Elliptic Boundary Problems and Heat Kernels

Ellipticity, boundary conditions, and the bottom of the spectrum answer different parts of an inverse problem. Ellipticity controls high-frequency spatial behavior. A boundary condition turns the differential expression into a particular operator. The resulting spectrum decides whether that operator is invertible and whether its heat evolution decays at large time. The heat kernel then packages the evolution, the inverse, and the local short-time geometry in one object.

This page develops that chain for scalar Laplace-type operators on smooth compact Riemannian spaces, with free Euclidean space as the QFT-facing example.

Required background. Symbols, Characteristics, and PDE Type supplies principal- and boundary-symbol reasoning; Fundamental Solutions and Green Operators supplies typed inverses, distribution kernels, and zero-mode logic.

Helpful background. Sturm–Liouville Problems and Eigenfunction Expansions supports the interval mode expansions; Spectra, Resolvents, Spectral Measures, and Functional Calculus supports the functional-calculus, semigroup, and proper-time arguments.

The differential expression is not yet the boundary problem

Let $(M,g)$ be a smooth compact connected $d$-dimensional Riemannian manifold with smooth boundary, and let $\mathrm d\mu_g$ be its volume measure. A clean model is

$$L=-\Delta_g+V,$$

where $\Delta_g=\operatorname{div}\operatorname{grad}$ and $V$ is a smooth real potential. Thus $-\Delta_g$ is nonnegative before lower-order and boundary contributions are considered. On scalar functions, the principal symbol is

$$\sigma_2(L)(x,\xi) = g^{ij}(x)\xi_i\xi_j.$$

It is positive for every nonzero covector $\xi$, so $L$ is elliptic. This statement concerns the interior, highest-derivative part. It does not choose the behavior of a field at $\partial M$.

Three standard homogeneous choices are

Name	Boundary operator $B$	Classical condition
Dirichlet	trace	$u\!\restriction_{\partial M}=0$
Neumann	normal derivative	$(\partial_nu)\!\restriction_{\partial M}=0$
Robin	normal derivative plus multiplication	$((\partial_n+\beta)u)\!\restriction_{\partial M}=0$

Here $n$ is the outward unit normal and $\beta$ is a real smooth function on the boundary. In a Sobolev formulation these equations are trace conditions on the operator domain. For example, on a smooth domain the Dirichlet realization has the schematic domain

$$\mathcal D(L_D) = H^2(M)\cap H_0^1(M),$$

whereas the Robin realization restricts $H^2(M)$ by $(\partial_n+\beta)u=0$.

An elliptic boundary problem is therefore the pair consisting of the elliptic differential expression and compatible boundary data, interpreted as an operator with a declared domain. The stationary equation $L_Bu=f$ is elliptic. Once the same realization generates $(\partial_t+L_B)u=0$, that evolution problem is parabolic; heat time is not a Lorentzian causal time.

Boundary data must also be compatible with the elliptic principal part. The relevant high-frequency check freezes the leading coefficients at a boundary point, Fourier transforms in tangential directions, and asks whether the boundary symbol uniquely controls the normal modes that decay into the interior. This is the complementing condition for the ordinary elliptic boundary problem. Heat-kernel theory uses strong ellipticity, a parameter-dependent strengthening that includes an allowed complex spectral parameter in the frozen normal problem. Standard scalar Dirichlet and Robin conditions, including Neumann as $\beta=0$, pass both tests for a Laplace-type operator. An arbitrary condition does not. Interior ellipticity alone therefore does not guarantee a well-behaved boundary spectrum or heat kernel. Grubb 1996, Chapter I, §§1.4–1.5, PDF separates elliptic realizations satisfying the Shapiro–Lopatinski condition from the parameter-ellipticity needed for resolvent and parabolic analysis; Vassilevich 2003, §5.4 gives the Laplace-type heat-kernel version of the latter condition.

