L\mu }.$$

(26)

In particular, a non-zero $L^p$ solution exists only if ${\Delta }_{\lambda }$ has a zero in $\Re \mu <0$.

Proof. 

We recall the reduction, because this is the point where the half-line domain enters. Choose $a>0$ and multiply u by a smooth cut-off which is equal to one on $[a,\infty )$. For $\Re z>0$ the one-sided Laplace transform

$$U_a\left(z\right)={\int }_a^{\infty }{\mathrm{e}}^{-zx}u\left(x\right)\mathrm{d}x$$

(27)

is well defined after the harmless exponential damping $u\left(x\right)\mapsto {\mathrm{e}}^{-\epsilon x}u\left(x\right)$ and passage to the limit $\epsilon \downarrow 0$. Transforming (24) on $[a,\infty )$ gives

$${\Delta }_{\lambda }(-z)U_a\left(z\right)=E_a\left(z\right),$$

(28)

where $E_a$ is an entire function depending only on the finite interval $[a,a+L]$ and on boundary terms at a. Thus the possible singularities of $U_a$ are poles at zeros of ${\Delta }_{\lambda }(-z)$, equivalently at exponential exponents $\mu =-z$ satisfying ${\Delta }_{\lambda }\left(\mu \right)=0$. The standard Carleman representation theorem for autonomous linear functional-differential equations makes this residue calculation precise: after moving the inversion contour to the left, the solution is a finite sum of residues $x^k{\mathrm{e}}^{{\mu }_{j}x}$ plus a remainder which vanishes by the homogeneous equation and the $L^p$ condition; see [5, Chapter 7] and [6, Chapter IV]. Since ${\Delta }_{\lambda }$ is an exponential polynomial and is not identically zero because of the term $-v\mu$, all relevant zeros have finite multiplicity. Finally, $x^k{\mathrm{e}}^{\mu x}\in L^p(0,\infty )$ only when $\Re \mu <0$, which removes all non-decaying terms and gives (25). □

Theorem 1 

(Point spectrum). For every $1\le p<\infty$,

$${Spec}_p\left({\mathcal{L}}_{+}\right)=m\left({\mathbb{H}}_{-}^{\circ }\right).$$

(29)

If ${\Delta }_{\lambda }$ has a zero μ of multiplicity r in $\Re \mu <0$, then the corresponding root vectors are generated by ${\partial }_{\mu }^k{\mathrm{e}}^{\mu x}$, $0\le k<r$.

Proof. 

The inclusion $m\left({\mathbb{H}}_{-}^{\circ }\right)\subset {Spec}_p\left({\mathcal{L}}_{+}\right)$ follows from (23). Conversely, if ${\mathcal{L}}_{+}u=\lambda u$ with $u\ne 0$, then u solves (24); by Lemma 1, ${\Delta }_{\lambda }\left(\mu \right)=0$ for some $\Re \mu <0$, which is equivalent to $\lambda =m\left(\mu \right)$. Differentiating $({\mathcal{L}}_{+}-m\left(\mu \right)){\mathrm{e}}^{\mu x}=0$ with respect to $\mu$ gives the root-vector statement at multiple zeros. □

Returning to the original variable, a modal solution has the form $Q\left(x\right)={\mathrm{e}}^{-zx}$, so $\mu =-z$. The equation $m(-z)=0$ is

$$Az+B+C{\mathrm{e}}^{-Lz}=0.$$

(30)

The condition $Q\in L^p(0,\infty )$ is $\Re z>0$. In terms of the original variable $t={\mathrm{e}}^{-x}$, this corresponds to the Mellin-type power $q\left(t\right)=t^z$, which is why (30) is the usual characteristic equation for pantograph modes.

