Gelfand-Yaglom theorem for functional determinants What is the 'Gelfand-Yaglom' Theorem?  I have heard that it is used to calculate Functional determinants by solving an initial value problem of the form
$Hy(x)-zy(x)=0$ with $y(0)=0$ and $y'(0)=1$.  Here $H$ is the Hamiltonian and $z$ is a real parameter.
Is it that simple?  If $H$ is a Hamiltonian, could I use the WKB approximation to solve the initial value problem and to be valid for $z$ big?
 A: *

*Another application of the Gelfand-Yaglom formula is the van Vleck determinant:

$$\begin{align}\langle q_f,& t_f | q_i,t_i \rangle \cr
~=~& \int_{q(t_i)=q_i}^{q(t_f)=q_f} \! {\cal D}q~\exp\left(\frac{i}{\hbar} S[q]\right)\cr
~\sim~&\sqrt{\det\left(\frac{-1}{2\pi i \hbar}\frac{\partial^2 S_{\rm cl}}{\partial q_f \partial q_i} \right)}
\exp\left(\frac{i}{\hbar} S_{\rm cl}\right) \cr
&\quad\text{for}\quad \hbar~\to~0, \end{align}\tag{1}$$

where
$$S[q]~:=~ \int_{t_i}^{t_f}\! dt ~  L(q(t),\dot{q}(t),t) \tag{2}$$
is the off-shell action functional, and
$$ S_{\rm cl}~:=~S[q_{\rm cl}] \tag{3}$$
is the Dirichlet on-shell action function for a classical path $q_{\rm cl}:[t_i,t_f]\to \mathbb{R}$. (In this answer, we assume for simplicity that the classical path exists and is unique, i.e. no instantons.)


*Example: The harmonic oscillator
$$ L~=~\frac{m}{2}\dot{q}^2 -\frac{m}{2}\omega^2 q^2 \tag{4}$$
has classical path
$$\begin{align} q_{\rm cl}(t)~=~&\frac{q_f\sin \omega (t-t_i)+q_i\sin \omega (t_f-t)}{\sin (\omega \Delta t)}, \cr \Delta t~:=~&t_f-t_i, \end{align}\tag{5} $$
on-shell action
$$ S_{\rm cl}~\stackrel{(4)+(5)}{=}~m\omega\frac{(q_f^2+q_i^2)\cos(\omega\Delta t)-2q_fq_i}{2\sin(\omega\Delta t)},   \tag{6}$$
and Feynman amplitude/kernel
$$\begin{align}\langle q_f,& t_f | q_i,t_i \rangle\cr
~\stackrel{(1)+(6)}{=}&~\sqrt{\frac{m\omega}{2\pi i \hbar\sin(\omega\Delta t)}}
\exp\left(\frac{i}{\hbar} S_{\rm cl}\right).\end{align}\tag{7}$$
It is remarkable that the full quantum amplitude (7) can be derived from the classical on-shell action (6) alone!