Green’s identity fixes symmetry and the quadratic form

Use the site’s convention that the first slot of the inner product is conjugate-linear:

$$\langle u|v\rangle = \int_M \overline{u}\,v\,\mathrm d\mu_g.$$

For smooth $u$ and $v$, integration by parts gives

$$\begin{aligned} \langle u|Lv\rangle-\langle Lu|v\rangle = \int_{\partial M} \left( v\,\partial_n\overline u - \overline u\,\partial_nv \right) \mathrm d\mu_{\partial}. \end{aligned}$$

The real potential cancels. Dirichlet data kill both traces in the boundary form. Neumann data kill both normal derivatives. If $u$ and $v$ obey the same Robin condition with real $\beta$, then

$$\partial_nu=-\beta u, \qquad \partial_nv=-\beta v,$$

and the two boundary terms cancel again. This proves formal symmetry on those domains. Self-adjointness additionally requires equality between the operator domain and the adjoint domain; cancelling the displayed form is a necessary calculation, not by itself the full domain theorem.

The integration-by-parts identity and its boundary-value consequences are reviewed in Hunter 2014, §§2.5 and 4.8–4.10, PDF.

The same integration by parts gives the energy identity

$$\langle u|Lu\rangle = \int_M \left( |\nabla u|^2+V|u|^2 \right) \mathrm d\mu_g - \int_{\partial M} \overline u\,\partial_nu\, \mathrm d\mu_{\partial}.$$

For Robin data this becomes

$$\langle u|L_Ru\rangle = \int_M \left( |\nabla u|^2+V|u|^2 \right) \mathrm d\mu_g + \int_{\partial M} \beta|u|^2\, \mathrm d\mu_{\partial}.$$

Consequently $V\geq0$ and $\beta\geq0$ are simple sufficient conditions for nonnegativity. They are not necessary conditions: a negative part can sometimes be controlled by an inequality. The point is that positivity is a property of the realization, including its boundary term, rather than of the interior symbol alone.

A robust construction starts from the sesquilinear form

$$\begin{aligned} q_R[u,v] =& \int_M \left( \nabla\overline u\mathbin{\cdot}\nabla v + V\overline u v \right) \mathrm d\mu_g \\ &+ \int_{\partial M} \beta\overline u v\, \mathrm d\mu_{\partial}. \end{aligned}$$

For Dirichlet data its form domain is $H_0^1(M)$ and the boundary integral is absent; for Neumann or Robin data it is $H^1(M)$. Under the smoothness and semiboundedness assumptions above, the representation theorem for closed forms produces a self-adjoint lower-bounded operator. The displayed $H^2$ domains then describe that operator when elliptic boundary regularity applies. This order of reasoning matters: a real Robin coefficient can give a self-adjoint operator even when it has negative eigenvalues, whereas $\beta\geq0$ is the simple condition used here to guarantee a nonnegative form. The form construction and the Neumann and Robin realizations on smooth bounded Euclidean domains are developed in Arendt et al. 2015, §§5–7, especially Theorems 7.13–7.15, PDF. That source takes the Hilbert inner product linear in its first slot; its form/operator pairing is translated here to the site’s conjugate-linear first-slot convention. Its outward-normal Robin sign agrees with the one derived above.

Compact resolvent, modes, and the zero-mode test

For a self-adjoint, lower-bounded, strongly elliptic realization $L_B$ on compact $M$, the resolvent is compact. Its eigenvalues, repeated according to multiplicity, may be ordered as

$$\lambda_0\leq\lambda_1\leq\lambda_2\leq\cdots, \qquad \lambda_j\longrightarrow+\infty,$$

and there is an orthonormal basis $\{\phi_j\}$ of $L^2(M,\mathrm d\mu_g)$ satisfying

$$L_B\phi_j=\lambda_j\phi_j, \qquad B\phi_j=0.$$

Arendt et al. prove the corresponding compact-resolvent statements on bounded smooth Euclidean domains. On a compact manifold, the same conclusion follows by localization, elliptic regularity, and compact Sobolev embedding; Grubb 1996, Chapter I, §§1.4–1.7, PDF provides the required manifold realization, adjoint, parameter-elliptic, and semibounded framework.

This conclusion changes on a noncompact space, where continuous spectrum can occur. It also depends on the boundary realization: changing Dirichlet to Neumann data usually changes both eigenfunctions and eigenvalues.

