What would be the special character for this mathematical notation?

\[DoubleVerticalBar] behaves like a binary operator (often used to indicate parallel lines in geometry). You probably want

Subsuperscript[\[LeftDoubleBracketingBar] h/3 \[RightDoubleBracketingBar], 0, H] // 
  TraditionalForm

which gives

form


Bob Hanlon's comment is closer to what I was expecting, but it may be that m_goldberg's notation may be more correct. In doubt, I'll post this:

(V == HoldForm[A/H^2] Subsuperscript[Abs[HoldForm[h^3/3]], 0, H])

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