# Weyl law for (non-semiclassical) Schrodinger operator

The Weyl law for a semiclassical Schrodinger operator $$A_h\ := \ -h^2\Delta+V(x)$$ on an $$d$$-dimensional complete Riemannian manifold $$M$$ says that the number $$N(A_h,1)$$ of eigenvalues of $$A_h$$ which are smaller than 1 has asymptotic behavior $$N(A_h,1)\ \sim \ \frac1{(2\pi h)^d}\ \text{Vol}\ {\Large(}(x,\xi)\in T^*M:\ |\xi|^2+V(x)\le 1{\Large)}, \quad h\to 0.\ \ \ (\ast)$$ I am interested in a non-semiclassical Schrodinger operator $$A\ := \ -\Delta+V(x).$$ I believe that the number $$N(A,\lambda)$$ of eigenvalues of $$A$$ smaller than $$\lambda$$ has a similar asymptotic $$N(A,\lambda) \sim \ \frac1{(2\pi)^d} \text{Vol}\ {\Large(}(x,\xi)\in T^*M:\ |\xi|^2+V(x)\le \lambda{\Large)}, \qquad \lambda\to \infty. \quad(\ast\ast)$$ This is, of course, true on a compact manifolds, due to the classical Weyl's law. It is also not difficult to verify (**) for operator $$-\Delta+|x|^n$$ on $$\mathbb{R}^d$$ (where $$A_h$$ and $$A$$ can be related by rescaling of $$x$$).

So my question: is (**) true and, if it is true, where can I find it?

PS. Victor Ivrii in his review article https://arxiv.org/abs/1608.03963 mentions (page 5) that, using Birman-Schwinger principle, one can obtain a Weyl law for $$N(A,\lambda)$$ from (*). But I don't see how it can be done.

• Victor Ivrii is somewhat responsive to emails. One could try emailing the question to him for a response. Apr 14 '19 at 6:44
• I don't think completeness alone will suffice to say anything this general. E.g. there are complete riemannian manifolds with purely continuous spectrum (such as the upper half-plane $\mathbb{H}^2$ has spectrum $[1/4,\infty)$.) If you're not considering closed manifolds, and you don't assume something about how your potential $V(x)$ behaves at infinity I don't think you can hope to get a Weyl Law. You might look at H\"{o}rmander's "spectral function of an elliptic operator" since he proves a weyl law very generally, though at the cost of some FIO theory being necessary. Apr 14 '19 at 17:24
• @HadrianQuan Clearly, I assume that $V(x)$ grows at infinity, so that the volume in (**) is finite. I did look at Hormander. I don't think the result I need follows from it. Apr 16 '19 at 15:21
• Apologies for not realizing you meant this. I think you're right, as the principal symbol in classical calculus he considers will not include the potential $V(x)$ (I suppose this is the point of considering semiclassical operators!) I am glad you were able to find some references. Apr 16 '19 at 16:54

We are talking about a non-compact Riemannian manifold, right? Then Weyl's law may be incorrect, at least out of the box. Let us consider $$H = -\Delta$$ (so $$V(x)=0$$) in the domain $$X \subset \mathbb{R}^d$$ with the Dirichlet or Neumann boundary condition (the case of the closed manifold is very similar to the Neumann case).

Let non-compact part of $$X$$ be $$\{x\colon x_1 > c, \ x'\in \rho(x_1) \Omega\}$$ with $$x'=(x_2,\ldots, x_d)$$, $$\Omega$$ a compact domain with a smooth boundary in $$\mathbb{R}^{d-1}$$ and $$\rho \to 0$$ as $$x_1\to \infty$$.

Observe that depending on the rate of decay of $$\rho$$ the volume of $$X$$ could be finite or infinite (and the area of $$\partial X$$ also could be finite or infinite).

The literal Weyl's law says: if $$X$$ has a finite volume then the spectrum is discrete and Weyl's law holds and if $$X$$ has an infinite volume then there is an essential spectrum.

However the reality is more nuanced:

Dirichlet Laplacian In this case spectrum is discrete for sure; simply in the case of the infinite volume the correct asymptotic expansion is different. Let $$\rho (t)= t^{-\mu}$$, $$\mu >0$$. Volume is finite for $$\mu (d-1)> 1$$ and infinite otherwise.

1. Literal Weyl's law holds for $$\mu (d-1)> 1$$.
2. For $$\mu (d-1)=1$$ (logarithmic divergence) $$N(\lambda)\asymp \lambda ^{(d-1)/2}\ln (\lambda)$$ and is given by the same formula but with integration over $$x_1\colon \rho(x_1)>\lambda^{-1/2}$$.
3. For $$\mu (d-1)<1$$ (power divergence) $$N(\lambda)\sim \sum_{j} n_j(\lambda) \tag{D}$$ where $$n_j(\lambda)$$ is a Weyl's expression for $$1$$-dimensional Schrödinger operator $$\mathsf{h}_j = -\partial_1^2 +\nu_j \rho(x_1)^{-2}$$ and $$\nu_j >0$$ are eigenvalues of $$(d-1)$$-dimensional Dirichlet Laplacian in $$\Omega$$.

Neumann Laplacian Then things change drastically. There is an essential spectrum unless $$\rho\to 0$$ really fast. To understand why look at (D) but instead of a Dirichlet Laplacian in $$\Omega$$ one should consider a Neumann Laplacian, and $$\nu_1=0$$. Does it mean essential spectrum?

1. If $$\rho =x_1^{-\mu}$$ or even $$\rho =\exp (-c x_1)$$ then yes.
2. But if $$\rho = \exp(-x_1^{k+1})$$ with $$k>0$$ then no, and $$N(\lambda) \sim N^W(\lambda) + N^c(\lambda) \tag{N}$$ where $$N^W(\lambda)$$ is a Weyl expression for an original operator, and $$N^c(\lambda)$$ is a Weyl expression for $$1$$-dimensional Schrödinger $$\mathsf{h}_0= -\partial_1^2 + W(x_1)$$ with a potential $$W =\frac{1}{4} (\partial_{x_1} \log \rho (x_1))^2$$ and depending on $$k$$ either the first or the second term in (N) may prevail.

For Dirichlet/Neumann case in more general setting see subsections 3.2.2/3.2.3 of 100 years of Weyl's law either in arXiv or in Bull Math Sci

• As someone mentioned, the best way to reach me is by email Aug 17 '19 at 10:25

I talked to several experts. Here is the punchline: apparently, only the case of $$\mathbb{R}^n$$ can be found in the literature (I would be glad to be wrong and would highly appreciate a reference). For $$\mathbb{R}^n$$ the asymptotic (**) holds if the potential does not oscillate too much and otherwise is reasonable. Probably the first result in this direction is in

de Wet, J. S.; Mandl, F. On the asymptotic distribution of eigenvalues. Proc. Roy. Soc. London. Ser. A. 200, (1950). 572–580.

where only the cases on $$n=1,2,3$$ are considered and the potential $$V(x)$$ is supposed to grow "fast enough". One of the best results is in

Levendorskii. Spectral asymptotics with a remainder estimate for Schr ̈odinger operators with slowly growing potentials. Proc. Roy. Soc. Edinburgh Sect. A, 126(4):829–836, 1996

where very slowly growing potentials are allowed. Another interesting paper is

G. V. Rozenbljum. Asymptotic behavior of the eigenvalues of the Schr ̈odinger operator. Mat. Sb. (N.S.), 93 (135):347–367, 487, 1974. (in Russian)

where very singular potentials are allowed.