Before writing an inverse, inspect $\ker L_B$. For a self-adjoint elliptic realization,

$$L_Bu=f$$

is solvable only when $f$ is orthogonal to the kernel; when it is solvable, the answer is unique only modulo that kernel. On a connected $M$, the Neumann Laplacian has the normalized constant zero mode

$$\phi_0 = \frac{1}{\sqrt{\operatorname{Vol}(M)}}.$$

The compatibility condition is visible without spectral theory. If

$$-\Delta u=f \quad\text{in }M, \qquad \partial_nu=g \quad\text{on }\partial M,$$

then the divergence theorem requires

$$\int_M f\,\mathrm d\mu_g + \int_{\partial M}g\,\mathrm d\mu_{\partial} = 0.$$

For homogeneous Neumann data, $f$ must have zero mean. The solution is then fixed, for example, by also requiring $\int_Mu\,\mathrm d\mu_g=0$.

If zero is absent from the spectrum, the Green kernel has the spectral representation

$$G_B(x,y) = \sum_j \frac{\phi_j(x)\overline{\phi_j(y)}}{\lambda_j}.$$

This expression is interpreted in the operator or distributional sense appropriate to the problem; it need not converge pointwise on the diagonal. If zero modes are present, the reduced inverse instead omits them and obeys

$$L_{B,x}G_B^\perp(x,y) = \delta_g(x,y)-\Pi_0(x,y),$$

where

$$\Pi_0(x,y) = \sum_{\lambda_j=0} \phi_j(x)\overline{\phi_j(y)}$$

is the kernel of the orthogonal projection onto $\ker L_B$. The subtraction is not optional bookkeeping: it states the exact data subspace on which the inverse exists.

Heat evolution packages the spectrum

Assume first that $L_B\geq0$. Functional calculus defines the heat semigroup

$$T(t)=e^{-tL_B}, \qquad t\geq0,$$

which is strongly continuous, self-adjoint, and contractive:

$$T(t)T(s)=T(t+s), \qquad T(0)=I, \qquad \|T(t)\|\leq1.$$

For $t>0$, ellipticity makes $T(t)$ smoothing. It has a smooth integral kernel relative to $\mathrm d\mu_g$:

$$\bigl(e^{-tL_B}f\bigr)(x) = \int_M K_B(t;x,y)f(y)\,\mathrm d\mu_g(y).$$

The kernel is characterized by

$$\begin{aligned} (\partial_t+L_{B,x})K_B(t;x,y)&=0, \\ B_xK_B(t;x,y)&=0, \\ \lim_{t\downarrow0}K_B(t;x,y)&=\delta_g(x,y), \end{aligned}$$

where the last line is a distributional initial condition. The semigroup law becomes the composition identity

$$K_B(t+s;x,y) = \int_M K_B(t;x,z)K_B(s;z,y)\,\mathrm d\mu_g(z).$$

Self-adjointness gives the Hermitian symmetry

$$K_B(t;x,y) = \overline{K_B(t;y,x)}.$$

The eigenfunction expansion is

$$K_B(t;x,y) = \sum_j e^{-t\lambda_j} \phi_j(x)\overline{\phi_j(y)}.$$

For every $t>0$, the heat operator is trace class and

$$\operatorname{Tr}e^{-tL_B} = \int_MK_B(t;x,x)\,\mathrm d\mu_g(x) = \sum_j e^{-t\lambda_j}.$$

It displays two distinct time regimes:

• As $t\to\infty$, the lowest eigenvalues dominate. If $L_B\geq\lambda_*>0$, then $\|e^{-tL_B}\|_{L^2\to L^2}\leq e^{-t\lambda_*}$. If zero modes are present, $e^{-tL_B}$ approaches $\Pi_0$ in operator norm. Negative eigenvalues instead grow exponentially.
• As $t\downarrow0$, arbitrarily high eigenvalues contribute. This is the regime controlled by local elliptic geometry and the short-time expansion.

