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

Abstract

We study an Eulerian pantograph equation with proportional delay by means of a logarithmic change of variables, which transforms the problem into a constant-shift equation on the half-line. The main object is the maximal left-transport realization on $L^p(0,\infty )$, with derivative domain $W^{1,p}(0,\infty )$ and no boundary condition at the endpoint. For the characteristic function $m\left(\mu \right)=v\mu +B+C\exp \left(L\mu \right)$, we prove that both the spectrum and the approximate point spectrum equal the closure of the image of the open left half-plane under m. Points obtained from the open half-plane are genuine eigenvalues, while the image of the boundary line supplies approximate eigenvalues. The resolvent outside this closed image is constructed directly as a one-sided convolution operator. The auxiliary semigroup generated by the half-line operator has the sharp norm identity $\parallel T\left(\tau \right)\parallel =\exp \left(\right(\mathrm{Re}B+\left|C\right|\left)\tau \right)$ and is exponentially stable exactly when $\mathrm{Re}B+\left|C\right|<0$; this concerns an additional evolution parameter, not stability in the original pantograph variable. We transfer the spectral theorem back to $L^p((0,1),\mathrm{d}t/t)$: interior modes decay at $t=0$, while boundary approximate modes are Mellin waves concentrating at the singular endpoint. As a secondary representation result, logarithmic pullback of the Hermite basis gives a Mellin–Hermite basis on $L^2({\mathbb{R}}_{+},\mathrm{d}t/t)$ with explicit Galerkin matrices for the Euler operator and dilations. The corresponding translation truncations converge strongly, and the shifted Gaussian admits an exact truncation-error formula.

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. Unless stated otherwise, we assume throughout that $A\in \mathbb{R}$ with $A<0$, $B,C\in \mathbb{C}$, and that all $L^p$ spaces are complex. Thus, the transport speed $v=-A$ introduced below is real and positive.

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 $t=0$. More precisely, $x\ge 0$ corresponds to $0<t\le 1$, whereas the complete multiplicative half-line $t>0$ corresponds to $x\in \mathbb{R}$. 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

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

(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 pantograph literature addresses several questions that should be distinguished from the one studied here. Classical works such as [1,2,3] develop solution theory, asymptotics, and characteristic-function methods for equations with proportional arguments. Liu [4] constructs exponential-, sine-, and cosine-type entire functions and uses them to study first- and higher-order linear initial- and boundary-value problems. More recent papers emphasize computation: Rihan and Rihan [5] analyze a $\vartheta$-method, Rihan [6] studies continuous Runge–Kutta schemes, and Abbasbandy [7] treats linear, nonlinear, and multiple-delay equations by a hybrid quantum pseudo-spectral framework. These works concern solution construction or approximation for prescribed functional-differential problems. The present question is different: after the logarithmic change of variables, which closed unbounded operator on a half-line is being considered, and what are its spectrum, resolvent, and semigroup norm?

General functional-differential and characteristic-root theory is developed in [8,9], while [10,11] provide the semigroup and Hardy-space tools used below. Those general frameworks do not remove the need to specify the realization. In particular, the full-line Fourier multiplier, the maximal half-line derivative used here, and a right-transport realization with an inflow boundary condition are different operators. Their spectra need not agree. For the maximal left-transport realization (5), every point parameterized by $m\left(\mathrm{i}\mathbb{R}\right)$ is an approximate eigenvalue, while every point parameterized by the open left half-plane is a genuine $L^p$ eigenvalue. These two images need not be disjoint. Representation theory for bounded translation-structured Wiener–Hopf operators on the half-line is developed, for example, in [12]. The operator (5) is instead an unbounded differential–shift realization; the direct kernel construction below is tailored to this specific maximal domain.

Main New Results

The new results of the paper concern the specified maximal realization on $L^p(0,\infty )$, $1\le p<\infty$, with no trace condition at $x=0$:

(i)

We identify the modal and boundary spectral mechanisms: every point of $m\left(\right\{\Re \mu <0\left\}\right)$ is a genuine $L^p$ eigenvalue, including the multiplicity statement for multiple characteristic roots, and every point of $m\left(\mathrm{i}\mathbb{R}\right)$ is an approximate eigenvalue generated by wave packets escaping to infinity. The two images may overlap.

(ii)

For every parameter outside the closed symbol image, we construct the resolvent explicitly as a one-sided convolution operator with an $L^1$ kernel. The construction is valid for the full range $1\le p<\infty$, including the endpoint space $p=1$.

(iii)

We prove the exact spectral identity

$${\mathrm{Spec}}_{\mathrm{ap}}\left({\mathcal{L}}_{+}\right)=\mathrm{Spec}\left({\mathcal{L}}_{+}\right)=m\left(\{\Re \mu \le 0\}\right)=\overline{m\left(\right\{\Re \mu <0\left\}\right)}.$$

(7)

Thus, the approximate point spectrum exhausts the full spectrum.

(iv)

We derive the sharp norm identity $\parallel T\left(\tau \right)\parallel =\exp \left(\right(\Re B+\left|C\right|\left)\tau \right)$ and hence the exact growth and auxiliary-semigroup stability bounds, rather than only an upper growth estimate.

(v)

We transfer the spectral theorem back to the original multiplicative variable: on $L^p((0,1),\mathrm{d}t/t)$, the Eulerian pantograph operator is isometrically conjugate to ${\mathcal{L}}_{+}$. This identifies interior spectral points with decaying power–logarithmic pantograph modes at $t=0$ and boundary spectral points with approximate Mellin waves concentrating at that singular endpoint.

(vi)

As a secondary representation result, we construct the Mellin–Hermite basis on $L^2({\mathbb{R}}_{+},\mathrm{d}t/t)$, obtain explicit Galerkin matrices for the Euler operator and dilations, and quantify the convergence of the associated translation truncations.

