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Consider the Schroedinger operator

$$H = -\Delta + \frac{c}{\vert x \vert^2}$$ in two dimensions with $c >0$

This operator has a self-adjoint realization, since it is a positive symmetric operator on $C_c^{\infty}( \mathbb R^2 \setminus \{0\}).$ I would like to understand three things:

0.) What are possible domains $D(H)$ on which $H$ is self-adjoint?

1.) Does there exist a domain $D(H)$ of $H$ on which $-\Delta$ and $H$ are both self-adjoint.

2.) On these domains $D(H)$, can we expect $$\Vert \frac{f}{\vert x \vert^2}\Vert_{L^2} \le \alpha_1 \Vert f \Vert_{L^2} + \alpha_2 \Vert \Delta f \Vert_{L^2} \text{ for some }\alpha_i \ge 0?$$

I would appreciate insights on any of these questions. My main interest is in small $c>0.$

One way to think about it may be to follow Example 4 in Reed-Simon II on page 160.

There the authors prove, going via a 1D reduction, that if

$c \ge 1,$ then $H$ is essentially self-adjoint (there is a unique self-adjoint extension) on $C_c^{\infty}(\mathbb R^2 \setminus \{0\})$ and if

$c \in [ \frac{1}{4},1),$ then $H$ is not essentially self-adjoint on $C_c^{\infty}(\mathbb R^2 \setminus \{0\}).$

The case $c \in (0,\frac{1}{4})$ seems to be missing, but I think one quickly sees by analysing the argument that it is the same case as the last one above (not essentially self-adjoint).

In a separate question Iosif Pinelis has shown that the quadratic form generated by the potential is not relatively form bounded with respect to the Dirichlet form.

The reason I asked this question was that I suspected that $H$ was self-adjoint on the Dirichlet domain of $-\Delta$, i.e. $H_0^1(\mathbb R^2\setminus \{0\}) \cap H^2(\mathbb R^2\setminus \{0\})$, but I am now not so sure about my conjecture anymore.

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  • $\begingroup$ This physics literature has a number of papers about finding the correct self-adjoint realization of this operator, e.g. starting from here: pubs.aip.org/aapt/ajp/article/42/11/960/1045509/… $\endgroup$
    – Buzz
    Commented Feb 26 at 4:49
  • $\begingroup$ See also the more recent papers by Derezinski. Note that this operator is usually called the "Bessel operator" which might help when doing a literature search. The radial part is equivalent to the generator of a Bessel process which might also help with intuition. $\endgroup$ Commented Feb 26 at 9:40

1 Answer 1

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Taking advantage of the spherical symmetry to decompose this into a sum of one-dimensional problems sounds like the right approach. I will probably just be redoing what Reed-Simon had in mind here.

The Laplacian is given in polar coordinates by $$ \Delta=\frac{1}{r}\frac{\partial}{\partial r} r\frac{\partial}{\partial r} + \frac{1}{r^2}\frac{\partial^2}{\partial\theta^2} . $$ We identify $L^2(\mathbb R^2)=L^2((0,\infty); rdr)\otimes L^2(S^1)$ and decompose $L^2(S^1)=\bigoplus L(e^{in\theta})$ and also get rid of the weight in the radial component with the help of the unitary map $U:L^2(r\, dr)\to L^2(0,\infty)$, $(Uf)(r)=r^{1/2}f(r)$, then $H$ becomes (is unitarily equivalent to, to be precise) an orthogonal sum $H=\bigoplus H_n$, with each $H_n$ acting in $L^2(0,\infty)$ and $$ H_n f = -f''+\frac{n^2-1/4+c}{r^2}f . $$ This operator is essentially self-adjoint if and only if $n^2-1/4+c\ge 3/4$ or $c\ge 1-n^2$. See for example the corresponding discussion in my answer here.