These statements explain the proper-time representation. If $L_B\geq\lambda_*>0$, then

$$L_B^{-1} = \int_0^\infty e^{-tL_B}\,\mathrm dt.$$

The identity follows mode by mode from $\int_0^\infty e^{-t\lambda}\mathrm dt=\lambda^{-1}$ and holds in operator norm under the spectral-gap hypothesis. If $L_B\geq0$ has a kernel but has a positive gap above it, let $\mathcal H_\perp=(\ker L_B)^\perp$ and define the reduced inverse on the whole Hilbert space by

$$\begin{aligned} G_B^\perp &= \left( L_B|_{\mathcal H_\perp} \right)^{-1} (I-\Pi_0) \\ &= \int_0^\infty \left( e^{-tL_B}-\Pi_0 \right) \mathrm dt. \end{aligned}$$

It annihilates $\ker L_B$ and is the genuine inverse on $\mathcal H_\perp$.

Equivalently, a positive shift gives

$$(L_B+m^2)^{-1} = \int_0^\infty e^{-tm^2}e^{-tL_B}\,\mathrm dt, \qquad m^2>0.$$

For a merely lower-bounded $L_B$, the shift must be large enough that $L_B+m^2$ is strictly positive. These large-$t$ conditions are essential; formally integrating a heat kernel with an unsubtracted zero mode diverges.

The interval shows boundary and zero-mode effects exactly

Take $L=-\mathrm d^2/\mathrm dx^2$ on $[0,\ell]$. With Dirichlet conditions, the normalized eigenfunctions and heat kernel are

$$\begin{aligned} \phi_n^D(x) &= \sqrt{\frac{2}{\ell}} \sin\left(\frac{n\pi x}{\ell}\right), \qquad n\geq1, \\ K_D(t;x,y) &= \frac{2}{\ell} \sum_{n=1}^{\infty} e^{-t(n\pi/\ell)^2} \sin\left(\frac{n\pi x}{\ell}\right) \sin\left(\frac{n\pi y}{\ell}\right). \end{aligned}$$

With Neumann conditions,

$$\begin{aligned} \phi_0^N(x)&=\frac1{\sqrt{\ell}}, \\ \phi_n^N(x) &= \sqrt{\frac{2}{\ell}} \cos\left(\frac{n\pi x}{\ell}\right), \qquad n\geq1, \\ K_N(t;x,y) &= \frac1{\ell} + \frac{2}{\ell} \sum_{n=1}^{\infty} e^{-t(n\pi/\ell)^2} \cos\left(\frac{n\pi x}{\ell}\right) \cos\left(\frac{n\pi y}{\ell}\right). \end{aligned}$$

The constant $1/\ell$ is precisely the zero-mode projector. Hence $K_D\to0$ as $t\to\infty$, while $K_N\to1/\ell$.

Taking the trace and applying Poisson summation gives, up to terms exponentially small as $t\downarrow0$,

$$\begin{aligned} \operatorname{Tr}e^{-tL_D} &\sim \frac{\ell}{\sqrt{4\pi t}}-\frac12, \\ \operatorname{Tr}e^{-tL_N} &\sim \frac{\ell}{\sqrt{4\pi t}}+\frac12. \end{aligned}$$

The common leading term measures the one-dimensional volume. The constant term detects both boundary conditions and, in their exact spectral difference, the Neumann zero mode. Thus the local differential expression is the same in the two problems, but the heat trace is not.

Short time is local, with a boundary qualification

On flat $\mathbb R^d$, the heat kernel of $-\Delta$ is the Gaussian

$$K_0(t;x,y) = \frac{1}{(4\pi t)^{d/2}} \exp\left( -\frac{|x-y|^2}{4t} \right).$$

Its width is of order $\sqrt t$, so short heat time probes short distance. For a smooth Laplace-type operator on a closed manifold, or locally in the interior away from the boundary, the Gaussian is multiplied by an asymptotic series of smooth geometric coefficients. On the diagonal,

$$K(t;x,x) \sim \frac{1}{(4\pi t)^{d/2}} \left[ 1+t\,a_1(x)+t^2a_2(x)+\cdots \right].$$

The coefficients are local expressions in the metric, curvature, potential, and their derivatives. This is an asymptotic statement as $t\downarrow0$, not a claim that the series converges for fixed $t$.