The compatibility calculation for finite-interval data is a supporting observation. The paper does not propose a new nonlinear pantograph model, a higher-order solution theory, or a numerical solver; its primary novelty lies in the explicit spectral, resolvent, and semigroup analysis of the specified half-line operator.

Potential applications are twofold. The exact spectrum and resolvent can be used to analyze solvability and conditioning of forced Eulerian pantograph problems near the singular endpoint, while the Mellin–Hermite matrices provide a scale-adapted Galerkin representation for numerical parameter studies. Developing and benchmarking a complete solver is beyond the scope of the present operator-theoretic paper.

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

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

(8)

we prove

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

(9)

Here, $\mathrm{Spec}\left({\mathcal{L}}_{+}\right)$ is the full spectrum of the unbounded operator ${\mathcal{L}}_{+}$, and ${\omega }_0\left({\mathcal{L}}_{+}\right)$ is the exponential growth rate of the auxiliary semigroup generated by ${\mathcal{L}}_{+}$. Its evolution parameter, denoted below by $\tau$, is independent of both the original pantograph variable t and the logarithmic coordinate x. In particular, Equation (3) is the stationary equation ${\mathcal{L}}_{+}Q=0$; generating $T\left(\tau \right)={\mathrm{e}}^{\tau {\mathcal{L}}_{+}}$ introduces an additional evolution and does not turn (1) into an evolution equation in $\tau$. This is a half-line result: the open-half-plane parameterization produces genuine $L^p$ eigenvalues, the boundary-line parameterization produces approximate eigenvalues by wave packets escaping to infinity, and the closed image is both the approximate point spectrum and the full spectrum. The two parameterized subsets may overlap.

Stability questions for time-delay systems in their original evolution variables are treated, for example, in [13]. As emphasized below, our exact norm identity instead concerns the auxiliary semigroup generated by (5). The Mellin–Hermite construction is presented separately as an exact scale-adapted representation and is not used to prove (7).

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 $\mathrm{Spec}\left(A\right)$ denotes its spectrum, i.e., the set of complex numbers $\lambda$ for which $\lambda -A$ is not boundedly invertible. The point spectrum ${\mathrm{Spec}}_p\left(A\right)$ is the set of eigenvalues. The approximate point spectrum ${\mathrm{Spec}}_{\mathrm{ap}}\left(A\right)$ 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 $\mathrm{Dom}\left(A\right)$ for the domain of A, $\mathrm{Ran}\left(A\right)$ for its range, $\mathrm{Ker}\left(A\right)$ 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 $s\left(A\right)=\sup \{\Re \lambda :\lambda \in \mathrm{Spec}(A\left)\right\}$. These conventions are standard in semigroup theory [10], but they are recalled here because the paper compares several related spectral notions.

2. Logarithmic Conjugacy and the Full-Line Model

Let

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

(10)

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

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

(11)

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

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

(12)

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

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

(13)

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:

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

(14)

Consequently, by the standard spectral theorem for multiplication operators,

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

(15)

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. 

With $v=-A>0$, Equation (14) 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.

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

$$\left(T_0\left(s\right)f\right)\left(x\right)=f(x+vs),s\ge 0.$$

(16)

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

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

(17)

The operator $S_L$ is a bounded contraction, so the bounded perturbation theorem [10] shows that (5), with the same domain $W^{1,p}$, generates a $C_0$-semigroup. 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 $T_0\left(s\right)$ simply transports the graph to the left: the value observed at x after semigroup time s is the old value at $x+vs$.

Let $\mathcal{D}$ denote the restrictions to $[0,\infty )$ of functions in $C_c^{\infty }\left(\mathbb{R}\right)$.

Lemma 1 

(Maximal domain and core). The operator ${\mathcal{L}}_{+}$ is closed on $X_p$ with domain $W^{1,p}(0,\infty )$, and $\mathcal{D}$ is a core for ${\mathcal{L}}_{+}$. For every $\lambda \in \mathbb{C}$, the graph norm of $\lambda -{\mathcal{L}}_{+}$ is equivalent to the $W^{1,p}$ norm. More precisely,

$$\parallel (\lambda -{\mathcal{L}}_{+}){u\parallel }_p \le v\parallel u^{\prime }{\parallel }_p{ + \left(\right|\lambda -B|+|C\left|\right)\parallel u\parallel }_p,$$

(18)

$$v\parallel u^{\prime }{\parallel }_p \le \parallel (\lambda -{\mathcal{L}}_{+}){u\parallel }_p{ + \left(\right|\lambda -B|+|C\left|\right)\parallel u\parallel }_p.$$

(19)

Proof. 

The maximal derivative D is closed, $S_L$ is bounded, and hence ${\mathcal{L}}_{+}$ is closed by bounded perturbation. The displayed estimates give the norm equivalence. Finally, $\mathcal{D}$ is dense in $W^{1,p}(0,\infty )$, so this equivalence makes it a graph-norm core for ${\mathcal{L}}_{+}$. □

Proposition 1 

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

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

(20)

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

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

(21)

Proof. 

Solving $v{f}^{\prime }=\lambda f$ gives the eigenvalue statement. Formula (21) 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. Characteristic Modes

For $\lambda \in \mathbb{C}$, set

$${\Delta }_{\lambda }\left(z\right)=\lambda -m\left(z\right)=\lambda -B-vz-C{\mathrm{e}}^{Lz}.$$

(22)

The exponential modes immediately give a point-spectrum inclusion which requires no representation theorem for general solutions.

Proposition 2 

(Modal eigenfunctions). For every $1\le p<\infty$,

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

(23)

Indeed, if $\Re \mu <0$, then $f_{\mu }\left(x\right)={\mathrm{e}}^{\mu x}$ belongs to $W^{1,p}(0,\infty )$ and

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

(24)

If ${\Delta }_{\lambda }$ has a zero $\mu$ of multiplicity r in $\Re \mu <0$, then $x^k{\mathrm{e}}^{\mu x}$, $0\le k<r$, are linearly independent eigenfunctions for $\lambda$.

