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For equivalent minima with nearest-neighbour hopping magnitude $\Delta$, paths with $r$ right steps and $s$ left steps have $r-s=n-m$. Summing $(\Delta \tau /\hslash {)}^{r+s}/(r!s!)$ and applying the Fourier representation of a Kronecker delta gives the displayed modified Bessel function kernel in a normalized localized-site basis. Its Fourier exponent gives the tight-binding model energy $E(\theta )=E_{\mathrm{well}}-2\Delta \cos \theta$. Position-endpoint kernels can have an additional common local-wavefunction prefactor.

The differential equation defining modified Bessel functions of order $\nu$. For real fixed $\nu$ and positive $z$, its independent standard solutions are $I_{\nu }(z)$ and $K_{\nu }(z)$.

For the new periodic potential, let $\kappa$ be the single-neighbour hopping rate and $\Delta ϵ=\hslash \kappa$. It is determined by the barrier between adjacent minima; an unspecified periodic potential does not determine it from the preceding quartic potential's numerical action. The minimum spacing is now $a$.

A path with $r$ right hops and $s$ left hops has $r-s=n-m$. Summing the ordered-center weights and their direction choices givesInsert the Fourier representation of a Kronecker delta. The two exponential series sum independently, yielding the dilute-hopping lattice propagatorThe two Fourier-sign choices are equivalent by $\theta \mapsto -\theta$. Equivalently the integral is the modified Bessel function $I_{n-m}(2\kappa \tau )$.

The original PDF prints $e^{+\omega \tau /2}$ here. Its sign is inconsistent with both the previous double-well result and the requested positive on-site oscillator energy. The correct factor is $e^{-\omega \tau /2}$: a positive oscillator ground-state energy must decay under $e^{-H\tau /\hslash }$. The other factors and their derivation remain as displayed. The energy $\Delta ϵ$ is the positive nearest-neighbour quantum tunnelling matrix-element magnitude, exponentially small relative to the local well scale in the semiclassical regime.

Put $\varphi =R(r)\Theta (\theta )Z(z)$ in the Laplace equation in cylindrical coordinates. Separation of variables first gives ${\Theta }^{\prime \prime }+m^2\Theta =0$; $2\pi$-periodicity forces $m=0,1,2,\dots$, with angular factors $\cos m\theta ,\sin m\theta$ (only the constant for $m=0$). Taking $Z^{\prime \prime }={\kappa }^{2}Z$ givesFor $\kappa >0$ the separated modes are thereforeHere $J_m$ and $Y_m$ are the Bessel function of the first kind and the Bessel function of the second kind. For the opposite sign $Z^{\prime \prime }=-{\kappa }^{2}Z$, the modes instead involve $I_m(\kappa r),K_m(\kappa r)$, the modified Bessel functions, paired with $\cos \kappa z,\sin \kappa z$. At zero separation constant use $Z=C+Dz$ and $R=A{r}^m+B{r}^{-m}$ for $m\ge 1$, or $R=A+B\log r$ for $m=0$. Superpositions over the integer angular orders and appropriate sums or integrals over the separation parameter give the general separated expansion; boundary conditions select its spectrum and coefficients. Regularity at the axis excludes $Y_m,K_m,r^{-m},\log r$ terms.

For the specified decaying side data, bounded separated modes use $e^{-\kappa z}{J}_m(\kappa r)$. A particular solution isAll three denominators are nonzero. Each term satisfies the radial Bessel differential equation, is regular at the axis and bounded for $0\le r\le 1,z\ge 0$, and substitution at $r=1$ gives the stated side values. The modes behave as $r^m$ at the axis, so their apparent angular dependence there causes no singularity.

There is an actual nonuniqueness in the printed problem: no values are specified at $z=0$. If $j_{0,1}$ is the first positive zero of $J_0$, then for every real $a$,is another bounded solution with the same side values. It remains smooth at the axis and even decays as $z\to \infty$. This explicitly proves that side boundary data do not determine a bounded harmonic function in a half-cylinder. Thus the boxed expression supplies a bounded solution; the phrase “the bounded solution” is not justified without an additional base boundary condition. The PDF has $z\ge 0$, whereas the TeX transcription displays $z>0$.

Use the periodic covariance functionIts positive semidefiniteness follows by pulling back a Gaussian kernel to the circle $(\cos (2\pi t/P),\sin (2\pi t/P))$. A zero Gaussian process mean is an ensemble statement, whereas an exactly zero long-term average for every periodic light curve is stronger. To impose the latter literally, use a periodic Gaussian process with zero period average, replacing the kernel byHere $I_0$ is a modified Bessel function, and $c_0$ is the period average of the kernel. Subtracting each process's random period average produces this covariance, so it remains positive semidefinite. We use $k_0$ to enforce the stated mean subtraction; using $k_P$ gives the usual ensemble-zero-mean formulation and the same likelihood derivation.