On a smooth compact manifold with boundary and a strongly elliptic local boundary condition, the integrated heat trace has the more general form

$$\operatorname{Tr}e^{-tL_B} \sim \frac{1}{(4\pi t)^{d/2}} \sum_{r=0}^{\infty} A_{r/2}(L,B)t^{r/2}.$$

For the scalar problem,

$$A_0(L,B)=\operatorname{Vol}(M).$$

Integer-indexed coefficients contain bulk contributions, and boundary terms can also contribute to them; the half-integer-indexed coefficients are pure boundary contributions in this smooth local setting. The exact coefficients depend on the operator, geometry, and boundary condition. A pointwise interior expansion is not uniform all the way to the boundary: a boundary layer appears on the scale

$$\frac{\operatorname{dist}(x,\partial M)}{\sqrt t}.$$

Nonsmooth boundaries, singular coefficients, nonlocal conditions, or loss of strong ellipticity can change this structure and may introduce other terms. The standard series must not be transplanted to those settings without a new theorem. The closed-manifold expansion and the boundary-layer construction are given in Grieser 2004, Theorem 1.1 and §3, PDF; Vassilevich 2003, §§2.1–2.2 and 5.1–5.4 supplies the general Laplace-type and local Dirichlet/Robin coefficient setting.

Euclidean propagator as a proper-time integral

First consider a real scalar on a smooth compact Euclidean region $M$ with action

$$\begin{aligned} S_E[\phi;J] =& \frac12 \int_M \left( |\nabla\phi|^2+m^2\phi^2 \right) \mathrm d\mu_g \\ &+ \frac12 \int_{\partial M} \beta\phi^2\,\mathrm d\mu_{\partial} - \int_MJ\phi\,\mathrm d\mu_g, \end{aligned}$$

where $m^2>0$ and $\beta\geq0$. Varying an unrestricted boundary trace gives

$$\delta S_E = \int_M \delta\phi \left[ (-\Delta_g+m^2)\phi-J \right] \mathrm d\mu_g + \int_{\partial M} \delta\phi (\partial_n+\beta)\phi\, \mathrm d\mu_{\partial}.$$

Thus the natural boundary problem is

$$(-\Delta_g+m^2)\phi=J, \qquad (\partial_n+\beta)\phi=0.$$

Its Hessian is strictly positive, so its Euclidean Green operator is

$$G_{m,R} = (-\Delta_R+m^2)^{-1} = \int_0^\infty e^{-t(-\Delta_R+m^2)}\,\mathrm dt.$$

At the formal Gaussian level, $G_{m,R}$ is also the covariance, and the source-dependent ratio is

$$\frac{Z[J]}{Z[0]} = \exp\left( \frac12\langle J|G_{m,R}J\rangle \right).$$

This is a controlled use of the mathematical inverse; the determinant hidden in $Z[0]$ is deliberately left untreated below.

For the translation-invariant check, take

$$L_m=-\Delta_E+m^2 \quad\text{on }\mathbb R^d, \qquad m>0.$$

With the site’s Fourier convention,

$$\widetilde f(p) = \int\mathrm d^dx\, e^{+ip\cdot x}f(x), \qquad f(x) = \int\frac{\mathrm d^dp}{(2\pi)^d}\, e^{-ip\cdot x}\widetilde f(p),$$

the derivative maps to $\partial_\mu\mapsto-ip_\mu$. Therefore the heat operator has multiplier $e^{-t(p_E^2+m^2)}$, and

$$\begin{aligned} K_m(t;x,y) &= \int \frac{\mathrm d^dp}{(2\pi)^d}\, e^{-ip\cdot(x-y)} e^{-t(p_E^2+m^2)} \\ &= \frac{e^{-m^2t}}{(4\pi t)^{d/2}} \exp\left( -\frac{|x-y|^2}{4t} \right). \end{aligned}$$