Proof. 

Only the last assertion needs comment. Differentiate $({\mathcal{L}}_{+}-m\left(z\right)){\mathrm{e}}^{zx}=0$ with respect to z and evaluate at $z=\mu$. The derivatives of $m\left(z\right)-\lambda$ through order $r-1$ vanish, which gives $({\mathcal{L}}_{+}-\lambda )\left(x^k{\mathrm{e}}^{\mu x}\right)=0$ for $0\le k<r$. □

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

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

(25)

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 (25) is the usual characteristic equation for pantograph modes.

5. Full Spectrum from a One-Sided Resolvent Kernel

We now prove the full spectral formula directly. The proof avoids a Fredholm index for an unbounded symbol: outside the spectral image, the inverse is an explicit one-sided convolution operator. The argument constructs an $L^1$ kernel from the reciprocal symbol, verifies both inverse identities on a core, and then passes to the maximal domain. Thus, no additional half-line spectral points can remain outside the closed symbol image. Write

$${\mathbb{H}}_{-}=\{z\in \mathbb{C}:\Re z\le 0\}.$$

(26)

Lemma 2 

(Closed spectral image). The map m is proper on ${\mathbb{H}}_{-}$, and

$$m\left({\mathbb{H}}_{-}\right)=\overline{m\left({\mathbb{H}}_{-}^{°}\right)}.$$

(27)

Proof. 

For $z\in {\mathbb{H}}_{-}$,

$$\left|m\right(z\left)\right|\ge v\left|z\right|-\left|B\right|-\left|C\right|.$$

(28)

Hence, the inverse image in ${\mathbb{H}}_{-}$ of every compact subset of $\mathbb{C}$ is compact, so $m\left({\mathbb{H}}_{-}\right)$ is closed. Continuity gives $\overline{m\left({\mathbb{H}}_{-}^{°}\right)}\subset m\left({\mathbb{H}}_{-}\right)$. Conversely, each $m\left(\mathrm{i}\xi \right)$ is the limit of $m(-n^{-1}+\mathrm{i}\xi )$, and therefore belongs to $\overline{m\left({\mathbb{H}}_{-}^{°}\right)}$. □

Lemma 3 

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

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

(29)

Proof. 

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

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

(30)

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

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

(31)

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

Theorem 1 

(One-sided resolvent kernel). Let $\lambda \notin m\left({\mathbb{H}}_{-}\right)$. Then, there are $\eta >0$ and a kernel $k_{\lambda }\in L^1(0,\infty )$, with ${\mathrm{e}}^{\eta t}{k}_{\lambda }\left(t\right)\in L^2(0,\infty )$, such that

$${\displaystyle \frac{1}{{\Delta }_{\lambda }\left(z\right)}}={\int }_0^{\infty }{k}_{\lambda }\left(t\right){\mathrm{e}}^{zt}\mathrm{d}t,\Re z<\eta .$$

(32)

For every $1\le p<\infty$, the operator

$$\left(R_{\lambda }f\right)\left(x\right)={\int }_0^{\infty }{k}_{\lambda }\left(t\right)f(x+t)\mathrm{d}t$$

(33)

is bounded from $X_p$ into $\mathrm{Dom}\left({\mathcal{L}}_{+}\right)=W^{1,p}(0,\infty )$ and satisfies

$$(\lambda -{\mathcal{L}}_{+})R_{\lambda }=I_{X_p},R_{\lambda }(\lambda -{\mathcal{L}}_{+})=I_{\mathrm{Dom}\left({\mathcal{L}}_{+}\right)}.$$

(34)

Consequently, $R_{\lambda }={(\lambda -{\mathcal{L}}_{+})}^{-1}$ and $\lambda \in \rho \left({\mathcal{L}}_{+}\right)$.

Proof. 

Since ${\Delta }_{\lambda }$ has no zero in ${\mathbb{H}}_{-}$ and ${\Delta }_{\lambda }\left(z\right)=-vz+O\left(1\right)$ uniformly on every half-plane $\Re z\le {\eta }_0$, its zeros have positive distance from the imaginary axis. Hence, some $\eta >0$ satisfies

$${\Delta }_{\lambda }\left(z\right)\ne 0(\Re z\le \eta ).$$

(35)

Indeed, a contrary sequence of zeros with real parts tending to zero is either bounded, and then has a zero limit on the imaginary axis, or is unbounded, contradicting the linear asymptotic above.

Put $K_{\lambda }\left(z\right)=1/{\Delta }_{\lambda }\left(z\right)$ and

$$M_{\eta }:=|\lambda -B|+\left|C\right|{\mathrm{e}}^{L\eta }.$$

(36)

For $\Re z\le \eta$,

$$|{\Delta }_{\lambda }\left(z\right)|\ge v|z|-M_{\eta }.$$

(37)

Choose $R>2M_{\eta }/v$. On $\left|z\right|\ge R$, (37) gives $|K_{\lambda }\left(z\right)|\le 2/(v\left|z\right|)$. On the compact set $\{\Re z\le \eta ,|z|\le R\}$, the zero-free condition (35) gives $|K_{\lambda }\left(z\right)|\le c_R$ for some finite $c_R$. Splitting each vertical-line integral into the parts $|s+\mathrm{i}\xi |\le R$ and $|s+\mathrm{i}\xi |>R$, and using

$$\underset{s\in \mathbb{R}}{\sup }{\int }_{|s+\mathrm{i}\xi |>R}{\displaystyle \frac{\mathrm{d}\xi }{{|s+\mathrm{i}\xi |}^2}}<\infty ,$$

(38)

we obtain the uniform estimate

$$\underset{s<\eta }{\sup }{\int }_{\mathbb{R}}{\left|K_{\lambda }(s+\mathrm{i}\xi )\right|}^2\mathrm{d}\xi <\infty .$$