Let $K_{ij}=k_0(t_i,t_j)$ and $R=\mathrm{diag}({\sigma }_1^2,\dots ,{\sigma }_N^2)$. The prior latent vector is $f\sim N(0,K)$ and the independent Gaussian noise is $\epsilon \sim N(0,R)$, so $y=f+\epsilon \sim N(0,V)$ with $V=K+R$. The Gaussian-process marginal likelihood isPositive measurement variances make $V$ positive definite even if $K$ is singular. Maximize $\ell (\mathcal{H})=-\frac{1}{2}{y}^T{V}^{-1}y-\frac{1}{2}\log |V|-\frac{N}{2}\log (2\pi )$ over all three hyperparameters, using a broad search in $P$ to find competing aliases, followed by joint optimization. Cholesky decomposition evaluates the likelihood without explicitly forming an inverse.

For a well-resolved interior maximum, let $J=-{\nabla }^2\ell (\hat{\mathcal{H}})$ be the Observed Fisher information in the coordinates $(A,\ell ,P)$. Then $\boxed{\hat{P}=(\hat{\mathcal{H}}{)}_P,{\sigma }_P\approx \sqrt{(J^{-1}{)}_{PP}}}$. Equivalently, the profile-likelihood curvature in $P$ gives the same local nuisance-adjusted variance; keeping the other hyperparameters fixed would generally underestimate it. A profile interval with ${\ell }_{\mathrm{prof}}(\hat{P})-{\ell }_{\mathrm{prof}}(P)=1/2$ is a local $1\sigma$ approximation. If the period likelihood is multimodal or poorly constrained, fit the complete posterior with proper hyperparameter priors and report its modes and marginal credible intervals rather than a single Hessian error bar. The sampling pattern can create period aliases that a single local search misses.

The intended integral representation of the modified Bessel function isDifferentiation under the integral sign givesSinceintegration by parts and the vanishing endpoint term yield $x{I}_0^{\prime \prime }+I_0^{\prime }-x{I}_0=0$.

For large positive $x$, Laplace's method localizes the integral near the endpoint $\theta =0$. Put $\theta =u/\sqrt{x}$ and useExtending the exponentially small tail to infinity and evaluating the resulting Gaussian integrals givesand therefore

Now write $y=x^{-1/2}w$. Direct differentiation givesso substitution into the ordinary differential equation givesFor the Liouville–Green approximation, put $Q(x)=1-1/(4x^2)$. The two approximate solutions areso at leading order $y_{\pm }(x)\sim x^{-1/2}{e}^{\pm x}$. The growing solution agrees with the preceding asymptotic expansion, including its leading normalization $I_0(x)\sim e^x/\sqrt{2\pi x}$.

For an axisymmetric surface $r=a(z,t)$, twice the mean curvature with the outward-normal convention isWriting $a=a_0+\eta$ and retaining linear terms gives

At the unperturbed boundary $r=a_0$, the linearized kinematic boundary condition, tangential-stress condition, and Young–Laplace equation areFor a normal mode $e^{ikz+st}$ these become $u(a_0)=s\eta$, ${\sigma }_{rz}(a_0)=0$, and

The Papkovich–Neuber representation of body-force-free Stokes flow iswhere $\mathbf{\Phi }$ and $\chi$ are harmonic.

Let $x=kr$ and $E=e^{ikz+st}$. Substitution of the given radial vector potential and scalar potential givesUsing the modified Bessel function identity $(x{I}_1{)}^{\prime }=x{I}_0$, the tangential stress simplifies toThe stress-free condition at $r=a$ therefore yields

The Laplace integral method for a differential equation seeksDifferentiating under the integral and substituting intogivesSince $z{e}^{zt}={\partial }_t{e}^{zt}$, integration by parts yieldsIt is therefore enough to choose $f$ and $C$ so thatand the boundary term vanishes. Solving the first-order equation on $-1<t<1$ givesTake $C$ to be the interval $[-1,1]$. Although $f$ has integrable endpoint singularities,vanishes at both endpoints, so the boundary term is zero. ThusAt $z=0$, substituting $t=\sin \theta$ givesThe normalization $y(0)=1$ therefore sets $K=1/\pi$. By the defining regularity and normalization of the modified Bessel function, this proves the real integral representation of the modified Bessel function I0:

A zero-mean Gaussian process need not have zero average in each realization. For the periodic covariance function, the constant Fourier component iswhere $I_0$ is a modified Bessel function. Replacing $f(t)$ by $f(t)-P^{-1}{\int }_0^{P}f(u)du$ gives a Gaussian process with kernel $k_P(t,t^{\prime })-c_0$. This is a positive-semidefinite kernel because it is a covariance after a linear transformation, and each sample has zero period average. The real integral representation of the modified Bessel function I0 yields the displayed constant.

If $Z_1,Z_2$ are independent $N(0,1)$ random variables, their product has characteristic function $(1+t^2{)}^{-1/2}$ and density $K_0(|x|)/\pi$, where $K_0$ is a modified Bessel function. A scale multiple $a{Z}_1{Z}_2$ has the corresponding scaled distribution.