5. Full Spectrum by a Wiener–Hopf Criterion

Boundary spectral points arise from approximate eigenvectors localized far from the endpoint. The Wiener–Hopf terminology refers here to operators on a half-line whose coefficients are translation invariant away from the endpoint. The boundary at $x=0$ prevents a full Fourier diagonalization, but the classical scalar Wiener–Hopf criterion relates invertibility to the boundary symbol and its winding number, or index. For readers not using this terminology, the criterion below may be viewed as the half-line analogue of checking that a Fourier multiplier has no zeros, with an additional index condition.

Lemma 2 

(Boundary approximate spectrum). For every $\xi \in \mathbb{R}$,

$$m\left(\mathrm{i}\xi \right)\in {Spec}_{\mathrm{ap}}\left({\mathcal{L}}_{+}\right).$$

(31)

Consequently,

$$\overline{m\left({\mathbb{H}}_{-}^{\circ }\right)}\subset Spec\left({\mathcal{L}}_{+}\right).$$

(32)

Proof. 

Let $\chi \in C_c^1(-1,1)$ be non-zero and put

$$u_R\left(x\right)=R^{-1/p}\chi ((x-R)/R){\mathrm{e}}^{\mathrm{i}\xi x}.$$

(33)

Then $\parallel u_R{\parallel }_p$ is independent of R for all large R, while

$$\parallel (D-\mathrm{i}\xi )u_R{\parallel }_p=O\left(R^{-1}\right),{\parallel (S_L-{\mathrm{e}}^{\mathrm{i}L\xi })u_R\parallel }_p=O\left(R^{-1}\right).$$

(34)

After normalization, $({\mathcal{L}}_{+}-m\left(\mathrm{i}\xi \right))u_R\to 0$ in $L^p$. □

Lemma 3 

(Maximal half-line realization and scalar symbol). For each $\lambda \in \mathbb{C}$, the operator

$$A_{\lambda }:=\lambda -{\mathcal{L}}_{+},Dom\left(A_{\lambda }\right)=W^{1,p}(0,\infty ),$$

(35)

is the maximal half-line realization of the scalar expression

$$A_{\lambda }u=(\lambda -B)u-v{u}^{\prime }-C{S}_{L}u.$$

(36)

Its translation-invariant boundary symbol is

$${\Delta }_{\lambda }\left(\mathrm{i}\xi \right)=\lambda -B-v\mathrm{i}\xi -C{\mathrm{e}}^{\mathrm{i}L\xi }.$$

(37)

No boundary trace at $x=0$ is part of the domain.

Proof. 

The derivative part $vD$ is the generator of the left-translation semigroup on $L^p(0,\infty )$ with maximal domain $W^{1,p}(0,\infty )$, and $S_L$ is a bounded operator on $L^p(0,\infty )$. Hence ${\mathcal{L}}_{+}=vD+BI+C{S}_L$ and $A_{\lambda }$ are closed on exactly this domain by the bounded perturbation theorem. This is a maximal, not minimal, realization: $C_c^1(0,\infty )$ is not used as a core, and no condition such as $u\left(0\right)=0$ is imposed. For test functions localized away from the endpoint, substituting $u\left(x\right)={\mathrm{e}}^{\mathrm{i}\xi x}$ in (36) gives (37); this is the symbol seen by wave packets escaping to infinity and by the scalar Wiener–Hopf operator associated with the maximal realization. □

The following non-elementary input is the only place where scalar Wiener–Hopf Fredholm theory is used. For scalar first-order plus-shift half-line expressions of the form (36), with maximal derivative domain and bounded shift term, the classical theorem states that non-vanishing of the boundary symbol together with index zero implies Fredholmness on $L^p(0,\infty )$; the Fredholm kernel is the space of $W^{1,p}$ solutions of the homogeneous equation, and the Fredholm index equals the winding number of the boundary symbol. This is the specific functional-analytic hypothesis needed below; it is a standard scalar Wiener–Hopf result, not a formal symbolic calculation. References include [3].