(39)

Thus, $K_{\lambda }$ belongs to the Hardy space $H^2$ of the left half-plane $\Re z<\eta$. To fix the sign convention, define $F_{\lambda }\left(w\right)=K_{\lambda }(\eta -w)$ for $\Re w>0$. The half-plane Paley–Wiener theorem for the Laplace transform [11] (Section 1.2) gives $h_{\lambda }\in L^2(0,\infty )$ such that

$$F_{\lambda }\left(w\right)={\int }_0^{\infty }{h}_{\lambda }\left(t\right){\mathrm{e}}^{-wt}\mathrm{d}t,\Re w>0.$$

(40)

Setting $k_{\lambda }\left(t\right)={\mathrm{e}}^{-\eta t}{h}_{\lambda }\left(t\right)$ gives (32) and ${\mathrm{e}}^{\eta t}{k}_{\lambda }\left(t\right)=h_{\lambda }\left(t\right)\in L^2(0,\infty )$. In particular,

$$\parallel k_{\lambda }{\parallel }_1 \le \parallel {\mathrm{e}}^{\eta ·}{k}_{\lambda }{\parallel }_2{\parallel {\mathrm{e}}^{-\eta ·}\parallel }_2 < \infty .$$

(41)

This p-independent passage from weighted $L^2$ to $L^1$ is what allows the same kernel construction to cover the endpoint space $p=1$.

Let ${\tilde{k}}_{\lambda }$ be the extension of $k_{\lambda }$ by zero to the negative half-line. The Fourier–Laplace transform is injective on distributions supported in $[0,\infty )$. Under the convention in (32), the transform of ${\widetilde{k\prime }}_{\lambda }$ is $-z{K}_{\lambda }\left(z\right)$, while the transform of ${\tilde{k}}_{\lambda }(·-L)$ is ${\mathrm{e}}^{Lz}{K}_{\lambda }\left(z\right)$. Therefore, ${\Delta }_{\lambda }\left(z\right)K_{\lambda }\left(z\right)=1$ is equivalent to the distributional identity

$$v{\widetilde{k\prime }}_{\lambda }+(\lambda -B){\tilde{k}}_{\lambda }-C{\tilde{k}}_{\lambda }(·-L)={\delta }_0.$$

(42)

Indeed, on each open interval separated by the points $0,L,2L,\dots$, the right-hand side obtained by solving (42) for ${\widetilde{k\prime }}_{\lambda }$ belongs to $L_{\mathrm{loc}}^1$. Hence, $k_{\lambda }\in W_{\mathrm{loc}}^{1,1}$ on every such interval and has one-sided traces at the junctions. A jump at a positive multiple of L would produce a Dirac mass in ${\widetilde{k\prime }}_{\lambda }$, while the other terms in (42) are locally integrable there; since the right-hand side has no point mass at such a junction, all positive-junction jumps vanish. If $a=k_{\lambda }(0+)$, the distributional derivative of the zero extension contains $a{\delta }_0$; comparison of the coefficients of ${\delta }_0$ in (42) gives $va=1$. Thus, $k_{\lambda }(0+)=1/v$, and for almost every $t>0$,

$$v{k}_{\lambda }^{\prime }\left(t\right)+(\lambda -B)k_{\lambda }\left(t\right)-C{\mathbf{1}}_{[L,\infty )}\left(t\right)k_{\lambda }(t-L)=0\mathrm{for}\mathrm{almost}\mathrm{every}t>0.$$

(43)

Since $k_{\lambda }$ and its delayed copy belong to $L^1(0,\infty )$, (43) also gives $k_{\lambda }^{\prime }\in L^1(0,\infty )$; hence, the boundary term at infinity vanishes in the integrations by parts below.

By Minkowski’s integral inequality and the contraction property of left translations,

$$\parallel R_{\lambda }{f\parallel }_p \le \parallel k_{\lambda }{\parallel }_1{\parallel f\parallel }_p.$$

(44)

For $f\in \mathcal{D}$, differentiation under the integral and integration by parts in the t variable give

$$\begin{array}{ll}\left((\lambda -{\mathcal{L}}_{+})R_{\lambda }f\right)\left(x\right)& =v{k}_{\lambda }(0+)f\left(x\right)\\ & +{\int }_0^{\infty }\left[v{k}_{\lambda }^{\prime }\left(t\right)+(\lambda -B)k_{\lambda }\left(t\right)-C{\mathbf{1}}_{[L,\infty )}\left(t\right)k_{\lambda }(t-L)\right]f(x+t)\mathrm{d}t\\ & =f\left(x\right).\end{array}$$

(45)

The identical integration-by-parts calculation, now beginning with $R_{\lambda }(\lambda -{\mathcal{L}}_{+})u$, gives

$$R_{\lambda }(\lambda -{\mathcal{L}}_{+})u=u,u\in \mathcal{D}.$$

(46)

It remains to pass from $\mathcal{D}$ to the maximal domain, using Lemma 1. For arbitrary $f\in L^p$, choose $f_n\in \mathcal{D}$ with $f_n\to f$. Then, $R_{\lambda }{f}_n\to R_{\lambda }f$ and $(\lambda -{\mathcal{L}}_{+})R_{\lambda }{f}_n=f_n$; closedness of $\lambda -{\mathcal{L}}_{+}$ yields $R_{\lambda }f\in W^{1,p}$ and $(\lambda -{\mathcal{L}}_{+})R_{\lambda }f=f$. Together with (19) and (44), this also proves boundedness of $R_{\lambda }:X_p\to W^{1,p}$. For arbitrary $u\in W^{1,p}$, choose $u_n\in \mathcal{D}$ with $u_n\to u$ in $W^{1,p}$. By (18), $(\lambda -{\mathcal{L}}_{+})u_n\to (\lambda -{\mathcal{L}}_{+})u$, and boundedness of $R_{\lambda }$ allows passage to the limit in (46). Hence, $R_{\lambda }(\lambda -{\mathcal{L}}_{+})u=u$. These are precisely the two identities in (34). □