For $x\neq y$, integrating over heat time gives an ordinary kernel integral. Globally the same statement is an identity of tempered distributions; for $d\geq2$ the coincident-point integral diverges at its small-$t$ endpoint. With that interpretation,

$$\begin{aligned} G_E(x-y) &= \int_0^\infty K_m(t;x,y)\,\mathrm dt \\ &= \int \frac{\mathrm d^dp}{(2\pi)^d}\, \frac{e^{-ip\cdot(x-y)}}{p_E^2+m^2}, \end{aligned}$$

so

$$(-\Delta_E+m^2)G_E(x-y) = \delta^{(d)}(x-y).$$

No $i0$ prescription is needed: this is a positive Euclidean denominator, not a Lorentzian causal boundary value. Wick rotation and the selection of retarded, advanced, or Feynman distributions require additional analytic input, as explained in Hyperbolic Equations and Causal Propagators.

In a Gaussian functional integral, a positive fluctuation operator also leads formally to $\tfrac12\operatorname{Tr}\log L$. Its proper-time expression involves

$$-\frac12 \int_0^\infty \frac{\mathrm dt}{t}\, \operatorname{Tr}e^{-tL},$$

which is generally divergent and is not a definition without a regulator, zero-mode prescription, and normalization. The small-$t$ coefficients are the mathematical input that organizes local ultraviolet and heavy-mass terms. Zeta continuation and spectral determinants belong to Heat Kernels, Zeta Functions, and Spectral Determinants; the physical expansion after eliminating a heavy field belongs to Integrating Out Heavy Fields. The proper-time identity, its divergent endpoints, and its one-loop role are set out in Vassilevich 2003, equations (1.9)–(1.21).

Checking an elliptic boundary realization

For a new elliptic inverse or heat-kernel problem:

1. Specify the realization. State the measure, differential expression, function or bundle space, operator domain, and boundary condition.
2. Check the high-frequency problem. Verify interior ellipticity and the complementing or strong-ellipticity condition at the boundary.
3. Check the boundary form. Decide whether the realization is symmetric, self-adjoint, and lower bounded; do not infer these properties from the principal symbol.
4. Inspect the bottom of the spectrum. Identify negative and zero modes, compatibility conditions, and any projection used in a reduced inverse.
5. Choose the representation. Use modes for global spectral information, the heat equation for smoothing and composition, and short-time asymptotics only for local high-frequency information.
6. Check both ends of proper time. Small $t$ controls ultraviolet and diagonal singularities; large $t$ controls convergence through the lowest eigenvalues.
7. Validate independently. Check the heat PDE, boundary data, delta normalization, symmetry, and semigroup composition, then compare with a spectral, image, or Fourier representation. A full eigenbasis is a global and often expensive computation; a local short-time parametrix is cheaper but cannot answer low-spectrum or large-time questions.

Stop rule. Do not invoke the standard boundary heat-kernel expansion until strong ellipticity is established. Do not identify the unmodified proper-time integral with $L_B^{-1}$ until zero modes are projected out and negative modes are projected out or removed by a declared shift, with the resulting map stated explicitly. Stop using a local short-time parametrix when the target is low-spectrum or large-time information; switch to a spectral or other global method.

Common pitfalls

Calling an elliptic expression invertible. Ellipticity controls the principal symbol, not the kernel of a boundary realization. State the domain and test zero modes before writing $L^{-1}$.

Equating a cancelled boundary form with self-adjointness. The calculation shows symmetry on a proposed domain. Self-adjointness also requires the adjoint to have exactly that domain.

Integrating through a zero or negative mode. A zero mode makes $\int_0^\infty e^{-tL}\mathrm dt$ diverge, while a negative mode grows. Project out the kernel or shift the operator, and state which map is being inverted.

Treating the short-time series as a global formula. It is ordinarily asymptotic, and an interior expansion is not uniform through a boundary layer. Large-time behavior comes from low spectrum, not from local coefficients.

Importing Lorentzian pole language into the Euclidean problem. The positive Euclidean resolvent has no causal support choice. An $i0$ prescription arises only after a separate Lorentzian boundary-value problem has been specified.