Lemma 4 

(Scalar half-line Wiener–Hopf invertibility). Let $\lambda \notin \overline{m\left({\mathbb{H}}_{-}^{\circ }\right)}$. Then the boundary symbol $\xi \mapsto {\Delta }_{\lambda }\left(\mathrm{i}\xi \right)$ is non-vanishing and has index zero, where the index is the winding number of this curve around the origin. Consequently, $A_{\lambda }=\lambda -{\mathcal{L}}_{+}$ is boundedly invertible from $W^{1,p}(0,\infty )$ onto $L^p(0,\infty )$, $1\le p<\infty$.

Proof. 

By Lemma 3, the relevant concrete operator is $A_{\lambda }$ with maximal domain (35), and its boundary symbol is (37). The assumption $\lambda \notin \overline{m\left({\mathbb{H}}_{-}^{\circ }\right)}$ means ${\Delta }_{\lambda }\left(\mu \right)\ne 0$ for $\Re \mu \le 0$. Hence the boundary symbol has no zeros. On a large left half-disc, the argument principle gives zero zeros inside the contour; the contribution of the large semicircle is governed by ${\Delta }_{\lambda }\left(\mu \right)=-v\mu (1+o\left(1\right))$. Passing to the boundary limit gives index zero for ${\Delta }_{\lambda }\left(\mathrm{i}\mathbb{R}\right)$.

The scalar Wiener–Hopf Fredholm theorem quoted above gives Fredholmness of $A_{\lambda }$ with index zero. Its kernel is precisely the set of $u\in W^{1,p}(0,\infty )$ satisfying (24). By Lemma 1, a non-zero kernel would imply a zero of ${\Delta }_{\lambda }$ in $\Re \mu <0$, which is excluded. Since the Fredholm index is zero, the cokernel is also trivial, and the range is all of $L^p(0,\infty )$. Thus $A_{\lambda }$ is bijective. The closed graph theorem gives boundedness of the inverse. □

Theorem 2 

(Half-line spectral equality). Assume the scalar Wiener–Hopf Fredholm theorem stated above for the maximal realization (35). Then, for every $1\le p<\infty$,

$$Spec\left({\mathcal{L}}_{+}\right)=\overline{m\left({\mathbb{H}}_{-}^{\circ }\right)}.$$

(38)

Proof. 

The inclusion “⊃” follows from Theorem 1 and Lemma 2 and closedness of the spectrum. If $\lambda \notin \overline{m\left({\mathbb{H}}_{-}^{\circ }\right)}$, then Lemma 4 gives bounded invertibility of the concrete maximal operator $A_{\lambda }=\lambda -{\mathcal{L}}_{+}$ from $W^{1,p}(0,\infty )$ onto $L^p(0,\infty )$. Thus $\lambda$ belongs to the resolvent set. □

6. Exact Semigroup Norm and Stability

Since $S_L=T_0(L/v)$ commutes with the left-translation semigroup, the semigroup generated by (5) is

$$T\left(t\right)={\mathrm{e}}^{Bt}exp\left(Ct{S}_L\right)T_0\left(t\right)={\mathrm{e}}^{Bt}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{\left(Ct\right)}^n}{n!}}{T}_0\left(t+{\displaystyle \frac{nL}{v}}\right).$$

(39)

This formula is a variation-of-constants expansion in which the bounded shift term $C{S}_L$ is exponentiated separately from the transport part $vD$. Commutation is the key point: it turns the semigroup into an explicit positive-time series of translations.

Theorem 3 

(Exact spectral and growth bounds). For every $1\le p<\infty$,

$$s\left({\mathcal{L}}_{+}\right)={\omega }_0\left({\mathcal{L}}_{+}\right)=\Re B+\left|C\right|,$$

(40)

and

$$\parallel T\left(t\right)\parallel ={\mathrm{e}}^{(\Re B+|C\left|\right)t},t\ge 0.$$

(41)

Proof. 