Corollary 1 

(No additional half-line spectrum). For every $\lambda \notin m\left({\mathbb{H}}_{-}\right)$ and every $1\le p<\infty$,

$$\mathrm{Ker}(\lambda -{\mathcal{L}}_{+})=\left\{0\right\},\mathrm{Ran}(\lambda -{\mathcal{L}}_{+})=X_p,\parallel {(\lambda -{\mathcal{L}}_{+})}^{-1}{\parallel }_{\mathcal{B}\left(X_p\right)}\le {\parallel k_{\lambda }\parallel }_1.$$

(47)

Consequently, no point, continuous, or residual spectral component of the maximal half-line realization can occur outside $m\left({\mathbb{H}}_{-}\right)$.

Proof. 

The two identities in (34) give injectivity and surjectivity, while (44) gives the norm estimate. □

Theorem 2 

(Spectrum and approximate point spectrum). For every $1\le p<\infty$,

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

(48)

Proof. 

By Proposition 2, every point of $m\left({\mathbb{H}}_{-}^{°}\right)$ is an eigenvalue and hence an approximate eigenvalue. By Lemma 3, every point of $m\left(\mathrm{i}\mathbb{R}\right)$ is an approximate eigenvalue. Since ${\mathbb{H}}_{-}={\mathbb{H}}_{-}^{°}\cup \mathrm{i}\mathbb{R}$, this proves $m\left({\mathbb{H}}_{-}\right)\subset {\mathrm{Spec}}_{\mathrm{ap}}\left({\mathcal{L}}_{+}\right)$. Conversely, Theorem 1 shows that every $\lambda \notin m\left({\mathbb{H}}_{-}\right)$ belongs to the resolvent set. The general inclusion ${\mathrm{Spec}}_{\mathrm{ap}}\left({\mathcal{L}}_{+}\right)\subset \mathrm{Spec}\left({\mathcal{L}}_{+}\right)$ and Lemma 2 complete the proof. □

6. Pantograph Endpoint Interpretation in the Original Variable

The spectral theorem on the logarithmic half-line has a direct interpretation for the original pantograph equation near its singular endpoint. Define

$$Y_p=L^p\left((0,1),{\displaystyle \frac{\mathrm{d}t}{t}}\right)$$

(49)

and the maximal Eulerian pantograph operator

$$\begin{array}{ll}\left({\mathcal{P}}_{+}q\right)\left(t\right)& =At{q}^{\prime }\left(t\right)+Bq\left(t\right)+Cq\left(\alpha t\right),\\ \mathrm{Dom}\left({\mathcal{P}}_{+}\right)& =\left\{q\in Y_p:q\in W_{\mathrm{loc}}^{1,p}(0,1),t{q}^{\prime }\in Y_p\right\}.\end{array}$$

(50)

Here, $q\in W_{\mathrm{loc}}^{1,p}(0,1)$ means that $q\in W^{1,p}(a,b)$ for every $0<a<b<1$; equivalently, q has a locally absolutely continuous representative whose weak derivative belongs locally to $L^p$. Let

$$\left(Vq\right)\left(x\right)=q\left({\mathrm{e}}^{-x}\right),x\ge 0.$$

(51)

Theorem 3 

(Spectral and endpoint interpretation in the pantograph variable). For every $1\le p<\infty$, the map $V:Y_p\to L^p(0,\infty )$ is an isometric isomorphism, maps $\mathrm{Dom}\left({\mathcal{P}}_{+}\right)$ onto $W^{1,p}(0,\infty )$, and satisfies

$$V{\mathcal{P}}_{+}{V}^{-1}={\mathcal{L}}_{+}.$$

(52)

Consequently,

$${\mathrm{Spec}}_{\mathrm{ap}}\left({\mathcal{P}}_{+}\right)=\mathrm{Spec}\left({\mathcal{P}}_{+}\right)=m\left({\mathbb{H}}_{-}\right)=\overline{m\left({\mathbb{H}}_{-}^{°}\right)}.$$

(53)

More specifically:

(i) 

If $\Re \mu <0$, then

$$q_{\mu }\left(t\right)=t^{-\mu }\in \mathrm{Dom}\left({\mathcal{P}}_{+}\right),{\mathcal{P}}_{+}{q}_{\mu }=m\left(\mu \right)q_{\mu },$$

(54)

and $q_{\mu }\left(t\right)\to 0$ as $t\to 0^{+}$. If ${\Delta }_{\lambda }$ has a zero $\mu$ of multiplicity r in $\Re \mu <0$, then

$$q_{\mu ,k}\left(t\right)={(-\log t)}^k{t}^{-\mu },0\le k<r,$$

(55)

are linearly independent eigenfunctions of ${\mathcal{P}}_{+}$ for $\lambda$, and all of them tend to zero at $t=0$.

(ii) 

If $\xi \in \mathbb{R}$, the formal Mellin wave $t^{-\mathrm{i}\xi }$ has constant modulus and does not belong to $Y_p$. Nevertheless, for any non-zero $\chi \in C_c^1(-1,1)$ the functions

$$q_R\left(t\right)=R^{-1/p}\chi \left({\displaystyle \frac{-\log t-R}{R}}\right)t^{-\mathrm{i}\xi }$$

(56)

belong to $\mathrm{Dom}\left({\mathcal{P}}_{+}\right)$, their normalized residuals satisfy

$${\displaystyle \frac{\parallel ({\mathcal{P}}_{+}-m\left(\mathrm{i}\xi \right))q_R{\parallel }_{Y_p}}{\parallel q_R{\parallel }_{Y_p}}}⟶0,R\to \infty ,$$

(57)

and their supports concentrate at $t=0$: for every $\epsilon >0$, $\mathrm{supp}{q}_R\subset (0,\epsilon )$ for all sufficiently large R.

