Assumptions, results and research status
An explicit finite mathematical example with declared inputs, conventions and scope.
Assumptions
- The operator, state family or representations described on this page are supplied as model inputs.
- Every exported quantity uses the units and conventions printed beside the calculation.
Results
- The default example, tables and equations remain readable without JavaScript.
- With JavaScript, changing valid inputs recomputes the declared model and enables numerical downloads.
Open questions
- What additional source dynamics and physical matching would turn this example into a prediction of a specified physical system?
The declared model
The first example uses a real symmetric matrix H with low entry a, high entry d and off-diagonal b. Its two eigenvalues are explicit. At a threshold z, the high block d − z must be positive for the simple positivity equivalence H ≥ zI if and only if F(z) ≥ 0, where F(z) = a − z − b²/(d − z).
The second example is separate: a normalized vacuum Ω = (√p,√(1−p)), its orthogonal excitation v, and a transfer operator T = |Ω⟩⟨Ω| + t|v⟩⟨v|. For 0 < t < r < 1, the vacuum-adjusted operator G = rI − T + (1 − r)|Ω⟩⟨Ω| equals (r − t)|v⟩⟨v|. A low window P = diag(1,0) sees only part of the vacuum, but its high completion makes the Schur value vanish. [Vacuum obstruction, v2]
Run the calculation
Block reduction and the vacuum window
Inspect an exact two-block eigenvalue problem, then a separate two-state vacuum example. The second calculation shows why removing the vacuum must precede a coercivity estimate.
The first matrix uses a chosen energy unit. The vacuum transfer eigenvalues t and r are dimensionless, with one transfer step as the time unit.
Block eigenvalues 0.8844506 and 4.115549. Threshold comparison: above. Vacuum-window Schur value 0.
Example A: exact block eigenvalues
- Lower eigenvalue λ−
- 0.8844506
- Upper eigenvalue λ+
- 4.115549
- High block d − z
- 4
- Schur value F(z)
- 0.91
The declared matrix has both eigenvalues above z. The positive-high-block Schur criterion applies. The eigenvalue comparison uses the raw margin and floating-point tolerance shown below.
| Quantity | Value |
|---|---|
| det(H − zI) | 3.64 |
| (d − z)F(z) | 3.64 |
| Raw margin λ− − z | 0.8844506 |
| Floating-point threshold tolerance | 2.92e-14 |
| Threshold classification | above |
Example B: the completed vacuum in a low window
- Exact low-window Schur value
- 0
- Correct mixing contribution B²/D
- 0.3
- Mixing from the wrong off-diagonal block
- 0.5333333
- Wrong / correct mixing ratio
- 1.777778
| Quantity | Value |
|---|---|
| Low block A | 0.3 |
| High block D | 0.3 |
| Correct cross block B | -0.3 |
| Full excitation-space rank | 1 |
| Punctured low-window rank | 0 |
| Toy physical gap −log t, step⁻¹ | 1.609438 |
Here Ω = (√p, √(1 − p)) and v = (√(1 − p), −√p). The vacuum correction changes the off-diagonal block as well as the diagonal blocks. Substituting the uncorrected −PTQ block gives an inflated mixing term and breaks the exact cancellation. Although the full two-state system has a positive excitation gap, the punctured rank-one low window contains no vector.
Equations and conventions used
T = |Ω⟩⟨Ω| + t|v⟩⟨v|; G = rI − T + (1 − r)|Ω⟩⟨Ω| = (r − t)|v⟩⟨v|.
For P = diag(1,0), A − B²/D = 0.
Two finite-dimensional demonstrations. The threshold label uses a displayed scale-aware floating-point tolerance; the raw eigenvalues and residuals remain available. A Schur identity is not by itself a uniform lower bound. In the two-state vacuum example the punctured low window has rank zero, so it supplies no nontrivial low-window estimate.
Downloads include the chosen inputs, units and model scope. Values use standard floating-point arithmetic. Read the source and its version notes.
Try these comparisons
- In the first example, increase b while keeping a and d positive. Observe when the smaller eigenvalue becomes negative.
- Set d = z. The tool marks the Schur expression undefined while still calculating the two exact eigenvalues.
- Restore the default vacuum parameters p = 0.5, t = 0.2, r = 0.8. Check A = D = 0.3 and B = −0.3.
- Compare the correct mixing term B²/D with the expression obtained from the wrong off-diagonal block. Only the fully corrected blocks give the required zero Schur value.
The second example requires t < r; invalid entries pause the calculation and leave the last valid results labeled as such. The two panels are not two approximations to the same matrix: one demonstrates generic mixing, the other the exact vacuum obstruction.
Interpret the result
The vacuum correction changes off-diagonal entries whenever the vacuum has components in both windows. Reusing the uncorrected −PTQ block changes the Schur subtraction and can break the cancellation. The full operator identity is the reliable starting point.
Removing PΩ from the low space is called puncturing the window. In this two-state example the original low space is one-dimensional, so puncturing leaves rank zero. The full excitation space still has rank one, with toy physical gap −log t in inverse time-step units. The example demonstrates an obstruction and its correct algebra; a useful positive bound on a nontrivial punctured window needs further estimates. See form Feshbach reduction for those requirements.
Reproduce or extend the calculation
Download JSON to retain the inputs, units, model scope and full computed results. Download CSV for the displayed numerical family or ledger; its opening rows record the input settings and definitions. The tool uses local browser calculations and ordinary floating-point arithmetic.
For an independent check, use the worked example in the exercise collection and the linked source manuscript. A useful extension states which input or assumption changes and which comparison would test the change. See the research collaboration page for focused starting questions.
Continue reading
Sources for this guide
Source manuscripts, version notes, and standard references are listed in the reference library.