Dynamics of zeros

This is a sub-page of page on the De Bruijn-Newman constant, and assumes all the notation from that page.

The entire functions [math]\displaystyle{ H_t(z) }[/math] obey the backwards heat equation

[math]\displaystyle{ \displaystyle \partial_t H_t(z) = - \partial_{zz} H_t(z). }[/math]

Dynamics of a simple zero

If [math]\displaystyle{ H_t }[/math] has a simple zero at [math]\displaystyle{ z_j(t) }[/math], then by the implicit function theorem [math]\displaystyle{ z_j(t) }[/math] varies in a continuously differentiable manner (in fact analytic) for nearby times [math]\displaystyle{ t }[/math]. By implicitly differentiating the equation [math]\displaystyle{ H_t(z_j(t)) = 0 }[/math], we see that

[math]\displaystyle{ \displaystyle \partial_t z_j(t) = - \frac{\partial_t H_t(z_j(t))}{\partial_z H_t(z_j(t))} = \frac{\partial_{zz} H_t(z_j(t))}{\partial_z H_t(z_j(t))}. }[/math]

(See also [CSV1994, Lemma 2.1].) Being a simple zero, we have the Taylor expansion

[math]\displaystyle{ \displaystyle H_t(z) = a (z-z_j(t)) + b (z-z_j(t))^2 + O( |z-z_j(t)|^3 ) }[/math]

for some complex numbers [math]\displaystyle{ a,b }[/math] with [math]\displaystyle{ a \neq 0 }[/math], and for [math]\displaystyle{ z }[/math] close to [math]\displaystyle{ z_j(t) }[/math]. In particular

[math]\displaystyle{ \displaystyle \partial_z H_t(z) = a + 2 b (z-z_j(t))^2 + O( |z-z_j(t)|^2 ) }[/math]

[math]\displaystyle{ \displaystyle \partial_{zz} H_t(z) = 2 b + O( |z-z_j(t)| ) }[/math]

which implies that

[math]\displaystyle{ \frac{\partial_{zz} H_t(z_j(t))}{\partial_z H_t(z_j(t))} = \frac{2b}{a} }[/math]

and also that

[math]\displaystyle{ \frac{\partial_{z} H_t(z)}{H_t(z)} = \frac{1}{z-z_j(t)} + \frac{b}{a} + O( |z-z_j(t)| ) }[/math]

and thus

[math]\displaystyle{ \displaystyle \partial_t z_j(t) = 2 \lim_{z \to z_j(t)} \frac{\partial_z H_t(z)}{H_t(z)} - \frac{1}{z-z_j(t)}. }[/math]

As [math]\displaystyle{ H_t }[/math] is even, of order 1, and has no zero at the origin, we see from the Hadamard factorisation theorem that

[math]\displaystyle{ H_t(z) = C_t \prod_{k=1}^\infty (1 - \frac{z}{z_k(t)}) (1 + \frac{z}{z_k(t)}) }[/math]

for some constant [math]\displaystyle{ C_t }[/math], and hence

[math]\displaystyle{ \frac{\partial_z H_t(z)}{H_t(z)} = \sum_k \frac{1}{z - z_k(t)} }[/math]

where the sum is in a principal value sense. Thus we have

[math]\displaystyle{ \displaystyle \partial_t z_j(t) = - 2 \sum_{k \neq j} \frac{1}{z_j(t) - z_k(t)} }[/math]

where the sum is again in a principal value sense (cf. [CSV1994, Lemma 2.4]).

Dynamics of a repeated zero

Now suppose that at some time [math]\displaystyle{ t_0 }[/math] one has a repeated zero at [math]\displaystyle{ z_0 }[/math] of some order [math]\displaystyle{ k \geq 1 }[/math], thus

[math]\displaystyle{ \displaystyle H_{t_0}(z) = a_k (z-z_0)^k + O( |z-z_0|^{k+1} ) }[/math]

for some non-zero [math]\displaystyle{ a_k }[/math] and [math]\displaystyle{ z }[/math] close to [math]\displaystyle{ . Using the backwards heat equation we then have :\lt math\gt \displaystyle \partial_t H_{t_0}(z) = - k(k-1) a_k (z-z_0)^{k-2} + O( |z-z_0|^{\max(k-1,0)} ) }[/math]

and more generally

[math]\displaystyle{ \displaystyle \partial_t^j H_{t_0}(z) = (-1)^j k(k-1) \dots (k-2j+1) a_k (z-z_0)^{k-2j} + O( |z-z_0|^{\max(k-2j+1,0)} ) }[/math]

for any fixed [math]\displaystyle{ j }[/math]. Performing Taylor expansion in time, we conclude that in the regime [math]\displaystyle{ z - z_0 = O( |t-t_0|^{1/2} ) }[/math] and [math]\displaystyle{ t }[/math] close to but not equal to [math]\displaystyle{ t_0 }[/math], one has

[math]\displaystyle{ \displaystyle H_t(z) = a_k ((t-t_0)^{1/2})^k ( P_k( \frac{z-z_0}{(t-t_0)^{1/2}} ) + O( |t-t_0|^{1/2} ) ) }[/math]

where we use some branch of the square root and [math]\displaystyle{ P_k }[/math] is the degree [math]\displaystyle{ k }[/math] polynomial

[math]\displaystyle{ P_k(z) := \sum_{0 \leq j \leq k/2} (-1)^j k (k-1) \dots (k-2j+1) z^{k-2j}. }[/math]

We claim that the [math]\displaystyle{ k }[/math] zeroes [math]\displaystyle{ x_{k,1},\dots,x_{k,k} }[/math] of [math]\displaystyle{ P_k }[/math] are real and simple. If so, then by Rouche's theorem we conclude that for [math]\displaystyle{ t }[/math] close to [math]\displaystyle{ t_0 }[/math], the [math]\displaystyle{ k }[/math] zeroes of [math]\displaystyle{ H_t }[/math] close to [math]\displaystyle{ z_0 }[/math] take the form

[math]\displaystyle{ z_0 + (t-t_0)^{1/2}( x_j + O( |t-t_0|^{1/2} ) ) }[/math]

for [math]\displaystyle{ j=1,\dots,k }[/math]. In particular, the zeroes approach [math]\displaystyle{ z_0 }[/math] from an asymptotically vertical direction as [math]\displaystyle{ t \to t_0^- }[/math] and repel in an asymptotically horizontal direction as [math]\displaystyle{ t \to t_0^+ }[/math].