In particular, if $z=-\mu$ with $\Re z>0$, then the stationary condition $m\left(\mu \right)=0$ is exactly

$$Az+B+C{\alpha }^z=0,$$

(58)

and $q\left(t\right)=t^z$ is a non-zero solution of the original homogeneous pantograph equation that decays at the singular endpoint.

Proof. 

The change of variables $t={\mathrm{e}}^{-x}$ gives

$${\parallel Vq\parallel }_{L^p(0,\infty )}^p={\int }_0^{\infty }|q\left({\mathrm{e}}^{-x}\right){|}^p\mathrm{d}x={\int }_0^1{\left|q\left(t\right)\right|}^p{\displaystyle \frac{\mathrm{d}t}{t}}.$$

(59)

Moreover,

$$V\left(t{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\right)V^{-1}=-D,V\left(q(\alpha ·)\right)=S_{L}Vq.$$

(60)

These identities prove the domain correspondence and (52); (53) then follows from Theorem 2. Formulas (54) and (55) are the inverse images under V of the eigenfunctions in Proposition 2. Since $-\Re \mu >0$, powers of $-\log t$ do not prevent their decay at $t=0$.

For the boundary statement, $V{q}_R$ is exactly the wave packet used in Lemma 3. The isometry and conjugacy therefore give (57). Because $\mathrm{supp}\chi$ is a compact subset of $(-1,1)$, there is $c>0$ such that $-\log t\ge cR$ on $\mathrm{supp}{q}_R$; hence, the supports concentrate at $t=0$. Finally, substituting $\mu =-z$ and $\alpha ={\mathrm{e}}^{-L}$ into $m\left(\mu \right)=0$ gives (58). □

Remark 2. 

Theorem 3 concerns only the limit $t\to 0^{+}$, because the state space $x\ge 0$ corresponds to $0<t\le 1$. The behavior as $t\to \infty$ corresponds to $x\to -\infty$ and is not controlled by ${\mathcal{L}}_{+}$. Its analysis requires a separate half-line realization, in logarithmic coordinates $y=\log t\ge 0$, with a retarded shift and an appropriate history or boundary specification.

7. Exact Semigroup Norm and Operator-Semigroup Stability

The pantograph equation itself is the stationary equation ${\mathcal{L}}_{+}Q=0$. We now introduce an independent evolution parameter $\tau \ge 0$ and consider the auxiliary abstract Cauchy problem

$${\displaystyle \frac{\mathrm{d}U}{\mathrm{d}\tau }}\left(\tau \right)={\mathcal{L}}_{+}U\left(\tau \right),U\left(0\right)=U_0.$$

(61)

All growth and stability statements in this section refer to (61), not to variation of the original solution $q\left(t\right)$ with respect to t.

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

$$T\left(\tau \right)={\mathrm{e}}^{B\tau }\exp \left(C\tau S_L\right)T_0\left(\tau \right)={\mathrm{e}}^{B\tau }{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{\left(C\tau \right)}^n}{n!}}{T}_0\left(\tau +{\displaystyle \frac{nL}{v}}\right).$$

(62)

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 4 

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

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

(63)

and

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

(64)

Proof. 

For the upper bound, use (62) and the contraction property $\parallel T_0\left(s\right)\parallel \le 1$:

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

(65)

It remains to prove that no cancellation can improve this estimate. Commutativity provides approximate joint eigenvectors for all translations in (62), and their oscillatory phase can be chosen to align the complex coefficients. If $C=0$, the identity follows from $\parallel T_0\left(\tau \right)\parallel =1$, 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 $\eta L\equiv \theta \left(\mathrm{mod}2\pi \right)$. Let $\chi \in C_c^1(-1,1)$ be normalized by ${\parallel \chi \parallel }_p=1$ and set

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

(66)

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

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

(67)

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 (67) to the finitely many shifts $s_n=\tau +nL/v$, $0\le n\le N$, gives

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

(68)

Because $\eta L\equiv \theta$ and $C{\mathrm{e}}^{\mathrm{i}\theta }=\left|C\right|$, the scalar sum in (68) is ${\sum }_{n=0}^N{\left(\right|C\left|\tau \right)}^n/n!$. The remaining tail of (62) is bounded uniformly in R by

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

(69)

Therefore,

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

(70)

Letting $N\to \infty$ in (70) gives $\parallel T\left(\tau \right)\parallel \ge {\mathrm{e}}^{(\Re B+|C\left|\right)\tau }$. Together with (65), this proves (64).

The growth bound is then

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

(71)

Finally, by Theorem 2,

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

(72)

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 2 

(Sharp auxiliary-semigroup stability threshold). The auxiliary semigroup ${\left(T\left(\tau \right)\right)}_{\tau \ge 0}$ generated by ${\mathcal{L}}_{+}$ is exponentially stable if and only if

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

(73)

If equality holds, the auxiliary semigroup is bounded but not exponentially stable. This conclusion concerns (61); it does not by itself imply stability or decay of solutions $q\left(t\right)$ of (1) as $t\to 0$ or $t\to \infty$.