For the upper bound, use (39) and the contraction property $\parallel T_0\left(s\right)\parallel \le 1$:

$$\parallel T\left(t\right)\parallel \le {\mathrm{e}}^{\Re Bt}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{\left|C\right|}^n{t}^n}{n!}}={\mathrm{e}}^{(\Re B+|C\left|\right)t}.$$

(42)

It remains to prove that no cancellation can improve this estimate. If $C=0$, the identity follows from $\parallel T_0\left(t\right)\parallel =1$, so assume $C\ne 0$. Choose $\theta \in \mathbb{R}$ such that $C{\mathrm{e}}^{\mathrm{i}\theta }=\left|C\right|$, and choose $\eta \in \mathbb{R}$ with $\eta L\equiv \theta (mod2\pi )$. Let $\chi \in C_c^1(-1,1)$ be normalized by ${\parallel \chi \parallel }_p=1$ and set

$$f_R\left(x\right)=R^{-1/p}\chi ((x-R)/R){\mathrm{e}}^{\mathrm{i}\eta x}.$$

(43)

For R large enough the support of $f_R$ lies in $(0,\infty )$, and $\parallel f_R{\parallel }_p=1$. For each fixed $s\ge 0$,

$$\parallel T_0\left(s\right)f_R-{\mathrm{e}}^{\mathrm{i}\eta vs}{f}_R{\parallel }_p⟶0,R\to \infty ,$$

(44)

which follows by the change of variables $x=R+Ry$ and the continuity of translations in $L^p\left(\mathbb{R}\right)$ applied to the compactly supported profile $\chi$.

Fix $N\in \mathbb{N}$. Applying (44) to the finitely many shifts $s_n=t+nL/v$, $0\le n\le N$, gives

$$\begin{array}{cc}\hfill & {∥{\mathrm{e}}^{Bt}{\displaystyle \sum _{n=0}^N}{\displaystyle \frac{{\left(Ct\right)}^n}{n!}}{T}_0\left(t+{\displaystyle \frac{nL}{v}}\right)f_R-{\mathrm{e}}^{Bt}{\mathrm{e}}^{\mathrm{i}\eta vt}{\displaystyle \sum _{n=0}^N}{\displaystyle \frac{{\left(Ct\right)}^n{\mathrm{e}}^{\mathrm{i}\eta nL}}{n!}}{f}_R∥}_p⟶0.\hfill \end{array}$$

(45)

Because $\eta L\equiv \theta$ and $C{\mathrm{e}}^{\mathrm{i}\theta }=\left|C\right|$, the scalar sum in (45) is ${\sum }_{n=0}^N{\left(\right|C\left|t\right)}^n/n!$. The remaining tail of (39) is bounded uniformly in R by

$${\rho }_N\left(t\right)={\mathrm{e}}^{\Re Bt}{\displaystyle \sum _{n=N+1}^{\infty }}{\displaystyle \frac{{\left(\right|C\left|t\right)}^n}{n!}}.$$

(46)

Therefore

$$\underset{R\to \infty }{lim\; inf}{\parallel T\left(t\right)f_R\parallel }_p\ge {\mathrm{e}}^{\Re Bt}{\displaystyle \sum _{n=0}^N}{\displaystyle \frac{{\left(\right|C\left|t\right)}^n}{n!}}-{\rho }_N\left(t\right).$$

(47)

Letting $N\to \infty$ in (47) gives $\parallel T\left(t\right)\parallel \ge {\mathrm{e}}^{(\Re B+|C\left|\right)t}$. Together with (42), this proves (41).

