# Algebraic Structure of the Uncertainty Principle in Rational Exponential Function Spaces

Wenxiang Chen

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Wenxiang Chen. Algebraic Structure of the Uncertainty Principle in Rational Exponential Function Spaces. 2025. hal-05092225

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Distributed under a Creative Commons Attribution 4.0 International License
# Algebraic Structure of the Uncertainty Principle
in Rational Exponential Function Spaces

Wenxiang Chen

School of Physics and Materials Science, Guangzhou University

wxchen4277@qq.com

## Abstract

Let $A(z), B(z) \in \mathbb{C}[z]$ with $A \neq 0$ and $\deg A \leq 4$. For the rational exponent $m(z) = A(z)/B(z)$ we study the Hilbert space $\mathcal{H}_f = \overline{\mathcal{L}}^{\langle \cdot, \cdot \rangle}$, where $\mathcal{L} = f(z)\mathbb{C}[z, z^{-1}]$ and $f(z) = e^{m(z)}$. We prove that the operator algebra generated by the canonical pair $(z, -i\partial_z)$ exhibits a *strictly positive* Heisenberg-type uncertainty lower bound $\Delta z \Delta k \geq \frac{1}{2} + C_0 > 0$ iff the zero set of $B$ (counting multiplicities) strictly exceeds that of $A$. The proof combines fine-grained Laurent analysis, entire-function growth theory, and Paley-Wiener spectral bounds. We also show that when $\deg Z_B \leq \deg Z_A$ the bound can be made arbitrarily small by a family of truncated $f$-modes, recovering classical-quantum correspondence in the limit.

## Contents

<table><tr><td><b>1</b></td><td><b>Preliminaries and Notation</b></td><td><b>1</b></td></tr><tr><td>1.1</td><td>Variance and uncertainty</td><td>2</td></tr><tr><td><b>2</b></td><td><b>Laurent Expansion and Essential Singularities</b></td><td><b>2</b></td></tr><tr><td>2.1</td><td>Growth order and exponential type</td><td>2</td></tr><tr><td><b>3</b></td><td><b>Uncertainty Bound When |Z<sub>B</sub>| > |Z<sub>A</sub>|</b></td><td><b>3</b></td></tr><tr><td><b>4</b></td><td><b>Localization Sequence When |Z<sub>B</sub>| ≤ |Z<sub>A</sub>|</b></td><td><b>3</b></td></tr><tr><td><b>5</b></td><td><b>Main Theorem</b></td><td><b>4</b></td></tr><tr><td><b>6</b></td><td><b>Physical Interpretation</b></td><td><b>4</b></td></tr><tr><td><b>7</b></td><td><b>Conclusion</b></td><td><b>4</b></td></tr></table>

## 1 Preliminaries and Notation

**Definition 1.1** (Zero sets). For $P \in \mathbb{C}[z]$ let $Z_P = \{z \in \mathbb{C} : P(z) = 0\}$ denote the multiset of zeros counted with algebraic multiplicity, and write $\#Z_P$ for its cardinality.

Throughout we fix

$$m(z) = \frac{A(z)}{B(z)}, \quad f(z) = \exp\{m(z)\}, \quad \mathcal{L} = f(z)\mathbb{C}[z, z^{-1}], \quad (1.1)$$
with closure $\mathcal{H}_f$ in a suitable $L^2$-based inner product described below.
Canonical observables act on $\mathcal{L}$ by

$$O_1 = z, \quad O_2 = -i\partial_z, \quad [O_1, O_2] = i, \qquad (1.2)$$

and extend to densely defined symmetric operators on $\mathcal{H}_f$.

## 1.1 Variance and uncertainty

For a normalized $\psi \in \mathcal{H}_f$ set

$$\Delta z^2 := \langle \psi, (z - \langle z \rangle_{\psi})^2 \psi \rangle, \quad \Delta k^2 := \langle \psi, (-i\partial_z - \langle k \rangle_{\psi})^2 \psi \rangle, \qquad (1.3)$$

where $\langle \cdot \rangle_{\psi}$ denotes expectation. Standard arguments yield the Robertson inequality

$$\Delta z \Delta k \geq \frac{1}{2} |\langle [O_1, O_2] \rangle_{\psi}| = \frac{1}{2}. \qquad (1.4)$$

Our goal is to sharpen (1.4) within $\mathcal{H}_f$.