8. Finite-Interval Data, Compatibility, and Mellin Modes

The equation in logarithmic variables is advanced rather than delayed, so data on one interval are not an arbitrary initial history. The following standard continuation condition is recalled only to distinguish admissible finite-interval data from arbitrary profiles. If $C=0$, the equation is the ODE $v{Q}^{\prime }+BQ=0$ and no step construction is needed. If $C\ne 0$, set

$$\left(\mathcal{R}\varphi \right)\left(x\right):=-{\displaystyle \frac{v{\varphi }^{\prime }\left(x\right)+B\varphi \left(x\right)}{C}},Q(x+L)=\mathcal{R}Q\left(x\right).$$

(74)

Remark 3 

(Standard compatibility condition). For $\mathsf{\Phi }\in C^{\infty }[0,L]$, the formal step continuation

$$Q(nL+x)=\left({\mathcal{R}}^n\mathsf{\Phi }\right)\left(x\right),n=0,1,2,\dots ,0\le x<L.$$

(75)

defines a function in $C^{\infty }[0,\infty )$ satisfying (3) if and only if

$${\left({\mathcal{R}}^n\mathsf{\Phi }\right)}^{\left(j\right)}\left(L\right)={\left({\mathcal{R}}^{n+1}\mathsf{\Phi }\right)}^{\left(j\right)}\left(0\right),n,j=0,1,2,\dots .$$

(76)

In particular, continuity at the first junction requires

$$\mathsf{\Phi }\left(L\right)=-{\displaystyle \frac{v{\mathsf{\Phi }}^{\prime }\left(0\right)+B\mathsf{\Phi }\left(0\right)}{C}}.$$

(77)

For the modal datum ${\mathsf{\Phi }}_z\left(x\right)={\mathrm{e}}^{-zx}$, the Mellin characteristic Equation (25) is equivalent to $\mathcal{R}{\mathsf{\Phi }}_z={\mathrm{e}}^{-Lz}{\mathsf{\Phi }}_z$; hence, all conditions (76) hold automatically. Thus, characteristic roots generate compatible modal solutions.

The logarithmic interval $x\in [0,L]$ corresponds under $t={\mathrm{e}}^{-x}$ to $t\in [\alpha ,1]$, not to $[0,1]$. The endpoint $t=0$ corresponds to $x=\infty$ and is therefore unrelated to the first finite continuation interval. The following elementary observation concerns global continuous functions in the original variable only.

Proposition 3 

(Endpoint obstruction for global continuous data). Let

$${\mathcal{M}}_{\mathrm{mod}}=\mathrm{span}\{t^z:zsolves\left(25\right)and\Re z>0\}$$

(78)

in $C[0,1]$. Then,

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

(79)

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

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

(80)

Thus, no modal expansion made only of decaying Mellin modes can approximate such a global continuous function 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 (79). The distance estimate follows by evaluating $g-m$ at $t=0$. □

Remark 4. 

Proposition 3 is not an incompleteness statement for data on $[0,L]$, which correspond to $t\in [\alpha ,1]$, and it is not a density statement in $L^p(0,\infty )$. In the latter space, evaluation at the endpoint is not a continuous state-space functional. The proposition is only a uniform-approximation obstruction in $C[0,1]$ for global continuous functions having a prescribed trace at $t=0$.

9. Mellin–Hermite Representation and Galerkin Matrices

This section records a secondary, exact representation result. It is independent of the half-line spectral proof. Let

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

(81)

Then, ${\left({\psi }_n\right)}_{n\ge 0}$ is an orthonormal basis of $L^2\left(\mathbb{R}\right)$ [14] and, for the unitary Fourier transform, $\mathcal{F}{\psi }_n={(-\mathrm{i})}^n{\psi }_n$. Let

$$P_{N}f={\displaystyle \sum _{n=0}^N}⟨f,{\psi }_n⟩{\psi }_n$$

(82)

be the orthogonal projection onto the first $N+1$ Hermite modes.

Lemma 4 

(Differentiation matrix). The Hermite functions satisfy

$${\psi }_n^{\prime }=\sqrt{{\displaystyle \frac{n}{2}}}{\psi }_{n-1}-\sqrt{{\displaystyle \frac{n+1}{2}}}{\psi }_{n+1},$$

(83)

where the first term is absent for $n=0$. Hence, the Galerkin differentiation matrix $D_N={\left(d_{mn}\right)}_{m,n=0}^N$ is tridiagonal, with

$$d_{mn}=⟨{\psi }_m,{\psi }_n^{\prime }⟩=\sqrt{{\displaystyle \frac{n}{2}}}{\delta }_{m,n-1}-\sqrt{{\displaystyle \frac{n+1}{2}}}{\delta }_{m,n+1}.$$

(84)

Proof. 

This is the standard creation–annihilation relation, followed by orthogonal projection. □

Lemma 5 

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

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

(85)

Proof. 

This is the displacement-operator formula in the Hermite basis; see [15] (Chapter 1). □

Let $S_N={\left(s_{mn}\right)}_{m,n=0}^N$, where

$$s_{mn}=⟨{\psi }_m,S_L{\psi }_n⟩.$$

(86)

The entries of $S_N$ are given explicitly by Lemma 5. Consequently, the extended full-line differential–shift expression has the computable Galerkin matrix

$${\mathsf{L}}_N=v{D}_N+B{I}_{N+1}+C{S}_N.$$

(87)

This full-line Galerkin matrix is not a spectral compression theorem for the maximal half-line realization.

Proposition 4 

(Convergence and limitation of Hermite translation truncations). For every $f\in L^2\left(\mathbb{R}\right)$,

$$P_N{S}_L{P}_{N}f⟶S_{L}f\mathrm{in}{L}^2\left(\mathbb{R}\right).$$

(88)

However, this convergence is not in operator norm; for every N,

$$\parallel S_L-P_N{S}_L{P}_N\parallel \ge 1.$$

(89)

Proof. 

Since $P_N\to I$ strongly and $S_L$ is unitary on $L^2\left(\mathbb{R}\right)$,

$$\parallel P_N{S}_L{P}_{N}f-S_L{f\parallel }_2 \le \parallel P_N{S}_L(P_{N}f-f){\parallel }_2 + {\parallel (P_N-I)S_{L}f\parallel }_2$$

(90)

$$\le \parallel P_N{f-f\parallel }_2+{\parallel (P_N-I)S_{L}f\parallel }_2⟶0.$$

(91)

For the second assertion, choose a unit vector $f\in \mathrm{Ker}\left(P_N\right)$. Then, $P_N{S}_L{P}_{N}f=0$, whereas $\parallel S_L{f\parallel }_2=1$. □

Example 1 

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

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

(92)

Thus, even the shifted Gaussian has infinitely many Hermite coefficients. Truncating (92) after mode N gives the exact error

$${∥S_L{\psi }_0-P_N{S}_L{\psi }_0∥}_2^2={\mathrm{e}}^{-L^2/2}{\displaystyle \sum _{m=N+1}^{\infty }}{\displaystyle \frac{{(L^2/2)}^m}{m!}}.$$

(93)

Thus, the right-hand side gives an exact mode-selection criterion for any prescribed tolerance.

