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Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$, consider $L=\Delta+V$ where $\Delta=div\nabla$. According to section 8.12 in Gilbarg and Trudinger's book, if $V\in L^\infty(\Omega)$, then the Dirichlet spectrum of $L$ exists, and the first eigenvalue is simple.

(1) What is the weakest condition on $V$ such that the spectrum still exists?

(2) What is the condition on $V$ such that the first eigenvalue is positive?

I think this should be well studied, but I haven't been able to find any good reference. If you are familiar with this, please let me know. Thanks for your help.

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    $\begingroup$ What do you mean by weakest? (for question 1) // For your second question, as you wrote it, your operator $L$ is formally a negative operator when $V \equiv 0$. I just want to double check that you really meant to ask about the first eigenvalue being positive, or if rather you want the opposite direction? In any case, certainly the answer to the second question depends lots on $\Omega$ and details of $V$, if only just by thinking about the family $\lambda V$ of potentials and noting for all sufficiently small $\lambda$ the spectrum should be a small perturbation of that of $\Delta$. $\endgroup$
    – Willie Wong
    Commented May 4, 2020 at 19:32
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    $\begingroup$ Welcome to MathOverflow! What precisely do you mean by "the spectrum exists"? Every linear operator has a spectrum by definition. $\endgroup$
    – Jochen Glueck
    Commented May 4, 2020 at 19:33
  • $\begingroup$ @JochenGlueck: I deleted my previous comment because my interpretation was not correct. Turns out that particular section of G+T is talking about Rayleigh quotients, and bounded coefficients are convenient. // And also, never noticed this before, but there is a typo in the 2001 edition of G+T: the operator and quadratic form on the bottom of page 212 disagree by a sign (which further muddies my second question to the OP). $\endgroup$
    – Willie Wong
    Commented May 4, 2020 at 20:04
  • $\begingroup$ One possibility (after taking a quick look at G+T) of a reasonable question: what are the conditions on $V$ that guarantees the spectrum is discrete. Perhaps this is what the OP meant by question 1. $\endgroup$
    – Willie Wong
    Commented May 4, 2020 at 20:18
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    $\begingroup$ I would recommend the book by B. Davies: spectral theory and differential operators. $\endgroup$
    – Giorgio Metafune
    Commented May 4, 2020 at 20:39

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