The growth bound is then

$${\omega }_0\left({\mathcal{L}}_{+}\right)=\underset{t\to \infty }{lim}{t}^{-1}log\parallel T\left(t\right)\parallel =\Re B+\left|C\right|.$$

(48)

Finally, by Theorem 2,

$$s\left({\mathcal{L}}_{+}\right)=\underset{\Re \mu <0}{sup}\Re \left(v\mu +B+C{\mathrm{e}}^{L\mu }\right).$$

(49)

The upper bound is at most $\Re B+\left|C\right|$ because $|{\mathrm{e}}^{L\mu }|<1$ and $\Re \left(v\mu \right)<0$. The opposite inequality follows by taking $\Re \mu \uparrow 0$ and choosing $\Im \mu$ so that the phase of $C{\mathrm{e}}^{L\mu }$ approaches the positive real direction. Thus $s\left({\mathcal{L}}_{+}\right)=\Re B+\left|C\right|$ as well. □

Corollary 1 

(Sharp stability threshold). The semigroup generated by ${\mathcal{L}}_{+}$ is exponentially stable if and only if

$$\Re B+\left|C\right|<0.$$

(50)

If equality holds, the semigroup is bounded but not exponentially stable.

7. Initial Histories and Modal Incompleteness

The preceding spectral results describe the maximal half-line operator on $L^p(0,\infty )$. Initial-history problems use the same characteristic roots, but in a different way. If a history $\mathsf{\Phi }\in C^1[0,L]$ is prescribed for (3), the solution can be continued interval by interval by the step method. Exponential histories $Q\left(x\right)={\mathrm{e}}^{-zx}$ are compatible with this continuation precisely when z satisfies the Mellin characteristic equation (30). These are the modal histories. The word “history” means the part of Q already known on one delay interval; for equation (3), values on $[0,L]$ determine later values recursively.

This modal family is useful for describing the eigenfunction part of the half-line operator, but it should not be mistaken for a basis of all admissible initial data. The reason is elementary and is tied to the singular endpoint $t=0$: every decaying Mellin mode has zero trace there. Therefore modal expansions cannot uniformly approximate continuous initial data with non-zero endpoint trace. In applications this means that a finite-history continuation generally contains a residual component that is not captured by the discrete Mellin roots alone, even though those roots exactly describe the point spectrum in Theorem 1.

Proposition 2 

(Modal incompleteness for endpoint data). Let

$${\mathcal{M}}_{\mathrm{mod}}=span\{t^z:z\mathit{solves}\left(30\right)\mathit{and}\Re z>0\}$$

(51)

in $C[0,1]$. Then

$${\overline{{\mathcal{M}}_{\mathrm{mod}}}}^{C[0,1]}\subset C_0[0,1]:=\{f\in C[0,1]:f\left(0\right)=0\}.$$

(52)

Consequently, every continuous history $g\in C[0,1]$ with $g\left(0\right)\ne 0$ satisfies

$${dist}_{C[0,1]}(g,{\mathcal{M}}_{\mathrm{mod}})\ge \left|g\left(0\right)\right|.$$

(53)

Thus no modal expansion made only of decaying Mellin modes can approximate such an initial history uniformly on $[0,1]$.

Proof. 

If $\Re z>0$, then $t^z\to 0$ as $t\to 0^{+}$. Hence every finite modal sum vanishes at $t=0$. The space $C_0[0,1]$ is closed as the kernel of the continuous trace functional $f\mapsto f\left(0\right)$, which proves (52). The distance estimate follows by evaluating $g-m$ at $t=0$. □

Remark 2. 

The term “non-modal” is used here only in this approximation sense: it refers to the part of a prescribed history that is not uniformly approximable by the decaying Mellin modes. It is not an additional spectral type beyond the point and approximate spectra identified above.

8. Representation and Approximation Perspectives

The spectral results above are independent of any numerical discretization. We do not solve numerical approximation or learning problems in this paper. Nevertheless, the logarithmic transformation reduces proportional delay to translation, and this makes it natural to compare the exact operator theory with two common representation languages: Hermite expansions for histories and Fourier-based operator-learning architectures. The purpose of this section is therefore interpretive. It explains how the exact spectrum and growth bound constrain such representations, without claiming that these representations replace the half-line spectral theorem.