*Proof of eq. (1) for 1D. Firstly, expand the Lagrangian to quadratic order in fluctuations $q=q_{\rm cl}+y$:
$$\begin{align}L(q,\dot{q},t)~=~&L(q_{\rm cl},\dot{q}_{\rm cl},t) + L_1 \cr
&+ L_2 + {\cal O}(y^3),\end{align}\tag{8}$$
$$\begin{align} L_1~:=~& p_{\rm cl}(t)\dot{y}+F_{\rm cl}(t)y~\stackrel{\begin{array}{c}\text{int. by} \cr\text{parts}\end{array}}{\sim}~0, \cr 
p_{\rm cl}(t)~:=~&\left. \frac{\partial L}{\partial \dot{q}}\right|_{q=q_{\rm cl}(t)}, \cr 
F_{\rm cl}(t)~:=~&\left. \frac{\partial L}{\partial q}\right|_{q=q_{\rm cl}(t)},\end{align}\tag{9} $$
$$\begin{align} L_2~:=~~&\frac{m(t)}{2}\dot{y}^2+ b(t)y\dot{y} - \frac{k(t)}{2}y^2\cr
~\stackrel{\begin{array}{c}\text{int. by} \cr\text{parts}\end{array}}{\sim}&~ \frac{m(t)}{2}\dot{y}^2  - \frac{k(t)+\dot{b}(t)}{2}y^2 ,\end{align}\tag{10}$$
$$\begin{align} m(t)~:=~&\left. \frac{\partial^2 L}{\partial \dot{q}^2}\right|_{q=q_{\rm cl}(t)}, \cr 
b(t)~:=~&\left.\frac{\partial^2 L}{\partial q~\partial \dot{q}}\right|_{q=q_{\rm cl}(t)}, \cr 
k(t)~:=~&-\left.\frac{\partial^2 L}{\partial q^2}\right|_{q=q_{\rm cl}(t)}. \end{align}\tag{11}$$
In eq. (10) the $b$-term is integrated by parts. The boundary terms vanish because of Dirichlet boundary conditions (BCs) $y(t_i)=0=y(t_f)$.
Secondly, expand the momentum to linear order in fluctuations $q=q_{\rm cl}+y$:
$$\begin{align} p~:=~&\frac{\partial L}{\partial \dot{q}}\cr
~=~&p_{\rm cl}(t) +b(t)y+m(t)\dot{y} + {\cal O}(y^2).\end{align}\tag{12}$$


*Next use the WKB/stationary phase approximation for $\hbar \to 0$:
$$\begin{align}\langle q_f,& t_f | q_i,t_i \rangle \cr
~=~& \int_{q(t_i)=q_i}^{q(t_f)=q_f} \! {\cal D}q~\exp\left(\frac{i}{\hbar} S[q]\right)\cr
~\stackrel{\text{WKB}}{\sim}&~ {\rm Det}\hat{H}^{-1/2} 
\exp\left(\frac{i}{\hbar} S_{\rm cl}\right) \cr
~\stackrel{(29)}{=}&~\left(2\pi i\hbar \phi_0(t_i)\psi_0(t_f) \right)^{-1/2}\exp\left(\frac{i}{\hbar} S_{\rm cl}\right),\end{align}\tag{13}$$
where the Hessian operator reads
$$ \hat{H}~:=~\frac{d}{dt}m(t)\frac{d}{dt}+k(t) +\dot{b}(t).\tag{14} $$
In the last equality of eq. (13) was used eq. (29) from my other answer in this thread.
Here $\phi_0$ is a zero-mode with $\phi_0(t_i)\neq 0$, and
$$\begin{align} \psi_0(t)~:=~&\phi_0(t)\int_{t_i}^t\! \frac{dt^{\prime}}{m(t)\phi_0(t^{\prime})^2},\cr  \psi_0(t)~=~&0,\end{align} \tag{15}$$
is an independent zero-mode, cf. the Gelfand-Yaglom formula. Note for later that the Wronskian is
$$\begin{align} W(\phi_0,\psi_0)
~:=~&\phi_0\dot{\psi}_0-\dot{\phi}_0\psi_0\cr~=~&\frac{1}{m(t)}.\end{align} \tag{16}$$