# 2 Laurent Expansion and Essential Singularities

Let $z_0 \in Z_B$ have multiplicity $n := \mathrm{ord}_{z_0}(B) \geq 1$. Writing $A(z) = (z - z_0)^m h(z)$ with $h(z_0) \neq 0$ gives

$$m(z) = \sum_{j=-(n-m)}^{\infty} c_j (z - z_0)^j, \quad c_{-(n-m)} \neq 0. \qquad (2.1)$$

If $n > m$, then $m$ has a pole of order $N := n - m \geq 1$ at $z_0$. Expanding $\exp m$ termwise,

$$f(z) = \sum_{\ell=-\infty}^{\infty} a_{\ell} (z - z_0)^{\ell}, \qquad (2.2)$$

which contains *infinitely many* negative powers; hence $z_0$ is an *essential singularity* of $f$ [2].

**Proposition 2.1.** If $z_0$ is an unmatched zero of $B$ (i.e. $\mathrm{ord}_{z_0}(B) > \mathrm{ord}_{z_0}(A)$), then $f$ has an essential singularity at $z_0$. Consequently every $\phi \in \mathcal{L}$ inherits an essential singularity at $z_0$.

*Proof.* The first assertion follows from (2.2). For the second, note that $\phi(z) = p(z) f(z)$ with $p$ a Laurent polynomial. Multiplying by $(z - z_0)^k$ for finite $k$ cannot cancel the infinitely many negative coefficients in (2.2). $\square$

## 2.1 Growth order and exponential type

Define the *exponential type* of an entire function $g$ by

$$\tau(g) = \limsup_{r \to \infty} \frac{\log \sup_{|z|=r} |g(z)|}{r}. \qquad (2.3)$$

If $g$ extends analytically to the strip $|\Im z| < a$ with finite type $\tau$, then by the Paley-Wiener theorem [1] its Fourier transform $\hat{g}$ is supported in $[-\tau - a, \tau + a]$. Essential singularities preclude any such strip, forcing $\tau(g) = \infty$.
**Lemma 2.2** (Essential ⇒ exponential tail). Let g be analytic on $\mathbb{R}$ and extend meromorphically to $\mathbb{C}$. If g has an essential singularity at distance d from $\mathbb{R}$, then for sufficiently large $|k|$

$$|\hat{g}(k)| \gtrsim e^{-d|k|}, \qquad (2.4)$$

and no truncation of g can make $\hat{g}$ compactly supported.

*Proof.* The strip $|\Im z| < d$ is maximal by assumption. Analytic continuation of the Fourier inversion contour yields the bound (2.4); cf. [3]. □

### 3 Uncertainty Bound When $|Z_B| > |Z_A|$

**Theorem 3.1.** Suppose $\#Z_B > \#Z_A$. Then there exists $C_0 = C_0(A, B) > 0$ such that

$$\Delta z_{\psi} \Delta k_{\psi} \geq \frac{1}{2} + C_0 \quad \text{for all } \psi \in \mathcal{H}_f \text{ with } \|\psi\| = 1. \qquad (3.1)$$

*Proof.* By pigeonhole, there is $z_0 \in Z_B \setminus Z_A$, so Proposition 2.1 applies. Let $d = \text{dist}(z_0, \mathbb{R})$. For any $\psi = p(z)f(z)$ with $p \in \mathbb{C}[z, z^{-1}]$, Lemma 2.2 ensures an exponential tail of rate d for $\hat{\psi}$. Compute

$$\Delta k_{\psi}^2 \geq \int_{|k|>1/d} k^2 |\hat{\psi}(k)|^2 dk \geq \int_{1/d}^{\infty} k^2 e^{-2dk} dk = \frac{1}{2d^3}, \qquad (3.2)$$

where the implicit constant is uniform in $\psi$ of unit norm. Meanwhile $\Delta z_{\psi}^2 \geq 0$. Inserting into (1.4) yields $\Delta z_{\psi} \Delta k_{\psi} \geq \frac{1}{2} + \frac{1}{4d^{3/2}} =: \frac{1}{2} + C_0$. □

### 4 Localization Sequence When $|\mathcal{Z}_B| \leq |\mathcal{Z}_A|$

Assume now $\#Z_B \leq \#Z_A$. Write

$$m(z) = Q(z) + \frac{R(z)}{S(z)}, \quad \deg R < \deg S, \quad (R, S) = 1. \qquad (4.1)$$

All poles of $m$ are simple; let $\{a_j\}_{j=1}^J$ be their locations, none of which lie on the real axis by hypothesis. Choose $\chi \in C_c^\infty(\mathbb{R})$ with $\chi \equiv 1$ on $[-1, 1]$. Set

$$\chi_R(z) := \chi(z/R) \prod_{j=1}^{J} \left(1 - \frac{z-a_j}{\rho_j}\right), \qquad (4.2)$$

where $\rho_j > 0$ are small radii so the product kills $f$ near each pole $a_j$. Define

$$\psi_R(z) := \frac{\chi_R(z)}{\|\chi_R\|_{L^2}} f(z). \qquad (4.3)$$

**Lemma 4.1.** As $R \to \infty$,

$$\Delta z_{\psi_R} \sim R, \quad \Delta k_{\psi_R} \lesssim R^{-1}, \qquad (4.4)$$

hence $\Delta z_{\psi_R} \Delta k_{\psi_R} \to 0$.

