The curious case of operators with spectral density increasing as $Ω(E)\sim e^{\,\mathrm{Const.}\, E^2}$

Motivated by modeling black holes as quantum objects, this paper demonstrates that while operators with a spectral density growing as $e^{E^2}$ exist, their associated barely localized wave functions create a fundamental tension with the compact nature of black holes.

The Gist

The Big Picture: What is a Black Hole made of?

Imagine a black hole not as a cosmic vacuum cleaner, but as a giant, invisible quantum machine. Just like a piano has keys that produce specific notes, a black hole has "energy levels" or "notes" it can play.

Physicists have known for decades that black holes have a massive amount of entropy (a measure of how many different ways the machine can be arranged internally). For a black hole the size of our Sun, the number of possible internal arrangements is roughly $e^{10^{77}}$. That is a number so huge it makes the number of atoms in the universe look like a grain of sand.

The big question this paper asks is: What kind of "piano" (mathematical operator) could possibly have that many notes?

The Experiment: A Gas of Ghosts

To find the answer, the authors (Erik Aurell and Satya Majumdar) set up a thought experiment. They imagined a box filled with non-interacting bosons.

• The Analogy: Think of these bosons as "ghosts" floating in a room. They don't bump into each other or push each other away; they just occupy energy levels.
• The Goal: They wanted to see what kind of "room" (potential energy landscape) these ghosts would need to live in to create the massive number of arrangements ($e^{E^2}$) that a black hole requires.

The Discovery: The "High-Energy Condensation"

In most normal physics systems (like a standard gas in a box), the number of ways to arrange particles grows slowly. It's like counting how many ways you can stack Lego bricks; the number goes up, but not explosively.

However, the authors found that to get the black hole's explosive growth in possibilities, the ghosts need to do something weird called High-Energy Condensation.

• The Analogy: Imagine a concert hall. Usually, the audience is spread out. But in this "High-Energy Condensation" scenario, almost the entire audience suddenly rushes to sit in the very last row (the highest energy level).
• The Result: Because almost all the energy is concentrated in just one or two particles sitting at the very top, the math explodes. The number of possible arrangements skyrockets to match the black hole's entropy.

The Catch: The "Shallow Trap"

To make the ghosts rush to the very top, the "room" they are in must have a very specific shape. The authors calculated that the walls of this room must rise incredibly slowly.

• The Analogy: Imagine a bowl.
  • A normal bowl (like a harmonic oscillator) has steep sides. If you roll a marble in, it stays near the bottom.
  • The black hole bowl described in this paper is almost flat. It rises so slowly that it looks like a flat floor for a long time, only curving up very gently at the very edges.
  • Mathematically, this shape grows like the square root of the logarithm ($\sqrt{\ln r}$). That is an incredibly slow growth rate. It's like a hill that gets higher, but you have to walk for miles before you notice the slope.

The Problem: The "Fuzzy" Black Hole

Here is where the story hits a snag. The authors realized that because the "walls" of this room are so shallow and rise so slowly, the ghosts (the quantum states) don't stay neatly packed in the center.

• The Analogy: Because the trap is so weak, the ghosts spread out over a massive area. They are "barely localized."
• The Conflict: A real black hole is a compact object. It is incredibly dense and small (defined by its Schwarzschild radius). But the mathematical model required to get the right number of energy states creates a system that is spread out over a huge distance.

It's like trying to explain why a marble is heavy by saying, "Well, imagine a cloud of gas the size of a city that weighs as much as a marble." The math works for the weight, but it fails the "compactness" test.

The Conclusion: A Mathematical Curiosity

The paper concludes that while it is mathematically possible to build an operator (a quantum machine) that has the exact energy spectrum of a black hole, the physical object it describes is unrealistic.

1. It works: You can build a quantum system where the energy states explode in number ($e^{E^2}$).
2. The mechanism: It requires a "high-energy condensation" where particles pile up at the very top of a very shallow energy hill.
3. The failure: The resulting quantum states are too "fuzzy" and spread out. They don't fit inside the tiny, compact space of a real black hole.

The Takeaway:
If black holes are truly quantum objects with this specific spectrum, they must be made of something much stranger than simple non-interacting particles. Either the particles interact in a way that squeezes them back into a small space, or the "inside" of a black hole is a bizarre, vast, and chaotic place (a "bag of gold") that defies our normal intuition of space.

The authors have essentially found a mathematical key that fits the lock of black hole entropy, but when they try to turn it, the door opens to a room that is far too big to be a black hole.

Technical Summary

1. Problem Statement

The paper addresses a fundamental question in quantum gravity: What type of quantum mechanical operators possess a spectral density (number of states) that grows as $\Omega(E) \sim \exp(C E^2)$?

This specific growth rate is motivated by the Bekenstein-Hawking entropy formula for black holes, where entropy $S \sim A \sim M^2$ (since Area $A \propto M^2$). If a black hole is treated as a quantum object with mass $M \approx E/c^2$, its entropy corresponds to the logarithm of the density of states: $S = \ln \Omega(E) \propto E^2$. While the Bekenstein-Mukhanov model proposes that black hole masses are quantized with exponential degeneracy ($D_n \sim e^{\alpha n}$), it remains an open mathematical problem whether such spectra can arise from reasonable, well-defined quantum mechanical operators (Hamiltonians) in standard models.

2. Methodology

The authors employ a statistical mechanics approach using a model of non-interacting bosons to reverse-engineer the required single-particle spectrum.