The Hermite representation in logarithmic coordinates has an equivalent formulation directly on the multiplicative variable t. Define

$${\varphi }_n\left(t\right)={\psi }_n(-\log t),t>0,n=0,1,2,\dots .$$

(94)

We call $\left({\varphi }_n\right)$ the Mellin–Hermite system. For $f\in \mathcal{H}$, use the unitary Mellin transform convention

$$\left(\mathfrak{M}f\right)\left(\xi \right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_0^{\infty }f\left(t\right)t^{\mathrm{i}\xi }{\displaystyle \frac{\mathrm{d}t}{t}}.$$

(95)

Proposition 5 

(Mellin–Hermite conjugacy). The system ${\left({\varphi }_n\right)}_{n\ge 0}$ is an orthonormal basis of $\mathcal{H}=L^2({\mathbb{R}}_{+},\mathrm{d}t/t)$, and

$$\mathfrak{M}{\varphi }_n={(-\mathrm{i})}^n{\psi }_n.$$

(96)

Let $\left({\mathcal{D}}_{\alpha }f\right)\left(t\right)=f\left(\alpha t\right)$ and $\mathcal{E}=t\mathrm{d}/\mathrm{d}t$. Then,

$$U{\mathcal{D}}_{\alpha }{U}^{-1}=S_L,U\mathcal{E}{U}^{-1}=-D,L=\log (1/\alpha ).$$

(97)

Consequently, the Galerkin compression of the Eulerian pantograph expression $A\mathcal{E}+BI+C{\mathcal{D}}_{\alpha }$ to the first $N+1$ Mellin–Hermite modes has the matrix

$$-A{D}_N+B{I}_{N+1}+C{S}_N\left(L\right)=v{D}_N+B{I}_{N+1}+C{S}_N\left(L\right).$$

(98)

Proof. 

The basis assertion follows from the unitarity of U in (11). With $t={\mathrm{e}}^{-x}$, Formula (95) becomes the unitary Fourier transform of $Uf$. The standard Hermite Fourier identity gives (96). The two conjugacy identities are (12) and the logarithmic form of the dilation $t\mapsto \alpha t$. Projecting them onto the first $N+1$ modes gives (98). □

Thus, the Mellin transform and the Hermite expansion play complementary roles. The Mellin transform diagonalizes dilations, while the Mellin–Hermite coefficients retain localization in logarithmic scale and represent a dilation by the explicit matrix $S_N\left(L\right)$. This provides a scale-adapted alternative to using periodic Fourier modes on a truncated logarithmic interval.

The distinction between the two domains remains essential. The basis $\left({\varphi }_n\right)$ is a basis on the full multiplicative half-line $t>0$, equivalently on the full logarithmic line $x\in \mathbb{R}$, whereas the main realization ${\mathcal{L}}_{+}$ acts on $x\ge 0$, equivalently on $0<t\le 1$. Applying the Mellin–Hermite representation to the latter problem therefore requires either an explicit extension from $x\ge 0$ to $\mathbb{R}$ or a restriction/projection procedure adapted to $0<t\le 1$. The matrices above describe the resulting extended or projected representation; they are not a canonical discretization of the maximal half-line operator.

10. A Model Operator-Semigroup Example

Take

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

(99)

Then,

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

(100)

By Theorem 4,

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

(101)

Every point of $m(\Re \mu <0)$ is an eigenvalue, and the boundary parameterization gives the approximate eigenvalues

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

(102)

These two parameterized subsets are not asserted to be disjoint. This example illustrates operator-semigroup stability for the auxiliary parameter $\tau$. Its spectral description uses the full half-line image, not merely the zero set 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 auxiliary evolution semigroup. No conclusion about decay of the original function $q\left(t\right)$ as $t\to 0$ or $t\to \infty$ is asserted here.

11. 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 (64) is special to the single scalar shift commuting with the translation semigroup.

The Mellin–Hermite result is a full-line representation statement. Its application to the principal half-line realization requires a specified extension or projection, and no claim of operator-norm convergence or spectral convergence of the Galerkin matrices is made.

12. Conclusions

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 $\exp \left(\right(\Re B+\left|C\right|\left)\tau \right)$. These formulas give the sharp stability threshold $\Re B+\left|C\right|<0$ for the auxiliary semigroup generated by that half-line operator. The parameter $\tau$ is independent of the original pantograph variable, so this threshold is not, by itself, a stability criterion for $q\left(t\right)$. In contrast, the isometric conjugacy with the maximal pantograph operator on $L^p((0,1),\mathrm{d}t/t)$ gives a direct endpoint interpretation of the spectrum: interior modes $t^{-\mu }$, $\Re \mu <0$, decay as $t\to 0^{+}$, while boundary approximate modes are Mellin waves concentrating at the singular endpoint. The other viewpoints serve a different role: the full-line Fourier multiplier supplies a comparison boundary curve, Mellin roots identify compatible modal profiles, and the functions ${\varphi }_n\left(t\right)={\psi }_n(-\log t)$ provide a Mellin–Hermite basis adapted to the scale-invariant measure $\mathrm{d}t/t$. In the corresponding Galerkin representation, the Euler operator is tridiagonal and the dilation $t\mapsto \alpha t$ has the explicit Laguerre matrix $S_N\left(L\right)$. This exact representation is a secondary result and is logically independent of the spectral proof.