p-adic spectral zeta functions via the inverse Stieltjes transform

Introduction to the show: ident: Mathematical Physics. The day in mathematical physics: rigorous models of physical systems.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "p-adic spectral zeta functions via the inverse Stieltjes transform".

Mira: Here's the gist of the paper. Spectral zeta functions provide a standard tool for regularizing determinants of differential operators in quantum physics.

Kai: The paper extends the framework for spectral zeta functions from discrete spectra, where interpolation over the ring of p-adic integers is required, to continuous spectra by constructing them directly from resolvent data using Vishik’s inverse Stieltjes transform.

Mira: This approach allows for the study of arbitrary compact spectral sets and generalized distributions that do not require a prescribed interpolation scheme.

Kai: The framework establishes a symmetry between bosonic and fermionic cases, and its application is demonstrated through the position operator in p-adic quantum mechanics on an arbitrary compact subset of Cp.

Title and authors: Kai: We started by looking at the title, "p-adic spectral zeta functions via the inverse Stieltjes transform." It sounds like a very specific mathematical tool is being introduced for spectral analysis in this context.

Mira: Yeah, it suggests they are taking something established for discrete spectra and generalizing it to continuous ones by using a specific functional method, the inverse Stieltjes transform.

Lev: From an implementation viewpoint, that means we have to figure out how to actually compute that transform when the operator T is defined on a general compact set sigma inside Cp.

Kai: The paper explains that they start from the resolvent of a bounded operator and use Vishik’s inverse Stieltjes transform to recover a generalized distribution, called the p-adic spectral distribution muT.

Mira: So they aren't just guessing the measure; they are deriving it directly from the data provided by how that operator acts on vectors.

Lev: That sounds like a big computational step because we need a stable way to compute that resolvent and then apply an inverse transform to get the distribution muT.

Kai: They define this distribution as SµT(z) equals FT(z), where FT is a scalarization of the resolvent that belongs to H0(Cp minus sigma).

Mira: That means the scalarization step is essential because it needs to land in that specific space H0, which is what gives them the necessary analyticity properties.

Lev: If we can nail that scalarization, then we can get a well-defined spectral distribution muT, and that’s where the theory starts getting practical for physical systems.

Kai: Then they define the zeta functions as ζµp(s, λ) and ζνpE(s, λ) using this derived distribution muT.

Mira: And this is where the real structure comes in; they show that these zeta functions have specific analytic properties and special values we discussed earlier.

Lev: The analytic properties are key for us because they tell us what kind of mathematical objects we can expect to extract from these spectral data.

Kai: They establish analyticity for the bosonic zeta function in s on Zp minus one and local analyticity for lambda on an open set Omega <ref:2609.33660#pg1>.

Mira: And then they provide those special values at non-positive integers, which are important benchmarks for any physical theory we try to build.

Lev: If we can calculate those special values reliably, it means the framework has some predictive power even outside the continuous regime.

Kai: They also establish analogous analyticity properties for the fermionic zeta function with a simpler proof because of the bounded nature of the underlying measure.

Mira: That's good news for us because it suggests that for fermionic systems, we don't need as many tricky assumptions about boundedness in our initial construction.

Lev: So they’ve managed to simplify one side of the symmetry, which is helpful when designing error correction protocols that involve fermions.

The paper's summary: Kai: Now let’s summarize what this paper actually does, beyond just naming the tools. It extends the framework for spectral zeta functions from discrete spectra where you needed interpolation to continuous spectra where you use the inverse Stieltjes transform.

Mira: So instead of needing a specific interpolation function to handle every possible spectrum, they build it directly from the resolvent data itself to define a generalized distribution muT.

Lev: In practical terms, this means we are bypassing the need for that prescribed interpolation scheme and going straight to using the operator's spectral properties.

Kai: This approach allows them to study arbitrary compact spectral sets and generalized distributions that don't require a prescribed interpolation scheme at all.

Mira: It establishes a symmetry between bosonic and fermionic cases, which is an interesting feature for this framework, but we need to see if that symmetry holds up when we move beyond the simplest models.

Lev: For me, it means the theory is more flexible because it can handle situations where the spectral distribution isn't induced by some simple Haar distribution on Zp.

Kai: The application is shown through a concrete example: considering the position operator in p-adic quantum mechanics on an arbitrary compact subset of Cp.

Mira: This example is really illustrative because they show that the p-adic spectral distribution of this position operator is exactly the measure mu they chose, which is a bounded measure on that set.

Lev: That’s a big confirmation because it grounds the abstract construction in something we can actually calculate—a concrete measure mu.

Kai: They then use this distribution mu to compute all the zeta functions and functional determinants, leading to those Stirling-type expansions.

Mira: So the whole point is that you get a regularization of spectral data using a method tied directly to the operator's resolvent.

Lev: It’s about making sure we have a consistent way to calculate those determinants, which is vital for things like Casimir energies in p-adic quantum field theory.

The paper's improvements: Kai: The authors suggest several improvements to the framework, mostly focusing on how the construction can be made more general.

Mira: They are moving from a discrete setup where they had to specify an interpolation function to a continuous setup where they use the inverse Stieltjes transform instead.

Lev: From an engineering perspective, that means we’re replacing one type of required input with another, which is just trading one constraint for another.

Kai: They also show that the framework is embedded into the discrete case when a locally analytic interpolation function exists, which validates their approach in the discrete setting.

Mira: So they can check for consistency between their continuous and discrete and continuous constructions, which strengthens the whole theory.

Lev: That consistency check is what we need before we trust it for any real hardware application—it’s about validating that the tools work together.

Kai: They also show how to recover the bosonic and fermionic zeta functions from the interpolation-based framework when you have a locally analytic interpolation function around a point.

Mira: So they can check if their continuous construction matches what we get from older methods in those specific, well-behaved local regions.

Lev: That validation is important because it lets us know where our model is reliable and where we need to focus our testing efforts.

Kai: The main improvement they highlight is the direct association of the spectral distribution mu with the resolvent R(z) via the inverse Stieltjes transform, which handles continuous spectra more generally.

Mira: That means that even if you don't have a simple interpolation function, you can still get a distribution.

Lev: That’s powerful because it means we don't need to assume the spectral distribution is generated by some specific structure imposed by Zp.

Conclusion: Kai: So to wrap up, this paper introduces "p-adic spectral zeta functions via the inverse Stieltjes transform" as a way to regularize determinants for continuous spectra using resolvent data.

Lev: The main practical takeaway for us in error correction research is having a direct link between the resolvent, the spectral distribution, and those zeta functions, which lets us compute those functional determinants through Stirling expansions.

Kai: This paper shows how to handle genuinely continuous spectra in p-adic quantum mechanics more generally by building the spectral distribution from R(z) = (zI - T)−one via Vishik’s inverse Stieltjes transform.

Kai: We’re ready for the next paper now, moving on from spectral zeta functions into how this affects operator theory itself.

Lev: I want to talk about how that Stirling-type expansion works in practice when we try to simulate these things on real quantum hardware where we have finite precision and noise limits.