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Given a second linear differential operator,

$(Hf)(x)=-\frac{d^2}{dx^2}(x)+V(x)f(x)$,

where $V$ is a bounded and real valued function, $f$ lies in $L^2(\mathbb{R})$.

For an $z$ with $Im(z)\neq 0$, we define the Weyl-Titchmarsh function $m_{+}$ as the the unique solution of the following equation:

$(Hf)(z)=zf(z)$

$f(0)=1$

$\int_{0}^{\infty}f^2(x)dx<\infty.$

Similarly, we can define Weyl-Titchmarsh function $m_{-}$ as the the unique solution of the following equation:

$(Hf)(z)=zf(z)$

$f(0)=1$

$\int_{-\infty}^{0}f^2(x)dx<\infty.$

There are very deep theory for Weyl-Titchmarsh m function, also there are some natural links between m function and Green functions. It is very important also in the discrete schrondinger operator.

However, at least for me, Weyl-Titchmarsh m function is really mysterious. I really want to know some backgrounds about Weyl-Titichmarsh m function, especially, how Weyl first introduce such a powerful tool to us.

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    $\begingroup$ Why not read Weyl's own explanation: Weyl, Hermann Ramifications, old and new, of the eigenvalue problem. Bull. Amer. Math. Soc. 56, (1950). 115–139. (Freely available, English). $\endgroup$
    – Alexandre Eremenko
    Commented Sep 14, 2015 at 18:49
  • $\begingroup$ The $m$ functions are matrix elements of the resolvents (and you didn't quite finish the definition). The question seems too general to admit useful answers. $\endgroup$
    – Christian Remling
    Commented Sep 16, 2015 at 1:55

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