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Suppose $T,S:D(\mathcal H)\to \mathcal H$ are two unbounded operators with discrete spectrum consisting eigenvalues $0<\lambda_1(T)\leq\lambda_2(T)\leq\dots$ and $0<\lambda_1(S)\leq\lambda_2(S)\leq\dots$. Moreover assume that $T+S$ also has discrete spectrum and that

Q. How large $\lambda_1(T+S)$ can be compared to $\lambda_1(T)$ and $\lambda_1(S)$?

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    $\begingroup$ You could write the Question as: "Suppose $T,S:D(\mathcal H)\to D(\mathcal H)$ are two unbounded Hermetian operators with discrete spectrum and least eigenvalues $0$. If $T+S$ also has discrete spectrum, how large can it's least eigenvalue be?" (Just adding constants to $S,T$.) Also, did you mean $D(\mathcal H)\to \mathcal H$ ? $\endgroup$ – Keith McClary Nov 20 at 17:52
  • $\begingroup$ Also, do you know the answer for $2 \times 2$ matrices? $\endgroup$ – Keith McClary Nov 20 at 18:14
  • $\begingroup$ Yes, edited it. I'm so used to copy-pasting in LaTex. $\endgroup$ – BigM Nov 20 at 18:29
  • $\begingroup$ You have a better chance of getting an Answer if you write the Question in the simplest possible way. If you agree that the answer follows from that of the special case I have stated then you should ask that. Anyone who could answer knows that discrete eigenvalues can be arranged in ascending sequence, so you don't need to write that out. $\endgroup$ – Keith McClary Nov 22 at 17:55
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I answer this question, which I think implies the answer to your question:

Suppose $T,S$ are possibly unbounded Hermitian operators in a Hilbert space with discrete spectrum and least eigenvalues $0$. If $T+S$ also has discrete spectrum, how large can it's least eigenvalue be?

In the case of $2 \times 2$ matrices you can solve the problem explicitly.

For general operators you can solve the Rayleigh-Schrodinger equation, which is non-linear and involves calculating the inverse of one of the operators (With discrete spectrum, calculating the inverse not a problem.).

If you want an upper bound you can construct approximants to the eigenvalue by simply choosing an arbitrary sequence of vectors which spans the Hilbert space and minimizing $\frac{(\psi , (T+S) \psi}{(\psi,\psi)}$ in the subspace spanned by the first $n$ vectors.

In special cases there may be explicit solutions to the eigenvalue problem . In one case ($x^4$ anharmonic oscillator) there are approximants from below.

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