8.1. Hermite Representation of Translations and Histories

Hermite functions provide a useful orthonormal basis of $L^2\left(\mathbb{R}\right)$ for representing extended histories, but they do not diagonalize the shift operator or the half-line operator. Let

$${\psi }_n\left(x\right)={\displaystyle \frac{{(-1)}^n}{\sqrt{2^{n}n!\sqrt{\pi }}}}{\mathrm{e}}^{x^2/2}{\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{x}^n}}{\mathrm{e}}^{-x^2},n=0,1,2,\dots .$$

(54)

Then ${\left({\psi }_n\right)}_{n\ge 0}$ is an orthonormal basis of $L^2\left(\mathbb{R}\right)$. With the Fourier transform convention

$$\left(\mathcal{F}f\right)\left(k\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{\mathbb{R}}{\mathrm{e}}^{-\mathrm{i}kx}f\left(x\right)\mathrm{d}x,$$

(55)

one has

$$\mathcal{F}{\psi }_n={(-\mathrm{i})}^n{\psi }_n.$$

(56)

Thus a square-integrable extended history can be encoded by its Hermite coefficients. This is analogous to a Fourier series, but the Hermite basis is localized in both position and frequency, which is convenient for finite or rapidly decaying histories.

Lemma 5 

(Translation matrix). For $L\in \mathbb{R}$,

$$\begin{array}{cc}\hfill {\psi }_n(x+L)& ={\mathrm{e}}^{-L^2/4}{\displaystyle \sum _{m=0}^n}\sqrt{{\displaystyle \frac{m!}{n!}}}{\left({\displaystyle \frac{L}{\sqrt{2}}}\right)}^{n-m}{L}_m^{(n-m)}\left({\displaystyle \frac{L^2}{2}}\right){\psi }_m\left(x\right)\hfill \\ \hfill & +{\mathrm{e}}^{-L^2/4}{\displaystyle \sum _{m=n+1}^{\infty }}\sqrt{{\displaystyle \frac{n!}{m!}}}{\left(-{\displaystyle \frac{L}{\sqrt{2}}}\right)}^{m-n}{L}_n^{(m-n)}\left({\displaystyle \frac{L^2}{2}}\right){\psi }_m\left(x\right).\hfill \end{array}$$

(57)

Proof. 

This is the standard displacement-operator formula; see [2, Chapter 1]. Writing $S_L=exp\left((L/\sqrt{2})(a-a^{*})\right)$ in the Hermite basis gives the displayed Laguerre-polynomial matrix elements. □

Example 1 

(Shift of the ground-state Hermite function). For $n=0$, Lemma 5 gives

$${\psi }_0(x+L)={\mathrm{e}}^{-L^2/4}{\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{1}{\sqrt{m!}}}{\left(-{\displaystyle \frac{L}{\sqrt{2}}}\right)}^m{\psi }_m\left(x\right).$$

(58)

Thus even the shifted Gaussian has infinitely many Hermite coefficients. Truncating (58) gives a finite matrix approximation of the translation operator on the Hermite basis. This illustrates the role of Hermite functions in the present paper: they provide a computable representation of shifted histories, while the spectrum of the half-line operator is still determined by the Wiener–Hopf formula in Theorem 2.

If a finite history $\mathsf{\Phi }$ on $[0,L]$ is extended to a function $\tilde{\mathsf{\Phi }}\in L^2\left(\mathbb{R}\right)$, then

$$\tilde{\mathsf{\Phi }}={\displaystyle \sum _{n=0}^{\infty }}{c}_n{\psi }_n,c_n={\langle \tilde{\mathsf{\Phi }},{\psi }_n\rangle }_{L^2\left(\mathbb{R}\right)}.$$

(59)

Together with Lemma 5, this gives a concrete matrix representation of translations acting on the extended history. The representation depends on the chosen extension and is therefore a computational model for histories, not a canonical spectral resolution of (5).