*Proof.* supp $\psi_R \subset [-2R, 2R]$, so $\Delta z \leq 2R$. Fourier transforming (4.3), $\hat{\psi}_R = \hat{\chi}_R * \hat{f}$, and $\hat{\chi}_R(k) = R\hat{\chi}(Rk)$ decays super-algebraically. Therefore $\int k^2 |\hat{\psi}_R(k)|^2 dk \leq CR^{-2}$, giving the claimed bound. □

**Corollary 4.2.** If $\#Z_B \leq \#Z_A$, the infimum of $\Delta z \Delta k$ over unit vectors in $\mathcal{H}_f$ is 0.
# 5 Main Theorem

**Theorem 5.1** (Necessary and sufficient condition). Let $A, B \in \mathbb{C}[z]$ with $A \neq 0$, $\deg A \leq 4$, and set $f(z) = \exp\{A(z)/B(z)\}$. Define $\mathcal{H}_f$ and the operators $(O_1, O_2)$ as in (1.1)-(1.2). Then

$$ \Delta z \Delta k \geq \frac{1}{2} + C_0 \quad (\text{some } C_0 > 0) \qquad (5.1) $$

for all $\psi \in \mathcal{H}_f$, iff $\#Z_B > \#Z_A$.

*Proof.* The forward implication is Theorem 3.1. For the converse, assume $\#Z_B \leq \#Z_A$. Lemma 4.1 produces a sequence with $\Delta z \Delta k \to 0$, contradicting any $C_0 > 0$. □

# 6 Physical Interpretation

Polynomial $A$ (degree $\leq 4$) may be viewed as an *effective action* in semiclassical path integrals, while $B$ encodes additional zeros (or “defects”). An unmatched zero of $B$ forces an essential singularity in $f$, introducing high-frequency components that are kinematically inevitable, thus lifting the uncertainty floor. Conversely, when the zeros match, wavefunctions can be simultaneously compressed in configuration and momentum space, approaching a *phase-space point* in the classical limit (cf. coherent-state quantization [6]).

**Application to lattice Bloch-Wannier duality.** Taking $z = e^{i\theta}$, $k$ the discrete reciprocal, and choosing $A, B$ from tight-binding dispersion data, counting their zeros immediately predicts whether band-limited Wannier states can be constructed with arbitrary localization, a key design criterion for flat-band superconductivity [7].

# 7 Conclusion

We have linked the algebraic data $(A, B)$ to a sharp dichotomy in uncertainty behaviour: one extra zero of $B$ suffices to enforce a genuine “Heisenberg gap” in $\mathcal{H}_f$, whereas perfect cancellation of zeros restores the possibility of near-classical states. Future work includes extending the argument to matrix-valued $A, B$ relevant for multi-flavour models, and quantifying $C_0$ in terms of pole geometry via refined Phragmén-Lindelöf estimates.

# References

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[2] J. B. Conway, *Functions of One Complex Variable*, 2nd ed., Springer, 1978.

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[4] M. Reed and B. Simon, *Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness*, Academic Press, 1975.

[5] Todorov, Todor D. "Infinite-dimensional linear algebra and solvability of partial differential equations." arXiv preprint arXiv:2106.04039 (2021).

[6] Ali, Syed Twareque, Jean-Pierre Antoine, and Jean-Pierre Gazeau. Coherent states, wavelets and their generalizations. Vol. 3. New York: Springer, 2000.
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[8] B. C. Hall, *Quantum Theory for Mathematicians*, Springer, 2013.

[9] G. B. Folland and A. Sitaram, “The uncertainty principle: a mathematical survey,” *J. Fourier Anal. Appl.* **3** (1997): 207–238.

[10] L. Hörmander, “The Weyl calculus of pseudodifferential operators,” *Comm. Pure Appl. Math.* **32** (1979): 360–444.

[11] S. G. Krantz and H. R. Parks, *A Primer of Real Analytic Functions*, 2nd ed., Birkhäuser, 2002.

[12] W. Rudin, *Real and Complex Analysis*, 3rd ed., McGraw–Hill, 1987.

[13] P. Deift and X. Zhou, “A steepest descent method for oscillatory Riemann–Hilbert problems,” *Ann. Math.* **137** (1993): 295–368.