• The Model: They consider a system of $N$ non-interacting bosons. The total energy $E$ is the sum of single-particle energies $\epsilon_i$ weighted by occupation numbers $n_i$.
• Partition Function: They analyze the microcanonical partition function $\Omega(E)$, which counts the number of ways to distribute energy $E$ among the single-particle levels.
• Laplace Transform & Saddle Point: They compute the canonical partition function $Z(\beta)$ and invert it via a Laplace transform to find $\Omega(E)$. Using the saddle-point approximation (valid for large $E$), they relate the total energy $E$ to the inverse temperature $\beta^*$ and the single-particle density of states $\rho(\epsilon)$.
• Asymptotic Analysis: The authors analyze two regimes:
  1. Algebraic Growth: Where $\rho(\epsilon)$ grows as a power law ($\epsilon^\alpha$).
  2. Super-exponential Growth: Where $\rho(\epsilon)$ grows faster than an exponential, specifically targeting $\rho(\epsilon) \sim \exp(C \epsilon^2)$.

3. Key Contributions and Results

A. Impossibility of Algebraic Potentials

The authors first demonstrate that standard confining potentials (e.g., harmonic oscillators $V \sim r^2$ or particle in a box $V=0$) yield a single-particle density of states $\rho(\epsilon)$ that grows algebraically ($\rho \sim \epsilon^\alpha$).

• Result: For algebraic $\rho(\epsilon)$, the many-body density of states grows as $\Omega(E) \sim \exp(E^{\frac{\alpha+1}{\alpha+2}})$.
• Conclusion: Since the exponent $\frac{\alpha+1}{\alpha+2} < 1$, algebraic potentials cannot produce the required super-exponential growth $\Omega(E) \sim \exp(E^2)$.

B. The "High Energy Condensation" Mechanism

To achieve $\Omega(E) \sim \exp(E^2)$, the authors show that the single-particle density of states must itself grow super-exponentially: $\rho(\epsilon) \sim \exp(C \epsilon^2)$.

• Mechanism: This leads to a phenomenon termed "High Energy Condensation." Unlike standard Bose-Einstein condensation (where particles accumulate in the lowest energy state), here, one or a few particles occupy a very high energy level, carrying a macroscopic fraction of the total energy $E$.
• Implication: The integral for the total energy is dominated by the contribution near the upper cutoff $E$, leading to the relation $\Omega(E) \approx \rho(E)$.

C. Derivation of the Required Potential

The core mathematical contribution is identifying the specific confining potential $V(r)$ that yields $\rho(\epsilon) \sim \exp(C \epsilon^2)$.

• Using the semi-classical Bohr-Sommerfeld quantization rule, they relate the cumulative density of states $N(\epsilon)$ to the integral of the momentum:
  $$\int_0^\infty \sqrt{\epsilon - V(r)} \, dr \approx [N(\epsilon)]^{1/d}$$
• Setting $N(\epsilon) \sim \exp(C \epsilon^2)$, they derive the asymptotic form of the potential required at large distances ($r \to \infty$):
  $$V(r) \sim \sqrt{\frac{d}{C}} \sqrt{\ln r}$$
• Result: The potential must grow extremely slowly, proportional to the square root of the logarithm of the distance ($\sqrt{\ln r}$).

D. Analysis of Eigenfunctions (Localization)

The authors investigate whether such a shallow potential ($\sqrt{\ln r}$) actually supports discrete, localized eigenstates.

• WKB Approximation: They solve the Schrödinger equation for $V(x) = \alpha \sqrt{\ln |x|}$ in 1D.
• Finding: The wavefunctions decay as $\psi(x) \propto \exp\left[ - \text{const} \cdot x (\ln x)^{1/4} \right]$.
• Conclusion: The wavefunctions are indeed square-integrable and the potential is "confining." However, the decay is extremely slow. The states are "barely localized," meaning the wavefunction extends over a vast spatial region, far exceeding the scale of the energy level itself.

4. Significance and Discussion

• Tension with Black Hole Compactness: The paper highlights a critical physical tension. While such operators mathematically exist, the resulting quantum states are extremely delocalized (spatially extended). This contradicts the physical intuition of a black hole as a compact object with a well-defined Schwarzschild radius. If the "internal states" of a black hole correspond to these operators, the black hole's wavefunction would extend far beyond its event horizon.
• Role of Interactions: The authors suggest that to resolve this, one must move beyond the non-interacting boson model. Interactions (as proposed in the "quantum n-portrait" by Dvali and Gomez) might be necessary to simultaneously generate the correct spectral degeneracy and shrink the spatial extent of the states.
• Internal Geometry: The paper discusses alternative possibilities, such as the "bag-of-gold" scenario where the internal volume of a black hole is much larger than the exterior volume, or the existence of infinite-dimensional internal spaces (e.g., tree-like structures in disordered systems). However, no known mechanism currently explains how to achieve the specific $\exp(E^2)$ scaling in these complex geometries.

5. Conclusion

The paper concludes that while operators with a spectral density $\Omega(E) \sim \exp(E^2)$ do exist mathematically, they require extremely shallow confining potentials ($\sim \sqrt{\ln r}$) that produce highly delocalized states. Consequently, simple non-interacting quantum mechanical models are likely insufficient to describe black holes as compact quantum objects. The existence of such spectra implies that a realistic quantum black hole model must involve either strong interactions to localize these states or a radically different internal spacetime geometry.