Assume $\Omega$ is a non-compact region or manifold with dimension $\geq4$. Let $H=-\Delta+V$ be Schrodinger operator. Here $V$ is a (smooth)function.

I know that if $V\geq c>0$ or $V\to c>0$, then $0$ does not locate in the essential spectrum of $H$.s

Q : Is there any work to consider the negative case, i.e. $V>-c$, here $c>0$ is a constant, with what condition on $V$, we also have that $0$ locates in the discrete spectrum of $H$


1 Answer 1


Obviously, the potential need to be bigger than some positive constant "most of the time". I did not see this exact question to be addressed in the literature. However, a related question: what is the most general condition on $V$ which guarantees that the spectrum of $H$ is discrete, was intensively studied, see

Kondratʹev, V. A.; Shubin, M. A. Conditions for the discreteness of the spectrum for Schrödinger operators on a manifold. Funct. Anal. Appl. 33 (1999), no. 3, 231–232 (2000)

Roughly, the condition is that there is a "small" subset outside of which the potential grows to infinity as $x\to \infty$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.