Exercises

1. Verify symmetry and nonnegativity for the Robin Laplacian with $\beta\geq0$.

   Check

   If $u$ and $v$ satisfy $\partial_nu=-\beta u$ and $\partial_nv=-\beta v$, then

   $$v\,\partial_n\overline u - \overline u\,\partial_nv = -\beta v\overline u + \beta\overline u v = 0.$$

   Thus the boundary form vanishes. For $V=0$, the quadratic form is

   $$\langle u|-\Delta_Ru\rangle = \int_M|\nabla u|^2\,\mathrm d\mu_g + \int_{\partial M}\beta|u|^2\, \mathrm d\mu_{\partial} \geq0.$$

2. Derive the compatibility condition for the inhomogeneous Neumann problem and explain the nonuniqueness.

   Check

   Integrating $-\Delta u=f$ and using the outward normal gives

   $$\int_M f\,\mathrm d\mu_g = -\int_{\partial M}\partial_nu\, \mathrm d\mu_{\partial} = -\int_{\partial M}g\, \mathrm d\mu_{\partial}.$$

   Hence the two integrals must sum to zero. If $u$ is one solution, then $u+C$ has the same Laplacian and normal derivative. On connected $M$, fixing the mean of $u$ removes this constant ambiguity.

3. Explain why the Neumann and Dirichlet interval heat traces differ by exactly one for every $t>0$.

   Check

   Their positive eigenvalues are identical: $(n\pi/\ell)^2$ for $n\geq1$. The Neumann spectrum has one additional eigenvalue $\lambda_0=0$. Therefore

   $$\operatorname{Tr}e^{-tL_N} - \operatorname{Tr}e^{-tL_D} = e^{-t\cdot0} = 1.$$

   This also matches the difference between the two constant terms in their short-time expansions.

4. Check the proper-time formula in Euclidean momentum space.

   Check

   Since $p_E^2+m^2>0$,

   $$\int_0^\infty e^{-t(p_E^2+m^2)}\,\mathrm dt = \frac{1}{p_E^2+m^2}.$$

   Insert this identity after pairing with a Schwartz test function, or use a regulator and remove it distributionally. The inverse Fourier transform then produces $G_E$. Multiplication by $p_E^2+m^2$ gives $1$, whose inverse Fourier transform is $\delta^{(d)}(x-y)$. No assertion of absolute momentum integrability or a finite coincident-point kernel is needed.

References

• Wolfgang Arendt, Ralph Chill, Christian Seifert, Hendrik Vogt, and Jürgen Voigt (2015), Form Methods for Evolution Equations, and Applications, PDF, §§5–7, especially Theorems 7.13–7.15. This is the operator-domain authority for form-defined self-adjoint realizations, weak normal derivatives, Robin signs, compact resolvent, and the Neumann zero mode.
• Daniel Grieser (2004), Notes on Heat Kernel Asymptotics, PDF, Theorem 1.1 and §3. This is the structural source for the closed-manifold diagonal expansion, the boundary layer, and the half-integer Dirichlet heat-trace expansion.
• Gerd Grubb (1996), Functional Calculus of Pseudodifferential Boundary Problems, Chapter I, PDF, §§1.4–1.7. These sections establish realizations on compact manifolds with boundary, the Shapiro–Lopatinski condition, parameter-ellipticity, adjoints, and semiboundedness.
• John K. Hunter (2014), Notes on Partial Differential Equations, PDF, §§2.5, 4.8–4.10, 5.1, and 5.4. This is the teaching source for Green’s identities, compact resolvent and discrete spectral theory, the Euclidean heat kernel, semigroups, and the resolvent as a Laplace transform.
• D. V. Vassilevich (2003), Heat Kernel Expansion: User’s Manual, §§1, 2.1–2.2, 3.1, and 5.1–5.4, especially equations (1.9)–(1.21), (2.15)–(2.21), (3.7)–(3.9), and the interval examples (5.1)–(5.3). This is the structural and QFT-facing source for proper time, heat traces, local short-time coefficients, boundary half-powers, strong ellipticity, and one-loop scope. Vassilevich uses an inward-normal convention in the boundary sections; this page uses the outward normal and therefore fixes the Robin sign directly from Green’s identity rather than importing later $S$-dependent coefficients.