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Maxim Braverman
Maxim Braverman
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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 $$ N(A,\lambda) \sim \ \frac1{(2\pi)^d} \text{Vol}\ {\Large(}(x,\xi)\in T^*M:\ |\xi|^2+V(x)\le \lambda{\Large)}, \qquad h\to 0. \quad(\ast\ast) $$$$ N(A,\lambda) \sim \ \frac1{(2\pi)^d} \text{Vol}\ {\Large(}(x,\xi)\in T^*M:\ |\xi|^2+V(x)\le \lambda{\Large)}, \qquad \lambda\to \infty. \quad(\ast\ast) $$ 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.

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 $$ N(A,\lambda) \sim \ \frac1{(2\pi)^d} \text{Vol}\ {\Large(}(x,\xi)\in T^*M:\ |\xi|^2+V(x)\le \lambda{\Large)}, \qquad h\to 0. \quad(\ast\ast) $$ 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.

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.

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Maxim Braverman
Maxim Braverman
  • 658
  • 3
  • 14

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 $$ N(A,\lambda) \sim \ \frac1{(2\pi)^d} \text{Vol}\ {\Large(}(x,\xi)\in T^*M:\ |\xi|^2+V(x)\le \lambda{\Large)}, \qquad h\to 0. \quad(\ast\ast) $$ 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.