So $H$ on $C_0^{\infty}(\mathbb R^2\setminus\{ 0\})$ itself is essentially self-adjoint if and only if this happens for all $n\in\mathbb Z$, which is indeed equivalent to $c\ge 1$. For $0<c<1$, $H_n$ is still essentially self-adjoint on $C_0^{\infty}(0,\infty)$ for $|n|\ge 1$, while the closure of $H_0$ on this domain is symmetric with deficiency index $(1,1)$. So in this case there is a one-parameter family of boundary conditions, to be imposed on the elements of $D(H_0^*)$, and this gives all self-adjoint realizations.

As for (1), we can just set $c=0$ in this analysis and everything is still correct, and we are now discussing the self-adjoint extensions of $-\Delta$ on $C_0^{\infty}(\mathbb R^2\setminus \{ 0\})$. Since there is nothing to choose in the higher harmonics, the question is if $-d^2/dr^2+k/r^2$ can ever be given the same domain as $-d^2/dr^2-1/(4r^2)$ for $-1/4<k<3/4$. The discussion here strongly suggests that the answer is no, though it's not completely conclusive since we fixed specific boundary conditions there.

Added later (with one detail corrected later still): This seems quite clear actually: When $c=0$, the possible domains $D(H_0)$ can be obtained by putting an arbitrary solution of $-f''-f/(4r^2)=0$ into $D(H_0)$. This is an Euler equation with solution basis $f_1=r^{1/2}$, $f_2=r^{1/2}\log r$. Note also that when we undo the unitary transformation from above, these correspond to $U^*f_j=1, \log r$. On the other hand, $f$ can never be in $D(H_0)$ when $c>0$ because then $-f''+(c-1/4)f/r^2= cf/r^2\notin L^2$. So we never have $D(H)=D(\Delta)$.

For the record ("what is the correction?"), in an earlier version I put $f_1=r^{1/2}$ into $D(H_0)$ and claimed that this corresponds to $D_0=H_0^1\cap H^2$, which you suggested as a possible domain. This was nonsense of course (I mixed up my signs in the exponent when applying $U^*$), since in that case we put $g=1$ (near $r=0$) into $D(\Delta)$ and we simply obtain the plain Laplacian on $H^2(\mathbb R^2)$. Any other choice, namely put $g=b+\log r$ into $D(\Delta)$, is certainly closer to what you had in mind, but it's not really the domain $D_0$. In fact, $H_0^1$ on $\mathbb R^2\setminus \{0\}$ is simply $H^1$ since the function $h(r)=\log a^2/\log r^2$ gets us from $h(0)=0$ to $h(a)=1$ and $\|h\|_{H^1(r<a)}\to 0$ as $a\to 0$.

Finally, (2) is now moot, but it would be a consequence of (1) since if $D(S)\supseteq D(T)$ for closed operators, then $S$ is $T$-bounded (= Theorem 5.9 in Weidmann, Linear operators in Hilbert spaces).

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  • $\begingroup$ So you are saying the Dirichlet domain is indeed an admissible domain? If I read your answer correctly, then you are saying that we just have to know the self-adjoint extensions of the Laplacian and we will have a complete description for $c<1$? $\endgroup$ Commented Feb 25 at 21:12
  • $\begingroup$ Thanks Christian, I think I still stumble over the same point: You say "we can just set 𝑐=0 in this analysis and everything is still correct". Are you suggesting that the conditions are in some sense independent of setting $c=0$ as long as $c<1$? This is not obvious to me. $\endgroup$ Commented Feb 25 at 21:45
  • $\begingroup$ @AntónioBorgesSantos: No, I'm just saying that we can decompose $-\Delta=\bigoplus H_n$ and then extend this, and since the $H_n$, $n\not= 0$ are already essentially self-adjoint, it's only about finding a domain for $H_0$. This works for any $c\in\mathbb R$, including $c=0$. $\endgroup$ Commented Feb 25 at 21:49
  • $\begingroup$ ah, but you think the domains will be different for $c \neq 0$ from the ones you can obtain for $c=0.$ So both (1) and (2) are wrong. $\endgroup$ Commented Feb 25 at 21:50

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