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# Certificates for T488
The quantity is the eigenvalue of the biharmonic operator on the radius-one
disk with both displacement and outward normal derivative zero. It is the
fourth power of a radial characteristic root, not an engineering frequency.
## Root count and indices
For order m, put W_m(x)=J_m(x) I_(m+1)(x)+J_(m+1)(x) I_m(x).
Its kth positive root gamma_(m,k) lies strictly between the kth and (k+1)st
positive J_m zeros. Each interval contains exactly one positive root.
These interlacing inequalities, and strict increase of gamma_(m,k) with m,
are stated in Baricz, Ponnusamy and Singh, arXiv:1507.01104, equation (2.2)
and the discussion preceding equation (1.1). Separation of variables and
the complete eigenfunction basis are given in Mangoubi and Rosenblatt,
arXiv:2411.07564v2, section 1.1.
The J_m indices are certified, rather than assigned by a numerical solver.
The alternating series for J_m(x), after extracting its positive leading
factor, is positive on 0<x<=1: consecutive term magnitudes have ratio at
most 1/[4(m+1)]<=1/4. There are no zeros in that interval. For x>=1,
v=sqrt(x) J_m(x) satisfies
v'' + (1-(m*m-1/4)/(x*x)) v = 0.
The coefficient is at most 5/4 for m=0, and at most 1 for m>=1. If two zeros
were distance L apart, integration by parts and the Dirichlet Poincare
inequality give (pi/L)^2 <= 5/4. Thus successive zeros are more than 2 apart.
A unit mesh cannot hide two zeros. All positive zeros are simple (uniqueness
for this regular second-order ODE). Arb signs at x=1,2,...,18 therefore count
every J_m zero to 18, for each order 0<=m<=9. Rational bisection refines the
sign-changing intervals to width less than 10^-14.
For every relevant consecutive pair of J_m zeros, mpmath supplies a guess
for W_m's root. The guess has no evidentiary role: Arb at 513 bits proves
opposite signs at rational endpoints separated by 2*10^-112. The derivative
W_m'=2 J_m I_m-W_m/x excludes zero throughout the bracket. Each bracket is
strictly within the certified consecutive J_m root brackets. Hence its root
and its radial index are certified.
All roots below 51/4 are included. A root past this cutoff excludes the later
roots at its order. Where the next J_m zero is past the cutoff, interlacing
excludes all remaining roots. The first root of order 9 exceeds 51/4;
monotonicity in order excludes every order m>=9. The 17 distinct roots below
the cutoff are pairwise separated by their certified intervals. The order-0
roots have angular multiplicity one; the positive-order roots have angular
multiplicity two, for cosine and sine. The result is exactly 30 eigenvalues.
No claim about the absence of additional multiplicities at higher energies
is made: that is a separate, nontrivial question.
## Independent radial differential equation
For a mode of angular order m the radial operator is
D_m=d^2/dr^2+(1/r)d/dr-m^2/r^2. Its square gives the coefficient recurrence
c_j = Lambda*c_(j-2)/[16*j*(j-1)*(m+j)*(m+j-1)]
in u(r)=sum c_j r^(m+2j). The two regular solutions start with (c_0,c_1)
equal to (1,0) and (0,1). At r=1 their displacement-and-slope determinant
must vanish. This is implemented directly, with no Bessel-function calls,
in checks.py. The two nonzero controls are determinant(0)=2 and derivative
at zero = -1/[8(m+1)(m+2)(m+3)], for m=0,...,9.
The ball calculation sums through j=180. All coefficients are positive at
positive Lambda. Subsequent term ratios decrease, and both the value and
derivative ratios are certified less than 1/2. Twice the first omitted term
therefore bounds each tail. Arb at 768 bits certifies opposite determinant
signs at both exact endpoints of every displayed decimal interval. All
values are finite, their relative widths are explicitly checked, and a
nonzero sign must have radius less than half its midpoint magnitude.
A separate mpmath implementation solves this determinant at 140 and 180
decimal digits. The results agree to at least 125 relative decimal places
and lie inside every stored decimal interval. This check is repeated after
reading the table back. The first value also agrees with the independent
published approximation 104.363 in Kawohl and Sweers, Remark 4.3 of
https://www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/Grid23.pdf.
## Scope and precision
The first 30 modes form a compact reference for low clamped-disk modes,
ending after an entire eigenspace. Other domains of family #204 are omitted:
this computation does not establish their eigenvalues. Their absence is
stated in the completeness note.
Every value is serialized to 100 significant digits. The returned balls'
minimum measured accuracy is 371 bits (more than 111 decimal digits), and
the independently checked written intervals allow one unit in the last place.
The standalone dry run found 30 entries, maximum 130 characters, 6.3 KB.