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edited Jun 10 at 14:08

Recently I have studied the paper: Rozenbljum, G. V. (1974). Asymptotics of the eigenvalues of the Schrödinger operator. Mathematics of the USSR-Sbornik, 22(3), 349.

In there Rozenbljum gives conditions for a Weyl law for $-\Delta+V$ on $\mathbb{R}^d$ (i.e. non-compact domain) to hold true. One of the assumptions for $V$ is that the measure of the sublevel set of the potential behaves reasonably. Namely, that there exists a constant $C>0$ such that for sufficiently large $\lambda>0$ we have $$ \left\vert \left\{ x\in \mathbb{R}^d \ : \ V(x)\leq 2\lambda \right\} \right\vert \leq C \left\vert \left\{ x\in \mathbb{R}^d \ : \ V(x)\leq \lambda \right\} \right\vert \qquad (1) $$ I was wondering what methods one has to check that such an estimate holds for a given potential $V$. One obstacle to get $(1)$ is of course strong oscillations, thus, I think it is reasonable to restricrestrict to "rigid" function. My main question is the following:

Let $V\geq 0$ be a semi-algebraic function on $\mathbb{R}^d$ such that $\lim_{\vert x \vert \rightarrow \infty} V(x) = \infty$. Are there checkable conditions which ensure that $(1)$ holds? In fact, already the case $V(x_1, \dots, x_d) = \sum_{j=1}^N \vert P_j(x_1, \dots, x_d)\vert^{1/a_j}$ with $P_j$ (real) polynomials would be interesting to me.

Please feel free to add/change tags. I was not quite sure what to put.

Recently I have studied the paper: Rozenbljum, G. V. (1974). Asymptotics of the eigenvalues of the Schrödinger operator. Mathematics of the USSR-Sbornik, 22(3), 349.

In there Rozenbljum gives conditions for a Weyl law for $-\Delta+V$ on $\mathbb{R}^d$ (i.e. non-compact domain) to hold true. One of the assumptions for $V$ is that the measure of the sublevel set of the potential behaves reasonably. Namely, that there exists a constant $C>0$ such that for sufficiently large $\lambda>0$ we have $$ \left\vert \left\{ x\in \mathbb{R}^d \ : \ V(x)\leq 2\lambda \right\} \right\vert \leq C \left\vert \left\{ x\in \mathbb{R}^d \ : \ V(x)\leq \lambda \right\} \right\vert \qquad (1) $$ I was wondering what methods one has to check that such an estimate holds for a given potential $V$. One obstacle to get $(1)$ is of course strong oscillations, thus, I think it is reasonable to restric to "rigid" function. My main question is the following:

Let $V\geq 0$ be a semi-algebraic function on $\mathbb{R}^d$ such that $\lim_{\vert x \vert \rightarrow \infty} V(x) = \infty$. Are there checkable conditions which ensure that $(1)$ holds? In fact, already the case $V(x_1, \dots, x_d) = \sum_{j=1}^N \vert P_j(x_1, \dots, x_d)\vert^{1/a_j}$ with $P_j$ (real) polynomials would be interesting to me.

Please feel free to add/change tags. I was not quite sure what to put.

Recently I have studied the paper: Rozenbljum, G. V. (1974). Asymptotics of the eigenvalues of the Schrödinger operator. Mathematics of the USSR-Sbornik, 22(3), 349.

In there Rozenbljum gives conditions for a Weyl law for $-\Delta+V$ on $\mathbb{R}^d$ (i.e. non-compact domain) to hold true. One of the assumptions for $V$ is that the measure of the sublevel set of the potential behaves reasonably. Namely, that there exists a constant $C>0$ such that for sufficiently large $\lambda>0$ we have $$ \left\vert \left\{ x\in \mathbb{R}^d \ : \ V(x)\leq 2\lambda \right\} \right\vert \leq C \left\vert \left\{ x\in \mathbb{R}^d \ : \ V(x)\leq \lambda \right\} \right\vert \qquad (1) $$ I was wondering what methods one has to check that such an estimate holds for a given potential $V$. One obstacle to get $(1)$ is of course strong oscillations, thus, I think it is reasonable to restrict to "rigid" function. My main question is the following:

Let $V\geq 0$ be a semi-algebraic function on $\mathbb{R}^d$ such that $\lim_{\vert x \vert \rightarrow \infty} V(x) = \infty$. Are there checkable conditions which ensure that $(1)$ holds? In fact, already the case $V(x_1, \dots, x_d) = \sum_{j=1}^N \vert P_j(x_1, \dots, x_d)\vert^{1/a_j}$ with $P_j$ (real) polynomials would be interesting to me.

Please feel free to add/change tags. I was not quite sure what to put.

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asked Jun 10 at 10:27

Severin Schraven

Class of semi-algebraic potentials satisfying growth condition for the sublevel sets

Recently I have studied the paper: Rozenbljum, G. V. (1974). Asymptotics of the eigenvalues of the Schrödinger operator. Mathematics of the USSR-Sbornik, 22(3), 349.

In there Rozenbljum gives conditions for a Weyl law for $-\Delta+V$ on $\mathbb{R}^d$ (i.e. non-compact domain) to hold true. One of the assumptions for $V$ is that the measure of the sublevel set of the potential behaves reasonably. Namely, that there exists a constant $C>0$ such that for sufficiently large $\lambda>0$ we have $$ \left\vert \left\{ x\in \mathbb{R}^d \ : \ V(x)\leq 2\lambda \right\} \right\vert \leq C \left\vert \left\{ x\in \mathbb{R}^d \ : \ V(x)\leq \lambda \right\} \right\vert \qquad (1) $$ I was wondering what methods one has to check that such an estimate holds for a given potential $V$. One obstacle to get $(1)$ is of course strong oscillations, thus, I think it is reasonable to restric to "rigid" function. My main question is the following:

Let $V\geq 0$ be a semi-algebraic function on $\mathbb{R}^d$ such that $\lim_{\vert x \vert \rightarrow \infty} V(x) = \infty$. Are there checkable conditions which ensure that $(1)$ holds? In fact, already the case $V(x_1, \dots, x_d) = \sum_{j=1}^N \vert P_j(x_1, \dots, x_d)\vert^{1/a_j}$ with $P_j$ (real) polynomials would be interesting to me.

Please feel free to add/change tags. I was not quite sure what to put.

sp.spectral-theory real-algebraic-geometry