= Solution
Experimental tests of <BCS theory> compare several properties of the same superconducting state. In the conventional weak-coupling setting, phonon-mediated attraction pairs opposite momenta and spins, and the <BCS quasiparticle spectrum> is $E_k=\sqrt{\xi_k^2+\Delta^2(T)}$. Agreement with a single <energy> scale and its <temperature> dependence is more discriminating than observing zero resistance alone.
The <superconducting isotope effect> tests the <lattice>'s role. Replacing one <isotope> by another changes ionic masses while leaving the chemical valence nearly unchanged. <Phonon> frequencies scale approximately as $M^{-1/2}$, so a weak-coupling <phonon> mechanism predicts $T_c\propto M^{-\alpha}$ with $\alpha\simeq1/2$ when other parameters stay fixed. The shift was measured in mercury before the microscopic pairing theory was developed.<Isotope Effect in the Superconductivity of Mercury by Emanuel Maxwell>[1950 primary report, transition-temperature comparison of mercury <isotopes>]{ref} Deviations from $1/2$ can result from Coulomb, anharmonic and strong-coupling effects; the <isotope> observation supports lattice-mediated pairing without uniquely proving every assumption of the simplest model.
Thermodynamic measurements test the gapped excitations and condensation <energy>. An isotropic fully gapped <superconductor> has exponentially small low-temperature electronic <heat capacity>, rather than the normal linear law. At the transition the <weak-coupling BCS thermodynamic ratios> are
$$
\boxed{\frac{2\Delta(0)}{k_BT_c}=2\pi e^{-\gamma_E}\simeq3.528,
\qquad \frac{C_s(T_c^-)-C_n(T_c)}{C_n(T_c)}
=\frac{12}{7\zeta(3)}\simeq1.426.}
$$
Here $C_n$ means the electronic contribution after subtracting the <lattice> background. The gap closes at $T_c$, and the same framework predicts the <temperature> dependence of penetration depth and thermodynamic critical field. These quantities were quantitative comparisons in the original theory.<Theory of Superconductivity by John Bardeen and others>[Abstract and thermodynamic discussion: <energy> gap, specific heat and penetration depth]{ref}
<Electron> tunnelling measures the excitation spectrum more directly. A normal-metal–insulator–<superconductor> junction has, at sufficiently low <temperature> and nearly constant tunnelling matrix element, $dI/dV$ proportional to the superconducting <density of states> at $E=eV$:
$$
\frac{N_s(E)}{N_n(0)}=
\operatorname{Re}\frac{|E|}{\sqrt{E^2-\Delta^2}}.
$$
The suppression below the gap and enhancement at its edges test the <BCS electronic density of states>. Giaever's thin oxide-barrier junction measurements established such energy-gap structure.<Energy Gap in Superconductors Measured by Electron Tunneling by Ivar Giaever>[Physical Review Letters 5, 147–148; tunnel-current and differential-conductance measurements]{ref} For a junction between two <superconductors> the threshold involves the sum of their gaps, so it must not be read as a normal-metal single-gap spectrum without accounting for the two electrodes.
\Image[/past-exam-of-the-advanced-quantum-condensed-matter-physics-course-of-the-university-of-cambridge/2015/paper-1-bcs-density-of-states.png]
{title=Ideal BCS density of states illustrating gap-edge spectroscopy; model prediction, not experimental measurements}
Nuclear spin-lattice relaxation supplies a phase-coherence-sensitive test. In the conventional s-wave case, the <Hebel-Slichter coherence peak> just below $T_c$ is followed by a falling rate at lower <temperature>. <Density of states> alone is not enough: the <BCS coherence factors> enter the transition matrix elements. The contrast between nuclear relaxation and ultrasonic attenuation was explained using the paired state, unlike a simple two-fluid population model.<Nuclear Spin Relaxation in Normal and Superconducting Aluminum by L. C. Hebel and Charles Pence Slichter>[Abstract: field-cycling method and comparison of nuclear relaxation with ultrasonic absorption]{ref}
Finally, <magnetic flux> quantization in units $h/(2e)$ and <Josephson effects> support a coherent charge-$2e$ condensate. Those observations establish pairing and phase coherence more broadly; their existence alone does not specify a <phonon> mechanism or an isotropic weak-coupling gap. \b[The converging <isotope>, spectroscopic, thermodynamic and coherence-factor tests support conventional BCS <superconductivity>, with explicit limits to its simplest universal ratios.]
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