## Reconstruct an operator by its eigenfunctions $e^{-g(p \circ x)}$

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

Is there some well-known description of an operator $D$ (pseudodifferential of order greater than 0) for which $e^{-g(x_1 p_1,...,x_n p_n)}$, where $g \colon \mathbb{R}^n_+ \to \mathbb{R}_+$ is a homogeneous of order 1 function, will be eigenfunctions for any $(p_1,...,p_n) \in \mathbb{R}^n_+$? I.e. $$D e^{-g(p_1 x_1,...,p_n x_n)} = \lambda_p e^{-g(p_1 x_1,...,p_n x_n)}.$$

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 5, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.887657105922699, "perplexity_flag": "middle"}