8.2. Fourier Neural Operator Viewpoint

The same logarithmic transformation that turns proportional delay into translation also suggests why Fourier Neural Operator (FNO) architectures can be a natural computational language for learned input-output maps associated with (3). On the full line, the principal operator is represented by the Fourier multiplier (13); on finite history windows, Hermite coefficients such as (59) give a complementary localized representation of the history. An FNO is a neural-operator architecture in which part of the learned map is parametrized in Fourier variables; it is mentioned here only because translations are especially simple in frequency space. Thus an FNO-type model could use Fourier modes to parametrize the translation-invariant part of the continuation map, while separate channels or lifted variables encode the endpoint trace and non-modal residual history. This observation is only a modelling motivation: the exact spectrum, approximate spectrum, and growth bound are the analytic results proved above, not consequences of a neural approximation scheme; see [13,14,15] for the FNO and neural-operator framework.

9. A Model Stability Example

Take

$$v=1,L=1,B=-2,C=1.$$

(60)

Then

$$m\left(\mu \right)=\mu -2+{\mathrm{e}}^{\mu }.$$

(61)

By Theorem 3,

$$s\left({\mathcal{L}}_{+}\right)={\omega }_0\left({\mathcal{L}}_{+}\right)=-1,\parallel T\left(t\right)\parallel ={\mathrm{e}}^{-t}.$$

(62)

The point spectrum is $m(\Re \mu <0)$, while the boundary approximate spectrum is

$$m\left(\mathrm{i}\xi \right)=-2+{\mathrm{e}}^{\mathrm{i}\xi }+\mathrm{i}\xi ,\xi \in \mathbb{R}.$$

(63)

This example illustrates why stability should be formulated at the operator level, using the full half-line spectral image, rather than only through the zeros of m. The zeros of m describe stationary modal solutions of the homogeneous equation, whereas the spectrum of ${\mathcal{L}}_{+}$ describes the possible exponential rates of the whole evolution semigroup.

10. Scope and Limitations

The restriction $1\le p<\infty$ is essential for the present formulation. On $L^{\infty }(0,\infty )$ the left-translation semigroup is not strongly continuous in the usual norm, so the generator and growth-bound statements require a different state space, such as $C_0(0,\infty )$ or weak-star semigroup methods. The assumption $v>0$ is also structural: for the right-transport sign one must impose an inflow boundary condition at $x=0$, and that boundary condition changes both the resolvent and the spectrum. Complex transport speeds are outside the ordered half-line translation model, since $x+vt$ no longer defines a real half-line flow. Finally, multiple shifts or general bounded perturbations may still be treatable by Wiener–Hopf methods, but the scalar symbol is then replaced by a more complicated exponential polynomial or by an operator-valued symbol; the exact norm identity (41) is special to the single scalar shift commuting with the translation semigroup.

The representation and approximation discussion is also limited in scope. Hermite functions, and their half-line counterparts the Laguerre functions, form natural orthonormal bases for spectral approximations on $L^2(0,\infty )$ [1], while FNO-type models exploit frequency-domain multipliers and translation-invariant structure [13,14,15]. In both cases, a priori knowledge of the exact spectrum, such as Theorem 2, is useful for understanding approximation capabilities and possible convergence mechanisms in half-line domains. A detailed numerical analysis or learning theory for such schemes is beyond the scope of the present work.

11. Conclusion

The Eulerian pantograph equation becomes a constant-shift equation after a logarithmic change of variables. The main result of the paper concerns the left-transport maximal half-line realization on $L^p(0,\infty )$, $1\le p<\infty$: under this realization, the spectrum is $\overline{m\left(\right\{\Re \mu <0\left\}\right)}$ and the semigroup norm is $exp\left(\right(\Re B+\left|C\right|\left)t\right)$. These formulas give the sharp stability threshold $\Re B+\left|C\right|<0$ for that half-line operator. The other viewpoints serve a different role: the full-line Fourier multiplier supplies a comparison boundary curve, Mellin roots identify modal histories, and Hermite functions provide a practical basis for representing histories and translations. The link to FNO-style operator learning is therefore a modelling motivation rather than a source of the spectral theorem; neither the Hermite nor the FNO representation diagonalizes the half-line problem.