*On the other hand, the final momentum $p_f$ can be found from the on-shell formula
$$ p_f ~=~ \frac{\partial S_{\rm cl}}{\partial q_f},\tag{17} $$
see e.g. eq. (11) in my Phys.SE answer here. Therefore the $1\times 1$ van Vleck matrix can be found
$$ \frac{\partial^2 S_{\rm cl}}{\partial q_f \partial q_i} ~\stackrel{(17)}{=}~\frac{\partial p_f}{ \partial q_i}\tag{18}$$
by varying infinitesimally the initial position $\delta q_i= y(t_i)$ for fixed final position $\delta q_f= y(t_f)=0$, and such that the new path $q=q_{\rm cl}+y$ is also a classical solution. The EL eq. for the new path $q=q_{\rm cl}+y$ (i.e. the linearized EL eq. for $y$) implies that the infinitesimal variation $y$ is a zero-mode $\hat{H}y=0$, i.e. a linear combination
$$ y(t)~=~A\phi_0(t)+ B\psi_0(t),\tag{19} $$
where $A$ & $B$ are 2 infinitesimal constants determined by the Dirichlet BCs:
$$\begin{align}\delta q_i~=~~~& y(t_i)\cr
~\stackrel{(15)+(19)}{=}&~A\phi_0(t_i)\cr\qquad\Downarrow&\qquad\cr 
A~=~&\frac{\delta q_i}{\phi_0(t_i)} ,\end{align}\tag{20}  $$
$$\begin{align}0~=~&\delta q_f\cr
~=~ &y(t_f)\cr
~\stackrel{(19)}{=}~&
A\phi_0(t_f)+B\psi_0(t_f) 
\cr\qquad\Downarrow&\qquad\cr  
B~=~&-A\frac{\phi_0(t_f)}{\psi_0(t_f)}\cr
~\stackrel{(20)}{=}~&
-\frac{\delta q_i}{\phi_0(t_i)}\frac{\phi_0(t_f)}{\psi_0(t_f)} .\end{align}\tag{21}  $$
The change in the final momentum is
$$\begin{align}\delta p_f
~\stackrel{(12)}{=}~&m(t_f) \dot{y}(t_f)\cr
~\stackrel{(19)}{=}~&m(t_f)\left(A\dot{\phi}_0(t_f)+B\dot{\psi}_0(t_f)\right)\cr
~\stackrel{(21)}{=}~&m(t_f)A\left(\dot{\phi}_0(t_f)-\frac{\phi_0(t_f)}{\psi_0(t_f)}\dot{\psi}_0(t_f)\right)\cr
~\stackrel{(16)}{=}~&-\frac{A}{\psi_0(t_f)}\cr
~\stackrel{(20)}{=}~&-\frac{\delta q_i}{\phi_0(t_i) \psi_0(t_f)}.\end{align}\tag{22}$$
Therefore
$$ \frac{\partial^2 S_{\rm cl}}{\partial q_f \partial q_i} ~\stackrel{(18)+(22)}{=}~ -\frac{1}{\phi_0(t_i) \psi_0(t_f)}.\tag{23}$$
Comparing eqs. (13) & (23) yields the sought-for van Vleck formula (1). $\Box$
References:

*

*B.S. DeWitt, The Global Approach to QFT, Vol 1, 2003; Chapter 14.


*H. Kleinert, Path Integrals in QM, Statistics, Polymer Physics, and Financial Markets, 5th ed.; Section 2.4.


*M. Blau, Notes for (semi-)advanced QM: The Path Integral Approach to QM; App. C.


*R. Rattazzi, Lecture notes for QM IV: The Path Integral approach to QM; Section 3.1.
A: I was at a talk a while back by Gerald Dunne where he talked about the Gelfand-Yaglom theorem. He used it for calculating some Euler-Heisenberg type effective actions. A paper of his with Hyunsoo Min on the subject is A comment on the Gelfand–Yaglom theorem, zeta functions and heat kernels for PT-symmetric Hamiltonians and he's got some nice lecture notes: Functional Determinants in Quantum Field Theory (also see a wider spanning set of lectures of the same name).
Basically, it's a way of calculating the determinant of a 1-dimensional operator $\det(H)=\prod_i \lambda_i$ with out calculating, let alone multiplying, any of its eigenvalues $H \psi_i = \lambda_i \psi_i$.
To state the original theorem: assume that you have a Schrodinger operator (or Hamiltonian) 
$ H = -\frac{d^2}{d x^2} + V(x) $
on the interval $x\in[0,L]$ with Dirichlet boundary conditions:
$$ H \psi_i(x) = \lambda_i \psi_i(x) \,,
\quad \psi(0)=\psi(L)=0 \ .
$$
Then we can compute its determinant by solving the related initial value problem
$$ H \phi(x) = 0\,, \quad \phi(0)=0\,,\quad \phi'(0) = 1 \ ,$$
so that
$$ \det H \approx \phi(L) \,,$$
where the final result is only $\approx$ as we can only really calculate the ratio of two determinants.
This basic result can be generalised to more general boundary conditions, coupled systems of ODEs and higher order linear ODEs.
